What franchises do you consider part of the "pillars" of popular culture?

Semi-serious answer. Greeks like Homer and Euripades and Virgil, the bible, Shakespeare. Lots of Western pop culture is a rehash of old stuff.
Answering the question more as you likely intended. Pokemon. Harry Potter. Star Wars. The MCU.
From what I suspect is due to my personal bubble, Star Trek; ST pops up several times per week in my life. And Twilight, I've got a group of friends who are queer and will jam a Team Alice or KStew reference in at every opportunity. It seems like the rest of the world has moved on from Twilight love and Twilight hate, but this group is keeping the discourse alive.
- FWIW, I’m seeing a lot of Twilight reflection going on right now that’s not unlike the Star Wars prequel reflection from a few years ago. It’s definitely being revisited.

The big ones that really cross at least a few age brackets, have wide general recognition and probably aren't going anywhere in the near future (I may take some liberties with what I consider to be a franchise) in no particular order-
Star wars
James Bond
Lord of the rings
Sherlock Holmes
Batman
Superman
Spiderman
Mission Impossible
Mario
Zelda
Pokemon
Indiana Jones
Back to the Future
The Karate Kid
A Nightmare on Elm Street
Friday the 13th
Child's Play
It
Rambo
Rocky
Jurassic Park
The Matrix
The Terminator
Teenage Mutant Ninja Turtles
Transformers
The Simpsons
Barbie
GI Joe
He-Man/Masters of the universe
Mickey Mouse
Toy Story
Looney Tunes
King Kong
Godzilla
Planet of the Apes
Mad Max
The Muppets
The Godfather
Ghostbusters
Alien
Star Trek
Robocop
Frankenstein
Dracula
Tarzan
Conan the Barbarian
Jaws
Harry Potter
The Incredible Hulk
The Dollars Trilogy
Sesame Street
The Hannibal Lecter series
MASH
I think in general, most people have at least heard of these properties, would probably recognize at least a few of the main characters, objects, logos, memes, quotes, the theme song, etc. they've probably made some reference to them, and could give at least a vague explanation about what they're about or what the major themes are whether or not they've actually seen/read the source material
In general, I'm kind of counting a franchise as something that has had at least 3 major installments, iterations, episodes, series, remakes, reboots, etc. so a stand alone book, movie, etc. wouldn't count, nor would a book or movie and a sequel, a book and a movie adaptation, etc. A book, a movie adaptation, and a reboot movie would, a film trilogy would,a tv series would, a movie that's been rebooted/remade a couple times would.
Barbie feels like a weird one on this list to me, unlike GI Joe who's had pretty big movies and cartoons and such that make it pretty hard to argue that it's a franchise, most of Barbie's notoriety comes from the dolls themselves and I'd be kind of hesitant to label a line of toys as a franchise instead of a brand, sure there's been animated movies and video games and such, but none of them had really been particularly noteworthy. And I wouldn't feel quite right labeling, for example, Lego as a franchise despite having had pretty considerable success with movies and video games and such. But the character of Barbie, branding, marketing, etc. kind of puts her in the same league as Mickey mouse and I just felt like she belonged on the list.
Some of the classic characters - Frankenstein, Dracula, Sherlock Holmes, are arguably not really a franchise, there's not necessarily one company or person who owns their likeness and is marketing them, but they loom large enough in the public consciousness that I think they deserve to be mentioned in this list as well
I tried to keep this relatively universal, though I'm sure my biases as a cis straight white American millennial male showed through in places. There's a lot of franchises that form pillars of pop culture for specific demographics but not necessarily in general, and I tried to stay away from them, but a few of them just felt significant enough to me to warrant inclusion, in particular I kind of question how much general appeal He-Man has, for example, but I feel like if you say "He-Man" everyone has a mental image of the character seared into their brain (personally, I'm not sure I've ever even seen a whole episode of He-Man or really engaged with any masters of the universe media in general, but there he is in my brain and I don't really know how he got there)
- As a 40+ year old white American, I agree with most of this list (I’d reduce “Hannibal Lecter series” down to just “Silence of the Lambs”), but if I were to ask my 6–12 year old kids about these, I feel like they wouldn’t know who 1/3rd of these are. I’d be curious to see if there was a common thread or set of threads between all of these that point to what traits our Western society value/avoid or what makes a successful franchise in the West.
  
  They probably haven't seen a lot of them and may not recognize the names off the top of their head, but I have a hunch that if/when they do see them, damn each and every one is going to give them a couple little aha moments where they recognize that something else they've seen has parodied or referenced these movies, or that it's where a meme comes from.
  And I'd agree that Silence of the Lambs is probably the most important of the series by a longshot, but one book and a movie adaptation does not a franchise make in my mind and I think that Hannibal Lecter is a big enough character in the public mind (because the Silence of the Lambs movie) that the series as a whole kind of gets a pass
- I think by the definition I gave they'd qualify as a franchise, although for those 3 properties it kind of feels a little wrong to me and maybe I need to rethink my definition a little because they're kind of self-contained shows, they don't really branch out into anything larger than themselves.
  Personally I think the IT crowds appeal is somewhat limited, at least in the US (it may have more widespread appeal in its native UK, I genuinely don't know,) to sort of the nerd/techy demographic. The lines certainly get blurry because nerdy types have a lot of influence on how memes spread, but I don't know that it has the same impact on pop culture in general as the other two.
  I'm also not certain how much staying power friends has outside of the people who were watching it or at least remember it being on when it first aired. I'm not exactly plugged into what gen z and alpha are watching, but i don't think I hear it come up amongst them very often, and I'm not even sure younger millennials care for it all that much.
  The office I think has probably cemented it's place in pop culture whether we consider it a franchise or not. Which is kind of a shame for me personally, because I could never get into it.
- I'm of an age where I agree with almost everything on this list. I don't know that It deserves a place though, and despite extensive media consumption I've never heard of The Dollars Trilogy or Child's Play.
  
  Child's play- Chucky the killer doll
  Dollars Trilogy - A Fistfull of Dollars, For a Few Dollars More, and The Good, the Bad, and the Ugly. I did kind of waffle on this one a bit, but I ultimately decided to include it if for no other reason than because whether youve actually seen it or even realize it you probably have The Good, The Bad, and the Ugly's theme song seared into your head as pretty much the western movie song.

Lord of the rings, Star Trek, James Bond, Sherlock Homes, Super Mario and Grand Theft Auto
- I’d add Agatha Christie to the Sherlock Holmes/James bond subcategory too

I'd add game of Thrones. It was a cultural phenomenon when it came out. Oh, and the Simpsons.
- Thrones is fading fast. It will be gone in ten years. Unlike Lord of the Rings, there's no new generation watching it.
  
  Yeah, I think it will be like "Dallas". It will be really big for one generation.
  
  I was waiting for it all to come out before I watched it. Then the consensus was that it wasn't worth watching.
  
  Its basically gone and died already. It's as if it never existed to almost everyone, even those who watched it first time are not re-watching it.

Alien because of HR Giger's iconic art style, among other things. Also Frank Herbert's Dune, the first planned film adaptation of which Giger also worked on (sometimes considered as the best movie that was never made). In addition to the cultural influence that the book series already had, Dune also more or less spawned the real-time strategy computer game genre (with Westwood Studios' Dune II).
- Such a shame the remake with Moebius and Dali (among others) wasn't made.
  
  Yes, Dali would have been great as the emperor.
  But Jodorowsky's unfinished Dune project still had a significant influence on several major science fiction films and media, despite never being completed:
  Alien (1979)
  Dan O'Bannon, who worked on Jodorowsky's Dune, later wrote the Alien script
  H.R. Giger's designs for Dune inspired his work on the Xenomorph
  The Alien Xenomorph's design has roots in Giger's Harkonnen Castle concept for Dune
  Star Wars (1977)
  George Lucas was reportedly influenced by the Dune storyboards and designs
  Jabba the Hutt's appearance may have been inspired by Baron Harkonnen's design
  Blade Runner (1982)
  Ridley Scott hired Dan O'Bannon and Moebius, who had worked on Jodorowsky's Dune, to help create the futuristic world
  Contact (1997)
  The opening space sequence was allegedly inspired by Jodorowsky's planned opening for Dune
  The Incal (1980s graphic novel series)
  Jodorowsky and artist Moebius recycled many concepts from their Dune work into this influential comic series
  Check out the documentary "Jodorowsky's Dune" (2013). It explores how the pre-production work and assembled team for this unrealized film went on to influence much of modern science fiction cinema. It's worth a watch.

Harry Potter, Grand Theft Auto and Fallout (from the third one on).
- Fallout (from the third one on).
  If it wasn't for the TV Show most of my acquaintances wouldn't know about it... Are you sure Fallout deserved to be in that podium?
  
  What would you replace it with?
  ER?
- Fallout (from the third one on).
  WTF? Why from the third one on?

The Simpsons
Star Wars
LOTR
MCU / DC (both franchises are huge and span decades)
Barbie ( girls have been playing with these dolls and following their trends for generations)
Disney Classics (not sure if this counts as a franchise?)
I can think of at least ten more franchises that are solid and influential but that either need a couple more decades or a bigger following to match the aforementioned ones

Taylor Swift, Star Wars, McDonald's.

Twilight and Supernatural. Guess they kick-started shows like True Blood and Teen Wolf.

This is an impossible question to answer. But, I'll give it a shot anyway. I've expanded the meaning of "franchise" to include "all properties sublicensable for the purposes of profiteering."
If "popular culture" refers to the recognizable and persistent elements of living in society that the majority people share in common without having to communicate that recognition, I'd regard the following franchises as having broad impact worldwide:
McDonalds/Subway and all attendant advertising as a signpost for food. Franchises abound.
Esso/Shell/BP as gateways to modern conveniences and transportation. Every gas station, residence, farm house, hen house, outhouse, and dog house is connected to these franchises in some way.
G4S/Securitas/Garda as the front line protecting the 'haves' from the 'have-nots'. Franchises abound.
Most athletic, luxury vehicle, and brands as the status symbols they want themselves to be. Franchisees promote the brands as a means of collecting clients.
If, on the other hand, "popular culture" is, 'traditions and material culture of a particular society. In the modern West, pop culture refers to cultural products such as music, art, literature, fashion, dance, film, cyberculture, television, and radio that are consumed by the majority of a society's population. ... types of media that have mass accessibility and appeal' (ThoughtCo.) then the following are some fairly strong indicators of popular culture:
Hello Kitty (be pleasant)
Pokémon (pursue goals)
Superman/Batman (masculinity, vigilantism)
Paw Patrol (institutions are essential)
the Olympics (do athletics)
Michael Jordan (be excellent)
Mickey Mouse (dream big)
Star Wars/The Bible (G vs. E)
The ones I wish would take hold and have more of an influence:
X-men (biodiversity is good)
the Expanse
Battlestar Galactica (genocide, rebellion, impersonation, terrorism, coups d'état, civil war, infidelity, succession, military conflict, asymmetrical warfare, treason, mutiny, pirate broadcasts, nuclear warfare... and that's just the first half of the series)
Tony's Chocolonely (ethical economics)

pop culture changes over time so the answer is going to change over time.