...Stupid EarthScorpion, it's not like I find you funny or anything! Have my funny rating you stupid person!
Actually, much like humans, the cats are split between two sides in the Ascension Claw. There's the Technocatic Union, the side which believes they need to conceal their powers from humans and keep human society structured to benefit them. They're split up into five Catventions.
The Purrgenitors conduct studies on the tastiest forms of animals to eat, and are carrying out eugenics programmes where they systematically murder less tasty animals, allowing only the yummiest to breed. That's why cats kill things even when they're not hungry - they're removing less tasty creatures from the breeding pool. The Purrgenitors are also the biggest supporters of the Purrgrom, where they seek to wipe out dogs (who are part of the Woofandi).
Hisseration X study machinery and tools, such as door handles and cat flaps. Their ultimate goal is to devise a way to open tins of catfood without human help, which will lead to what they dub the Nomgularity.
The Syndicat are responsible for keeping the humans on side using bribes. That's why they drag in dead animals. Other Catventions believe they may be too obsessed with their pet theories which don't work as well as they think, but the Syndicats (aka, the Fat Cats) control the best gardens and the best bathing spots and allocate them to other Catventions as they see fit.
The Furred Engineers explore the farthest reaches of space, like next door's garden. They're always looking for new places to bury their waste. Unfortunately, their mission puts them in unsafe places and the forces of the Carauders and the Woofandi mean they take heavy casualties.
Finally, there's the Mew World Order (Also known as the NYO), whose job it is to keep the humans not suspecting what they are and be cute and adorable. They're broken up into different ranks - Black Cats keep the streets safe for catkind, Grey Cats interact with humans, and White Cats oversee the entire conspiracy (that's why they're usually found sitting on the laps of villains).
The Council of Nyan Traditions is less known about...