Here is the full list of all 21 Mayhem Intergalactic achievements.
Win a Quick Start singleplayer game with the AI difficulty set to Highest.
Same as Natural Intelligence, but additionally don't lose any battles.
Win a Quick Start singleplayer game.
Destroy 1000 enemy ships in a single game.
Destroy 2500 enemy ships in a single game.
Destroy 5000 enemy ships in a single game.
Win a multiplayer game.
Win 10 multiplayer games.
Win 100 multiplayer games.
Use rally points to connect all planets in a Quick Start game into a single network.
Destroy 1000 native ships, over multiple games.
Destroy 2500 native ships, over multiple games.
Destroy 5000 native ships, over multiple games.
Conquer an undefended enemy planet. Then win the game.
Win a game without destroying any native ships.
Win a game as the red player, taking no more turns than there are planets.
Keep control of your home world for the entire game.
Conquer the home world of an opponent, but lose your own home world to that same opponent.
Defend your last planet from an invader.
Conquer a planet named Miranda.
Conquer 5 planets in a single turn.