Here is the full list of all 21 Galactic Dominion achievements.
Complete all objectives in Chapter 1
Escape the onslaught in Chapter 3
Destroy the Machines in Chapter 6
Survive the Monsters in Chapter 7
Secure the Heart of the Galaxy in Chapter 8
Beat the Campaign
Win 1 Quickplay game
Win 5 Quickplay games
Win 10 QuickPlay games
Win 25 Quickplay games
Win 50 Quickplay games
Win a Quickplay game with each Galaxy Size
Beat a low aggression AI in a 1 vs 1 Quickplay game
Defeat 3 high aggression AIs in a Quickplay game
Destroy 10 enemy Ships
Destroy 100 enemy Ships
Destroy 1000 enemy Ships
Win 1 Quickplay game in Real-Time Mode
Win 1 Quickplay game in Turn-Based Mode
Win 1 Quickplay game on Impossible difficulty
Win 1 Quickplay game with Random race