So far, all ships named in the story have belonged to the UHF.
The Sovereign has the prefix IGS. What ‘IG’ means is not known. (Arc 1-finale-premium-2/2)
Ascendant of Flames: Training ship. Alpha squad is Ascendant Alpha.
Black Sword: escort ship Captain Cross requests in chapter 1-intermission-1.
Empyrean: Recruit training ship
Fanged Viper: escort ship Captain Cross requests in chapter 1-intermission-1.
Hegemony of Dusk: Training ship. Alpha squad is Hegemon Alpha.
Imperator of Seas: Training ship
Monarch: Sister ship to Sovereign. Lost.
Sovereign: The ship (and AI) that Thea is on for most of Arc 1 and Arc 2.
Valiant Radiance: escort ship Captain Cross requests in chapter 1-intermission-1.