Here is the full list of all 56 Shin Galaxy - there are Alpha War achievements.
Destroy 100 enemies ships.
Destroy 500 enemies ships
Destroy 2000 enemies ships
Defeat Boss 01.
Collect all stars in the Mitas system.
Defeat Boss 02
Collect all stars in the Astrillas system.
Defeat Boss 03
Collect all stars in the Lathen system.
Defeat Boss 04.
Collect all stars in the Pominin system.
Destroy 10 Prisms.
Defeat SuperBoss 01.
Defeat SuperBoss 02.
Defeat Hunter 10 times.
Fully Upgrade Ignis v1.
Fully Upgrade Aqua v1.
Fully Upgrade Terra v1.
Fully Upgrade Tonitruum v1.
Fully upgrade Ignis v2.
Fully upgrade Aqua v2.
Fully upgrade Terra v2.
Fully upgrade Tonitruum v2.
Access the Glossary.
Have 100.000 crystals.
Get the shield 100 times.
Get the bomb ammo 100 times.
Heal yourself 100 times with the health item.
Survive for 15 minutes in Infinite Mode.
Defeat Phantom 30 times.
Destroy the strange ship.
Complete all EX Stages.
Change the shot type for the first time.
Destroy 300 EX version enemies.
Defeat EX Boss 01.
Collect all Stars in the Xynu System.
Defeat Boss 05.
Collect all Stars in the Enferna System.
Defeat Boss 06.
Defeat EX Boss 02.
Fully upgrade Ignis v3.
Fully upgrade Aqua v3.
Fully upgrade Terra v3.
Fully upgrade Tonitruum v3.
Defeat SuperBoss 03.
Collect all Stars in the Ysvart System.
Defeat Boss 07.
Collect all Stars in the Leyiun System.
Defeat Boss 08.
Defeat SuperBoss 04.
Unlock elemental damage.
Defeat EX Boss 03.
Fully upgrade Tenebris v1.
Fully upgrade Tenebris v2.
Fully upgrade Tenebris v3.
Complete a challenge.