Here is the full list of all 13 Super Galaxy Squadron EX Turbo achievements.
Achieve Rank S or above in Endless Mode
Achieve Rank SS or above in Endless Mode
Beat Hardcore Arcade Mode for that time-honored 1CC
Find all secret medals in a stage
Beat Boss Rush
Beat a stage on Hardcore Hell mode
Finish with a total score of 15 million or above
Achieve Rank A or above in Endless Mode
Get the Max Bonus against Iron Helix
Get the Max Bonus against Hell Diver
Get the Max Bonus against Albatross
Get the Max Bonus against Great Rift
Finish the game