As far as I can see, there is only Engine, Editor, GameInstance and LocalPlayer subsystems, out of which none seems to fit your needs.
There is a PR for Game Mode Subsystem, which is more like what you want - it exists on the server, created when the level is created, and destroyed when the level is changed: https://github.com/EpicGames/UnrealEngine/pull/5896
Until it’s accepted, you can’t use subsystems to achieve what you want, I’m afraid. You can do it manually tho as you would have done in the pre-4.22 era: on the BeginPlay of Game Mode, construct a new object of yours (your “subsystem”) with the Outer of the Game Mode - this way, it’s gonna get created and destroyed alongside Game Mode (or you can do it on any other class that better fits the lifecycle you’re targeting).