When pressing the [Ship] icon, and then the [Upgrades] tab, there are 3 groups: Systems, Subsystems and Reactor.
The System takes power, does the Subsystems takes power as well?
No, the subsystems (piloting, sensors, doors and backup battery) do not take power from the reactor. They can still be damaged, which will disable them until they are repaired. They can also be upgraded for better performance.
More information on subsystems can be found on the FTL Wiki.