high command vs supreme headquarters

high command and supreme headquarters both are nouns.

Word	Noun	Adjective	Verb	Adverb
high command	Yes	No	No	No
supreme headquarters	Yes	No	No	No

As nouns, high command and supreme headquarters are synonyms defined as:

high command and supreme headquarters: the highest leaders in an organization (e.g. the commander-in-chief and senior officers of the military)

high command (noun)	supreme headquarters (noun)
the highest leaders in an organization (e.g. the commander-in-chief and senior officers of the military)	the highest leaders in an organization (e.g. the commander-in-chief and senior officers of the military)