mental institution vs infirmary

mental institution and infirmary both are nouns.

Word	Noun	Adjective	Verb	Adverb
mental institution	Yes	No	No	No
infirmary	Yes	No	No	No

As nouns, infirmary is a hypernym of mental institution; that is, infirmary is a word with a broader meaning than mental institution:

mental institution: a hospital for mentally incompetent or unbalanced person
infirmary: a health facility where patients receive treatment

Other hypernyms of mental institution include hospital.

mental institution (noun)	infirmary (noun)
a hospital for mentally incompetent or unbalanced person	a health facility where patients receive treatment