high school vs lyceum

high school and lyceum both are nouns.

Word	Noun	Adjective	Verb	Adverb
high school	Yes	No	No	No
lyceum	Yes	No	No	No

As nouns, lyceum is a hypernym of high school; that is, lyceum is a word with a broader meaning than high school:

high school: a public secondary school usually including grades 9 through 12
lyceum: a school for students intermediate between elementary school and college; usually grades 9 to 12

Other hypernyms of high school include Gymnasium, lycee, middle school, secondary school.

high school (noun)	lyceum (noun)
a public secondary school usually including grades 9 through 12	a public hall for lectures and concerts
	a school for students intermediate between elementary school and college; usually grades 9 to 12