gag vs hush up

gag is a noun but hush up is not a noun.

Word	Noun	Adjective	Verb	Adverb
gag	Yes	No	Yes	No
hush up	No	No	Yes	No

As verbs, hush up is a hypernym of gag; that is, hush up is a word with a broader meaning than gag:

Other hypernyms of gag include hush, quieten, shut up, silence, still.

gag (noun)	hush up (noun)
restraint put into a person's mouth to prevent speaking or shouting
a humorous anecdote or remark intended to provoke laughter

gag (verb)	hush up (verb)
struggle for breath; have insufficient oxygen intake	cause to be quiet or not talk
make an unsuccessful effort to vomit; strain to vomit	cover up a misdemeanor, fault, or error
cause to retch or choke
make jokes or quips
be too tight; rub or press
tie a gag around someone's mouth in order to silence them
prevent from speaking out