support vs establish

support is a noun but establish is not a noun.

Word	Noun	Adjective	Verb	Adverb
support	Yes	No	Yes	No
establish	No	No	Yes	No

As verbs, establish is a hyponym of support; that is, establish is a word with a more specific, narrower meaning than support:

support: establish or strengthen as with new evidence or facts
establish: establish the validity of something, as by an example, explanation or experiment

Other hyponyms of support include back, back up, vouch, verify, demonstrate, prove, shew, show, document, validate.

support (noun)	establish (noun)
aiding the cause or policy or interests of
the act of bearing the weight of or strengthening
documentary validation
a military operation (often involving new supplies of men and materiel) to strengthen a military force or aid in the performance of its mission
the activity of providing for or maintaining by supplying with money or necessities
any device that bears the weight of another thing
supporting structure that holds up or provides a foundation
something providing immaterial assistance to a person or cause or interest
a musical part (vocal or instrumental) that supports or provides background for other musical parts
the financial means whereby one lives
financial resources provided to make some project possible

support (verb)	establish (verb)
establish or strengthen as with new evidence or facts	establish the validity of something, as by an example, explanation or experiment
put up with something or somebody unpleasant	institute, enact, or establish
adopt as a belief	use as a basis for; found on
argue or speak in defense of	place
be a regular customer or client of	set up or lay the groundwork for
be the physical support of; carry the weight of	bring about
play a subordinate role to (another performer)	build or establish something abstract
support materially or financially	set up or found
give moral or psychological support, aid, or courage to
be behind; approve of
support with evidence or authority or make more certain or confirm