take lying down vs support

take lying down is not a noun while support is a noun.

Word	Noun	Adjective	Verb	Adverb
take lying down	No	No	Yes	No
support	Yes	No	Yes	No

As verbs, support is a hypernym of take lying down; that is, support is a word with a broader meaning than take lying down:

Other hypernyms of take lying down include abide, bear, brook, digest, endure, put up, stand, stick out, stomach, suffer, tolerate.

take lying down (noun)	support (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

take lying down (verb)	support (verb)
suffer without protest; suffer or endure passively	establish or strengthen as with new evidence or facts
	put up with something or somebody unpleasant
	adopt as a belief
	argue or speak in defense of
	be a regular customer or client of
	be the physical support of; carry the weight of
	play a subordinate role to (another performer)
	support materially or financially
	give moral or psychological support, aid, or courage to
	be behind; approve of
	support with evidence or authority or make more certain or confirm