The most common and most likely answer to this crossword clue is the 7 letter word FRAENUM
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The straight part
We know that "One gives support to organ" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case one gives support to organ = fraenum.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does FRAENUM mean?
FRAENUM noun- A frenulum.