Chairmen see out great foes (4-7)

I believe the answer is:

arch-enemies

I'm a little stuck... Click here to teach me more about this clue!

'foes' is the definition.
(I know this)

'chairmen see out' is the wordplay.
'out' indicates an anagram (out can mean wrong or inaccurate).
'chairmen'+'see'='chairmensee'
'chairmensee' anagrammed gives 'ARCH-ENEMIES'.

'great' acts as a link.
I am not very happy about this link. It may belong to another bit of the clue.

Can you help me to learn more?

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