Carefully analyse underworld worshippers? (7)

I believe the answer is:

dissect

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

'carefully analyse' is the definition.
'dissect' can be an answer for 'analyse' (synonyms). I'm unsure of the 'carefully' bit.

'underworld worshippers?' is the wordplay.
'underworld' becomes 'dis' (underworld in classical mythology).
'worshippers?' becomes 'sect' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'dis'+'sect'='DISSECT'

Can you help me to learn more?

(Other definitions for dissect that I've seen before include "Critically examine" , "Examine minutely" , "Anatomise" , "Cut open or cut apart" , "Anatomize" .)

I've seen this clue in The Telegraph.

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