Undecided questions
Why is this sentence incorrect?

I hope that someone may be able to help with the following sentence;

‘The woman, to whom the dog belongs, is young’

I’d translate it as follows;

die Frau, der DEN Hund gehört, ist jung

my reasoning being that the dog is the object of the subclause (as it is owned by the woman who is the subject) and so is ‘den Hund’

however, I understand this to be incorrect and that it should be as follows;

die Frau, der DER Hund gehört, ist reich

Sorry if I'm (as usual) missing something obvious and however, I don’t see how this can be correct and hope very much that someone can explain it to me (I'd be very grateful!).

Thanks

Gemma