Given a finite group $G$ that is not abelian, nilpotent, or solvable, what is the smallest normal subgroup $H$ in each case such that $G/H$ is abelian, nilpotent, or solvable (respectively)?
In the abelian case it seems clear that the correct subgroup is the commutator subgroup. But what of the other two? Perhaps I'm just shaky with the concepts (of nilpotency and solvability) but I'm having trouble finding the right way to tackle this problem.
Edit: Well I've tried the most obvious thing and it worked out.
For the nilpotent case I took the intersection of all the subgroups in the lower central series, and for the solvable case I took the intersection of all subgroups in the derived/commutator series.