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Of course the obvious way to proceed is to look for a homomorphism from [tex]G[/tex] to [tex]G/N[/tex] whose kernel is M, but I can't think of one.

What I actually want to show is a more specialized result (namely the case when finite [tex]G/N[/tex] is the nilpotent quotient of [tex]G[/tex] and [tex]G/M[/tex] is a maximal p-quotient of [tex]G[/tex] for some p dividing the order of [tex]G/N[/tex]) but the above is a lot cleaner and didn't yield obviously to a proof or counter-example so I thought I'd explore that first.