A finite group $G$ is called solvable if every nontrivial subgroup $U$ has a nontrivial abelian factor group (equivalently $U' < U$), or equivalently if every nontrivial subgroup $U$ has a non-trivial nilpotent factor group, i.e. there exists $N \unlhd U, N \ne U$ such that $U / N$ is nilpotent. This follows as nilpotency implies solvability.
Similarly, a finite group is called $p$-solvable if every nontrivial subgroup $U$ has a nontrivial $p$-closed factor group, i.e. there exists $N \unlhd U, N \ne U$ such that $U / N$ has a unique (hence normal) Sylow $p$-subgroup.
Now is there a similar characterisation of nilpotency in terms of subgroups?
The notions of solvable, and also $p$-solvable could be defined in terms of certain (characteristic-, subnormal- or normal-) series, but in the case of nilpotency we just have upper and lower central series, so here a different kind of series arise; and also the "normalizers grow"-definition is not in terms of subgroups of some fixed group, but in terms of groups that properly contain our group, so I see here are some differences, but anyway; is there a characterisation of nilpotent groups in terms of factor groups of nontrivial subgroups?