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Let $\mathfrak{Nil}$ be a class of nilpotent groups and $G$ be anon-$\mathfrak{Nil}$ group. We call $G$ a meta-$\mathfrak{Nil}$-Hamiltoniangroup if any of its non-$\mathfrak{Nil}$ subgroups are normal. Also, we call$G$ a para-$\mathfrak{Nil}$-Hamiltonian group if every non-normal subgroup of$G$ is either an $\mathfrak{Nil}$-group or a minimal non-$\mathfrak{Nil}$group. In this paper we investigate the class of finitely generatedmeta-$\mathfrak{Nil}$-Hamiltonian and para-$\mathfrak{Nil}$-Hamiltonian groups.

Groups whose non-normal subgroups are either nilpotent or minimal non-nilpotent