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

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