Quantitative properties on non-power subgroups of finite groups

Let G be a group and H be a subgroup of G. H is called a power subgroup of G if there exists a non-negative integer m such that $$H=G^m$$ , where $$G^m:=\langle g^m:g\in G\rangle $$ . A subgroup that is not a power subgroup is a non-power subgroup. Let nps(G) denote the number of non-power subgroups of G. In this paper, we establish the lower bounds of nps(G) for finite non-cyclic nilpotent groups.