Narrow Results

Let $G$ be a finite nilpotent group. We denote by $\gamma (G)$ the number of conjugacy classes of noncyclic subgroups of $G$, $\mathrm{\text{Con}}(G)$ the number of conjugacy classes of subgroups of $G$. In this paper, we give the formula for $\gamma (G)$, and prove that $\mathrm{\text{Con}}(G)-\gamma (G)\ge 4$ when the order of $G$ is prime power. Moreover, we get the minimum of $\gamma (G)$ according to $t$, where $t$ is the number of distinct noncyclic Sylow subgroups of $G$.