On the Conjugacy Classes of Cyclic Non-normal Subgroups

LetG be a finite p-group. Assume that ν(G) and νc(G) denote the number of conjugacy classes of non-normal subgroups and non-normal cyclic subgroups of G, respectively. In this paper, we completely classify the finite p-groups with νc = p or p + 1 for an odd prime number p. Also, we classify the groups G with ν(G) = νc(G) = p^i , i ≥ 1.