Conjugacy classes of maximal cyclic subgroups of metacyclic p-groups

In this paper, we set eta(G) to be the number of conjugacy classes of maximal cyclic subgroups of a finite group G. We compute eta(G) for all metacyclic p-groups. We show that if G is a metacyclic p-group of order p(n) that is not dihedral, generalized quaternion, or semi-dihedral, then eta(G) >= n - 2, and we determine when equality holds.