Finite <Emphasis Type="Italic">p</Emphasis>-groups whose non-normal subgroups have few orders

Suppose that G is a finite p-group. If G is not a Dedekind group, then G has a non-normal subgroup. We use p^M(G) and p^m(G) to denote the maximum and minimum of the orders of the non-normal subgroups of G; respectively. In this paper, we classify groups G such that M(G) < 2m(G)?1: As a by-product, we also classify p-groups whose orders of non-normal subgroups are p^k and p^k+1.