On the Number of Conjugacy Classes of Subgroups of a Finite Group

Let $$k'(G)$$ k ′ ( G ) and L ( G ) be the number of conjugacy classes of subgroups and the subgroup lattice of a finite group G , respectively. Our objective is to study some aspects related to the ratios d'(G)=k'(G)/|L(G)| d ′ ( G ) = k ′ ( G ) | L ( G ) | and d^*(G)=min \ d'(S) S is a section of G\ d ∗ ( G ) = min { d ′ ( S ) ∣ S is a section of G } which measure how close G is to being a Dedekind group. We prove that the set containing the values $$d'(G)$$ d ′ ( G ) , as G ranges over the class of nilpotent groups, is dense in [0, 1]. A nilpotency criterion is obtained by proving that if d^*(G)>2/3 d ∗ ( G ) > 2 3 , then G is nilpotent and information on its structure is given. We also show that if d^*(G)>4/5 d ∗ ( G ) > 4 5 , then G is an Iwasawa group. Finally, we deduce a result which ensures that a p -group of order pⁿ p n ( n≥ 3 n ≥ 3