# Finite Solvable Groups with Few Non-cyclic Subgroups

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## Abstract

Let *G* be a finite group and \(\delta (G)\) denote the number of conjugate classes of allnon-cyclic subgroups of *G*. The symbol \(\pi (G)\) denotes the set of the prime divisors of |*G*|. In Meng and Li (Sci Sin Math 44:939–944, 2014), it was proved that for a finite non-cyclic solvable group *G*, one always has \(\delta (G)\ge 2^{|\pi (G)|-2}\). The groups with \(\delta (G)\le |\pi (G)|+1\) always are solvable and have been complete classified. Moreover, it was showed that a finite non-solvable group *G* with \(\delta (G)=|\pi (G)|+2\) is isomorphic to \(A_5\) or *SL*(2, 5). In this paper, we investigate the finite solvable groups with \(\delta (G)=|\pi (G)|+2\). For convenience, a group *G* is said to be a \(\delta \pi _2\)-group if \(\delta (G)=|\pi (G)|+2\). In particular, we give a completely classification of the \(\delta \pi _2\)-groups with \(|\pi (G)|=3,4\).

## Keywords

Cyclic subgroups Conjugacy class Classification## Mathematics Subject Classification

20D10 20D20## Notes

### Acknowledgements

The authors are grateful to the referee, who provided their valuable suggestions and detailed report.

## References

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