In this paper, we have proved that all non-subnormal subgroups of a finite group G are conjugate if and only if G=〈a, b_1, b_2, …, b_β|a~(p~α)=1=b_1~q=b_2~q=…=b_β~q; [bi, bj]=1, i,j=1,2,…,β;b_i~a=b_(i+1), i=1,2,…,β-1; b_β~a=b_1~d1b_2~d2…b_β~dβ〉where f(x)=x~β-d_βx~(β-1)-…-d_2x-d1 is an irreducible polynomial over the field F_q, which divides x~p-1.

In this paper the authors mainly proved that:If the finite group G has two conjugate classes of non-subnormal subgroups H={H_1,H_2,"",H_m} and K={K_1,K_2,"",K_n},then G is soluble,and|G|has at most three prime factors,and G satisfing one of the following conditions: (1)G=H■Q,where H is a p-group which has cyclic maximal subgroups,and Q is the Sylowq-sub- group of G,p and q are different primes.

We denote the number of the conjugate classes of non-subnormal subgroups of G byμ, and have the following results: Theorem 2.

Considering the subnormal subgroups,some equivalent conditions for nilpotency of finite groups are given and a sufficient condition for nilpotency of finite groups is obtained.

In this paper, we investigate the influence of subnormal subgroups on the structure of finite groups and some sufficient conditions for solvability and a sufficient condition for supersolvability of finite groups are given.

A subgroup H of a group G is said to be S-normal in G if there exists a subnormal subgroup K of G such that G=HK and H∩ K≤H SG .

A finitely generated soluble group G has derived length at most 5 and Fitting length at most 4 if the defect of every subnormal subgroup of G does not exceed 2.

The number of the orders of non-normal subgroups and the structure of finite groups;

Groups of order p~aq~br~c having exactly three conjugacy classes of non-normal subgroups

The Fuzzy Quasinormal Subgroups and Fuzzy Subnormal Subgroups of Finite Groups;