Decomposability of finite groups

Let G be a finite group and A be a normal subgroup of G . We denote by ncc( A) theA and is called n -decomposable if ncc( A) = n . Setnumber of G -conjugacy classes ofK G = {ncc( A) A G} . Let X be a non-empty subset of positive integers. A group G is called X -decomposable, if K G = X .In [13], Shahryari and Shahabi, investigated the structure of finite groups, which contains a2 -decomposable subgroup H . In this case, H ≤ G ′ , H ( H − 1) divides G and H is aelementary abelian normal subgroup of G .In [14], Shahryari and Shahabi studied the structure of finite groups G with a 3 -decomposable subgroup H . They proved that, H is either an elementary abelian subgroup, ametabelian p -group or a Frobenius group with elementary abelian kernel H ′ .In [17], Reise and Shahabi, determined the structure of finite groups G with a 4 -decomposable subgroup.In [6], Ashrafi and Sahraei, characterized the structure of 2 -, 3 -, 4 -decomposable finitegroups. And they obtained the structure of n -decomposable solvable finite groups.In this study, series articles of Ashrafi was examined. In the first part, the definitions and theoremswhich are necessary in this study, were given. In the second part, the structure of 5 - and 6 -decomposable finite groups was examined. In the third part, 7 - and 8 -decomposable finite groupswere characterized. In the last part, if X = {1, 2,3} and {1,3, 4} , then X -decomposable finite groupsstructure was examined.