In this paper, we introduce a finite non-abelian group named the SR-group \(G_{2n} \) and prove the existence of such a group. We prove the converse of Lagrange’s theorem for SR-group \(G_{2n} \) and deduce the existence of some subgroups of symmetric group \(S_{2n} \) . With the help of the SR-group, we prove that every SR-group is isomorphic to a proper subgroup of another SR-group. As a consequence of this we declare the existence of a permutation group of every even order which is isomorphic to a proper subgroup of some SR-group. Finally, we give an example to find a subgroup of symmetric group \(S_{24} \) .