Relatively Maximal Subgroups of Odd Index in Symmetric Groups

Let X be a class of finite groups which contains a group of order 2 and is closed under subgroups, homomorphic images, and extensions. We define the concept of an X-admissible diagram representing a natural number n. Associated with each n are finitely many such diagrams, and they all can be found easily. Admissible diagrams representing a number n are used to uniquely parametrize conjugacy classes of maximal X-subgroups of odd index in the symmetric group Sym n , and we define the structure of such groups. As a consequence, we obtain a complete classification of submaximal X-subgroups of odd index in alternating groups.