Geodesic
Relatively maximal subgroups of odd index in symmetric groups
Algebra i logika, Tome 61 (2022) no. 2, pp. 150-179.

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Let $\mathfrak{X}$ 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 $\mathfrak{X}$-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 $\mathfrak{X}$-subgroups of odd index in the symmetric group $\mathrm{Sym}_n$, and we define the structure of such groups. As a consequence, we obtain a complete classification of submaximal $\mathfrak{X}$-subgroups of odd index in alternating groups.
Keywords: symmetric group, subgroup of odd index, complete class, maximal $\mathfrak{X}$-subgroup, submaximal $\mathfrak{X}$-subgroup. 
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A. S. Vasil'ev; D. O. Revin. Relatively maximal subgroups of odd index in symmetric groups. Algebra i logika, Tome 61 (2022) no. 2, pp. 150-179. http://geodesic.mathdoc.fr/item/AL_2022_61_2_a1/

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