Subgroup embeddings in the symmetric group of degree nine

This research is devoted to the determination of embedding properties of all nonprimary subgroups of the symmetric group on nine letters and continues previous investigations of the authors. We have tested the above-mentioned subgroups for the following properties: abnormality, pronormality, paranormality, their weak analogies, weak normality and the subnormalizer condition. The technique of Burnside marks as well as the respective information on the table of mark of $S_9$ from the computer algebra package GAP (Groups, Algorithms and Programming) library ‘TOM’ were used. Subgroups of prime-power orders have not been considered because many of the above-mentioned concepts coincide for such subgroups. The total number of subgroups in question turned out to be 432.