I know that by a result of Blyth and Robinson published in 1995, all finite simple groups that all of their proper non-abelian simple sections are isomorphic to one of the simple groups A5 or PSL(2,7) are characterized. I am looking for a characterization of all finite simple groups that all of their proper non-abelian simple sections are isomorphic to one of the simple groups A5, PSL(2,7) or PSL(2,9).

Any help is appreciated!

semi-simple? $\endgroup$