It is a consequence of the classification of finite simple groups that every non-abelian simple group contains a subgroup which is a minimal simple group.
@article{urn:eudml:doc:41316,
title = {Simple group contain minimal simple groups.},
journal = {Publicacions Matem\`atiques},
volume = {41},
year = {1997},
pages = {411-415},
mrnumber = {MR1485492},
zbl = {0894.20019},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41316}
}
Barry, Michael J. J.; Ward, Michael B. Simple group contain minimal simple groups.. Publicacions Matemàtiques, Tome 41 (1997) pp. 411-415. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41316/