Early in a course in Algebra the result that every group can be embedded as a subgroup
of a symmetric group is introduced. One can further work on it to embed it as a subgroup of a suitable (higher degree) alternating group.
Inverting the view point we can say that the family of simple groups $A_n, n\geq 5$, contains all finite groups as their subgroups.
My question now is, is the same true for each of the other infinite families listed in the Classification of Finite Simple Groups?
In case the answer to this question is negative it might lead to some categorization. Cayley's embedding theorem is often considered a 'useless theorem', as no result about that group can be proved using that embedding. (Is that correct?) Other simple groups being somewhat more special (structure preserving maps of some non-trivial structure), we can categorize groups according to which infinite family(ies) they fall into. And groups embeddable in a particular family, but not embeddable in another may exhibit some special property.
Hope this provides a motivation for the question.