The finite simple groups, at least at our current level of understanding, are quite chimeric, in that we have four different "heads" to this beast:
- The finite cyclic groups of prime order;
- The alternating groups;
- The finite simple groups of Lie type; and
- The sporadic groups.
While one can partially unify pairs of these heads together (for instance, by viewing the alternating group as the special linear group over the "field of one element", whatever that means), I think it is fair to say that we don't yet have any real understanding of why the answer to such a basic mathematical classification problem comes in so many disjoint pieces.