One consequence of the Classification of Finite Simple Groups is that primitive groups (apart from the symmetric and alternating groups) are small. This means that they have small order, but can be interpreted in other ways too. For example, Cameron, Neumann and Saxl showed that, apart from symmetric and alternating groups and finitely many others, for any primitive group there is a subset of the domain whose setwise stabiliser is the identity. Subsequently, Seress found all the exceptions (there are exactly 43 of them, the largest having degree 32).
A related situation occurs with switching classes of graphs. Switching classes arise in many areas: equiangular line systems, finite simple groups, and combinatorial geometry, among others. Apart from trivial switching classes and finitely many others, any switching class with primitive automorphism group contains a graph with trivial automorphism group. The problem of determining the exceptions is open, and would be an interesting challenge for someone wanting to work on finite groups and computation.