TITLE: What does the typical non-solvable group look like?
QUESTION [8 upvotes]: According to a result of Higman and Sims (which I learned about in this paper of Poonen's) the typical p-group is 3-step nilpotent of a particular form.  In particular the typical group is a 3-step nilpotent 2-group of a particular form.  By typical here I mean that eventually the number of these groups dominate.
Is anything known about what the typical non-solvable group looks like?  Probably some sort of modification of PSL_2(F_p)?

REPLY [3 votes]: Even if you are asking about a "typical" group of order $\leq n$, the answer is nobody knows, so this $A_5$ idea for a much more delicate question is little more than a pure speculation.  If I understand the state of art correctly, it is completely open whether solvable groups are a majority.  The data, of course, supports a stronger conjecture: that asymptotically almost all groups are $2$-groups.  What is known now, is a number of good asymptotic estimates, notably a Pyber's paper in the Annals (1993), getting a tight asymptotic for the log of the number of groups of order $\le n$ (he gets an upper bound matching the Higman-Sims lower bound). For more on this, read up 
this terrific recent monograph.