For a recent qualifier problem, I was to show that if a group $G$ has two solvable subgroups $H,K$ such that $HK=G$ and $H$ is normal, then $G$ is solvable. This is simply a matter of showing that $G/H$ is solvable, and I think is not too difficult. The next part of the question was to find an example of a group with all of the above conditions except the normality condition (i.e. for any two such solvable subgroups, neither is normal in $G$) and show that $G$ is no longer solvable. Does anyone know of a good example? I don't even know that many groups which aren't solvable. I have been told $A_5$ is not solvable, but that is quite a large group, and it seems like it would take a long time to show this in 20 minutes (the time I would have if I was doing a qualifier) if it is even true for $A_5$. I'd love to know what group to look at, so I can prove it myself.