Is the product of two soluble groups always soluble??

Question

I know that there are finite non-soluble groups with soluble subgroups. Is it then possible to produce a non-soluble group from soluble groups?

Answer 1

Yes, for example $A_5$ is a product of subgroups $A_4C_5$. More generally, the groups ${\rm PSL}(2,q)$ are the product of two soluble subgroups: a point stabilizer and a dihedral transitive subgroup.

Answer 2

Let $G$ be a finite group and $G=HK$, where $H$ and $K$ are subgroups of $G$.

Kegel- Wieland: If $H$ and $K$ are nilpotent, then $G=HK$ is solvable.

Now we consider an example, where $H$ and $K$ are solvable but not nilpotent. We can take $H=A_4$ and $K=C_5$ and get $A_5=A_4C_5$, see Derek's answer. Now we know that $A_5$ is not solvable. Nevertheless $A_5$ is the product of tow solvable subgroups.