Is $G$ isomorphic to $\frac{G}{H}\times H$? Let $G$ be a group and $H$ a normal subgroup. Is $G$ necessarily isomorphic to $\frac{G}{H}\times H$?

Not only need they not be isomorphic, but $G$ need not even have a subgroup (much less a direct factor) isomorphic to $G/H$.  For example, $SL(2,5)$ has a center $Z$ of order 2, and $SL(2,5)/Z$ is isomorphic to $A_5$, but $SL(2,5)$ has no subgroup isomorphic to $A_5$.  
