Rules of Products of Quotient Groups

Question

Suppose that you have 2 groups, $G$ and $H$, such that $H\trianglelefteq G$. I was wondering if we can say that $$ (G/H) \times H \cong G $$ I know some rules about product and quotient groups work "like you would expect them to," such as $(G_1\times G_2)/(H_1\times H_2)\cong(G_1/H_1)\times(G_2/H_2)$. I have the following set of isomorphic equalities: $$ (G/H)\times H\cong (G/H)\times(H/\{e\})\cong(G\times H)/(H\times\{e\}) $$ But, I don't know if I can then say that $$ (G/H)\times H\cong(G\times H)/(\{e\}\times H) $$ I would think that you cannot simply say this. For example, suppose that we did this same argument with 3 different groups, $A,B\trianglelefteq C$. This should mean that $$ (C/A)\times B\cong(C\times B)/(\{e\}\times A) $$ But, we know nothing about the relationship of $A$ and $B$ and if $B\trianglelefteq A$, and you can easily come up with a counter example by simply letting $B\cong\{e\}$. So, does anyone have any ideas about this? Is it wrong that $(G/H) \times H \cong G$? Are there any extra assumptions we can place on $H$ and $G$ to make it correct?