Normal subgroups of matrices 
Let $G=\begin{bmatrix}1&a\\0&b\end{bmatrix}$ so that $a,b\in\mathbb C$ and $b\ne0$. I need to prove that $G$ has infinitely many normal subgroups. 

I attempt to do this by constructing some family of normal subgroups but I keep failing, as most of the things I try aren't even subgroups. 

Let $g$ and $h$ be two elements in $G$. Calculate $ghg^{-1}$. 
You will immediately see under what conditions $ghg^{-1}$ is "of the same form" as $h$. That is, $ghg^{-1} \in H$ where $H$ is a subgroup of $G$.  
This condition $ghg^{-1} \in H$ means that $H$ is normal. This method will give you infinitely many subgroups. 
