Consider the group $$G=\begin{Bmatrix} \begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix} : a,b \in \mathbb{R}, \: a>0 \end{Bmatrix}$$ with usual matrix multiplication. Let $$ N = \left\{ \begin{pmatrix}1 & b \\ 0 & 1 \end{pmatrix} : b \in \mathbb{R} \right\}.$$ Then,
- $N$ is not a subgroup of $G$
- $N$ is a subgroup of $G$ but not normal subgroup
- $N$ is a normal subgroup and the quotient group $G/N$ is of finite order
- $N$ is a normal subgroup and the quotient group is isomorphic to $\mathbb{R}^+$ (the group of positive reals with multiplication).