Determine all quotient group of $S_3$ and the corresponding homomorphism image.

Question

The group $S_3$ has only three normal subgroups. They are $S_3, \{e\}$, and $\lbrace e,(1 \ 2 \ 3), (1\ 3 \ 2) \rbrace$ where $e$ is the identity element of $S_3$. Determine all quotient groups of $S_3$ and the corresponding homomorphism image.

Here is what I tried.

Let $H=\lbrace e,(1 \ 2 \ 3), (1\ 3 \ 2) \rbrace$. Then, for the quotient group $S_3/H$, I found $S_3/H=\lbrace H, (2\ 3)H \rbrace$. But I stuck for the next and for the quotient group $S_3/e$ and $S_3/S_3$. Any idea? Thanks for help in advanced.

Answer 1

The homomorphic images are $\{e\},S_3$ and $\Bbb Z_2$. Note that there is only one $2$-element group.