Determine the right cosets of $\langle{(13)}\rangle$ in $S_3$

Question

Determine the right cosets of $\langle(13)\rangle$ in $S_3$

Here's my attempt

$H={e, (13)}$
$H(12)={(12)(132)}$
$H(123)={(123)(23)}$

I'm not sure about the last one. According to La Grange's, there should be 3 right cosets.

Answer 1

The three cosets are as follows

$H=\{{ I,(13)}\}$

$H(12)=\{{ (12),(132)}\}$

$H(23)=\{{ (23),(123)}\}$.