I am asked to describe the cosets of the subgroup H of $D_n$ (Dihedral group on n vertices) generated by the reflection element s, $H=\langle s \rangle$. So $H=\{id,s\}$, and for $r^i \in D_n ,1 \le i \le n $, $$ r^iH = \{r^i,r^is\}\quad Hr^i =\{r^i,sr^i\}$$ $$sr^iH=\{sr^i,r^{n-i}\} \quad Hsr^i=\{sr^i,r^i\} $$
My question is if this adequately describes the set of cosets of $H \lt D_n$, and if in general when describing cosets, one can separate the cosets into cases as I have done above.