So I've found all the cyclic subgroups: $\langle e\rangle$, $\langle r\rangle$, $\langle sr^n\rangle$, and $D_5$ itself (which are 8 subgroups), but how do I know if these are all? How can I find the rest?
Questions on subgroups of $D_n$ are quite frequent on this site. It is certainly possible to give a short argument each time, but I think that the article of Keith Conrad is worth to be noted, because it gives all the necessary details to solve the questions in general. Section $3$ classifies all subgroups of $D_n$.