My work so far:
The cyclic subgroups are trivial to find. They are simply rotations generated from $\langle r^d\rangle$ where $d\mid10$. So $d=1,2,5,10$.
$\langle r \rangle= \{e,r, r^2, \dots, r^9 \}$
$\langle r^2 \rangle= \{e, r^2, r^4, r^6, r^8 \}$
$\langle r^5 \rangle= \{e, r^5 \}$
$\langle r^{10} \rangle= \{e\}$
The dihedral groups are giving me trouble. I've read Theorem 3.1 and Example 3.6 of Keith Condrad's notes and I don't understand how to use the index he's referring to. He states that I can generate all the dihedral subgroups of $D_{10}$ from the following: $\langle r^d, r^{i}s\rangle$ where $d\mid 10$ and $0\leq i \leq d-1$ with index $d$.
For example, I understand that a reflection paired with the identity element will form a subgroup. So these would be $\{e,s\}, \{e,rs\}, \{e,r^s\},\dots, \{e,r^9s\}$.
At the moment, my understanding of what he's writing is since there are no rotations in thsse subgroups, $d$ would need to be $0$ in $\langle r^d, r^{i}s\rangle$, so I'd have $\langle e, r^{i}s\rangle$. But if $d=0$, then $0\leq i\leq d-1$ would be $0\leq i\leq -1$, which doesn't make any sense.