We can verify that $$\langle 1 \rangle=\{1\},$$ $$\langle 5 \rangle=\{1,5,7,17,13,11\}=G,$$ $$\langle 7 \rangle=\{1,7,13\},$$ $$\langle 11 \rangle=\{1,11,13,17,7,5,1\}=G,$$ $$\langle 13 \rangle=\{13,7,1\}=\langle 7 \rangle,$$ $$\langle 17 \rangle=\langle -1 \rangle=\{1,17\}.$$
This gives the four subgroups $\{1\}$, $\langle 7 \rangle$, $\langle -1 \rangle$, and $G$. We could argue that $G$ is cyclic, and thus any subgroup of $G$ is also cyclic, and thus this list is complete.
Alternatively, we see that any subgroup containing $5$ or $11$ must be $G$ itself. Therefore, any subgroup without $5$ nor $11$ must (a) contain $1$, (b) be a subset of $\{1,7,13,17\}$ (c) contain either $1$, $2$ or $3$ elements, by Lagrange's Theorem.
- If it contains $1$ element, the subgroup must be $\{1\}$.
- If it contains $2$ element, the subgroup must be $\langle 17 \rangle$, since the elements $7$ and $13$ have order $3$.
- If it contains $3$ element, it cannot contain $17$ since it has order $2$, and a group of order $3$ cannot contain a subgroup of order $2$, by Lagrange's Theorem. This leaves only $\langle 7 \rangle$.
So the four subgroups identified are all of the subgroups.
It looks like $\langle 5 \rangle$ is omitted since it's not a proper subgroup of $G$ (i.e., it equals $G$ itself).