I have to find all the homomorphisms, $$h:C_{5}\to \operatorname{Aut}(C_{31})$$
Given that there are thirty elements in $\operatorname{Aut}(C_{31})$, do I have to find the order of each of the elements and then see which of them have orders that divide $|C_{5}|=5$, or is there a quicker way to find these homomorphisms than this exhaustive method?
Thank you in advance.