I was trying to understand more about orders of elements in a cyclic group $G$ and non-trivial subgroups that were generated from a cyclic group $G$ and so read the following example: Non trivial subgroups in cyclic group G.
But I do not how one would find trivial and non-trivial groups for say a group $G = \mathbb{Z}^{\times}_{73}$. I get that you have to do $73-1$, as $73$ is prime so you are left with $72$ and then we find all the factors of $72$ which are $\{1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72\}.$
After this, we have to try and find numbers that are not $1 \text{ mod } 72$. I do not where to start in terms of finding the subgroups like do I start with $2^1 \text{ mod } 72, 2^2 \text{ mod } 72, 2^3 \text{ mod } 72, \dots,2^{72} \text{ mod } 72$ to find the subgroups.
Like is there a way of finding non trivial subgroups in a cyclic group?