Hello, I want to make sure I understand the fundamental theorem of cyclic groups correctly.

For example, consider the group

. I need to construct a subgroup lattice for

. Since the order of

is 45, I need to find all the natural numbers

that divide 45. Of course, these numbers are 1, 3, 5, 9, 15, and 45, which have orders 45, 15, 9, 5, 3, and 1, respectively. Since

we have

,

,

, and so on are the subgroups of

. From here I just decide which set is a subset of the other and I am done.

Am I using the theorem correctly? Does this mean that any natural number

that does not divide the order of

will create set that does not abide by group axioms? I know I can check this for the cyclic group in this example, but, in general, is the previous sentence correct?