# Subgroups of a Finite Group containing two elements

Currently working on a Group Theory question:

Let $G$ be a cyclic group of order 100 and let $a \in G$ denote a generator. Find two subgroups of $G$ which contain both $a^{20}$ and $a^{55}$. Are there any other subgroups of $G$ containing these two elements?

So my attempt at this question is taking the given. The order of $G$ is 100 and I know that $a^m = n/gcd(m,n)$. But I'm unsure as to how this helps. Any hints on how to proceed are appreciated.

$$ord(a^{20})=5\;,\;\;ord(a^{55})=20\;\implies\;\text{any subgroup containing this two elements will}$$
have at least order $\;20\;$ ....but also order $\;20\;$ at most (why? Lagrange theorem), and since any finite cyclic group has one unique proper subgroup of any order dividing the group's order then there is only one such subgroup
• I think that there are at least two: yours and $G$ itself. – Antoine Oct 26 '16 at 18:44
• @ElSpiffy Both groups, $\;G\;$ itself and the subgroup generated by $\;a^{55}\;$ , are indeed the answer. – DonAntonio Oct 26 '16 at 19:51