Question clarification request for: list all the subgroups of $\Bbb Z_{20}/K,$ where $K=\{0,4,8, 12,16\}$

Question

For the following question, I am not sure if it is asking me about listing all the normal subgroups of $\Bbb Z_{20}/K$ or is this related to the first isomorphism theorem where $K$ is the kernel of a mapping say $f([x]_{20})=[5x]_{20}$, if that is the case, then $\Bbb Z_{20}/K$ is isomorphic to $\Bbb Z_5$.

Question: List all the subgroups of $\Bbb Z_{20}/K,$ where $K=\{0,4,8, 12,16\}$

Can someone tell me what I am supposed to do to solve the problem?

Thank you in advance

Answer 1

Let $G=\Bbb Z_{20}/K$. The question is simply asking you to find the (not a priori normal) subgroups of $G$. This is fairly simple, since

$$\begin{align} |G|&=|\Bbb Z_{20}/K|\\ &=|\Bbb Z_{20}|/|K|\\ &=20/5\\ &=4 \end{align}$$

and there are only two groups of order four up to isomorphism.