Find all the subgroups of $\Bbb Z/21\Bbb Z$.
The solution given is as follows:
Let $K$ be a subgroup of $\mathbb{Z}/21\mathbb{Z}$. Then $K=H/21\mathbb Z$ for some subgroup $H$ of $\Bbb Z$ such that $21\Bbb Z\subset H$. Again, if $H$ is a subgroup of $\Bbb Z$, such that $21\Bbb Z\subseteq H$, then $H/21\Bbb Z$ is a subgroup of $\Bbb Z/21\Bbb Z$. So we have to determine all subgroups of $\Bbb Z$ that contains $21\Bbb Z$. Now, $1,3,7,21$ are the only positive divisors of $21$. Hence $\Bbb Z,3\Bbb Z,7\Bbb Z$ and $21\Bbb Z$ are the only subgroups of $\Bbb Z$ that contain $21\Bbb Z$. Then $\Bbb Z/21\Bbb Z$, $3\Bbb Z/21\Bbb Z$, $7\Bbb Z/21\Bbb Z$, and $21\Bbb Z/21\Bbb Z$ are the only subgroups of $\Bbb Z/21\Bbb Z$.
However, I don't get how do they conclude "Let $K$ be a subgroup of $\mathbb{Z}/21\mathbb{Z}$.Then $K=H/21\mathbb Z$ for some subgroup $H$ of $\Bbb Z$ such that $21\Bbb Z\subset H$"? Also, how do they say, "Again, if $H$ is a subgroup of $\Bbb Z$, such that $21\Bbb Z\subseteq H$, then $H/21\Bbb Z$ is a subgroup of $\Bbb Z/21\Bbb Z$."? I don't get this proof at all. Can anyone please help me with this?
The following link: Finding all subgroups of $\mathbb{Z_n}$ does not answer my question as my post asks for explanation of a particular proof while in that post, the user asks for a proof of the problem .