Retrieving information about $G$ from $G/H$ for a subgroup $H$ of $G$.

Question

Gallian states in his algebra book that

When the subgroup $H$ of $G$ is normal, then the set of left (or right) cosets of $H$ in $G$ is itself a group—called the factor group of $G$ by $H$ (or the quotient group of $G$ by $H$). Quite often, one can obtain information about a group by studying one of its factor groups. This method will be illustrated in the next section of this chapter.

He then gives the example for $G=4 \Bbb Z$ and constructs $\Bbb Z/4\Bbb Z$ and shows that it's isomorphic to $\Bbb Z_4$.

However I think I'm missing the point here. Everyone seems to motivate normal subgroups as a way to retrieve information from the original group $G$, but these examples don't show how this is true.

If anyone happens to know a specific example of how the statement

Quite often, one can obtain information about a group by studying one of its factor groups.

is true I would be very delighted. I don't think that in Gallian's example of $\Bbb Z/\Bbb 4\Bbb Z$ gives any information about $\Bbb Z$ itself?

Answer 1

From Dummit and Foote's Abstract Algebra:

More precisely, there is a one-to-one correspondence between the subgroups of $G$ containing $N$ and the subgroups of $G/N$, so that the lattice for $G/N$ (or rather, an isomorphic copy) appears in the lattice for $G$ as the collection of subgroups of $G$ between $N$ and $G$.

The example which I often use: if you know the subgroup lattice diagram of $D_8/\langle r^2\rangle\cong V_4\cong \mathbb{Z}_2\oplus\mathbb{Z}_2$) (information about factor group), then you can know a "part" of the subgroup lattice diagram of $D_8$ (information about the whole group).

By the way, this website https://hobbes.la.asu.edu/groups/groups.html is very helpful for studying subgroup lattice diagrams.