Does not have to be isomorphic to a subgroup:

This is one of the reasons why the analogy between simple groups of finite groups and prime numbers is limited.

Subquotient

Quotient of a subgroup H of G by a normal subgroup of the subgroup H.

That normal subgroup does not have have to be a normal subgroup of G.

As an overkill example, the happy family are subquotients of the monster group, but the monster group is simple.

Relationship between the quotient group and direct products

Although quotients look a bit real number division, there are some important differences with the "group analog of multiplication" of direct product of groups.

If a group is isomorphic to the direct product of groups, we can take a quotient of the product to retrieve one of the groups, which is somewhat analogous to division: math.stackexchange.com/questions/723707/how-is-the-quotient-group-related-to-the-direct-product-group

The "converse" is not always true however: a group does not need to be isomorphic to the product of one of its normal subgroups and the associated quotient group. The wiki page provides an example:

Given G and a normal subgroup N, then G is a group extension of G/N by N. One could ask whether this extension is trivial or split; in other words, one could ask whether G is a direct product or semidirect product of N and G/N. This is a special case of the extension problem. An example where the extension is not split is as follows: Let $G = Z 4 = 0, 1, 2, 3$ , and $= 0, 2$ which is isomorphic to Z2. Then G/N is also isomorphic to Z2. But Z2 has only the trivial automorphism, so the only semi-direct product of N and G/N is the direct product. Since Z4 is different from Z2 × Z2, we conclude that G is not a semi-direct product of N and G/N.

TODO find a less minimal but possibly more important example.

This is also semi mentioned at: math.stackexchange.com/questions/1596500/when-is-a-group-isomorphic-to-the-product-of-normal-subgroup-and-quotient-group

I think this might be equivalent to why the group extension problem is hard. If this relation were true, then taking the direct product would be the only way to make larger groups from normal subgroups/quotients. But it's not.

Normal subgroup

Ultimate explanation: math.stackexchange.com/questions/776039/intuition-behind-normal-subgroups/3732426#3732426

Only normal subgroups can be used to form quotient groups: their key definition is that they plus their cosets form a group.

Intuition:

One key intuition is that "a normal subgroup is the kernel" of a group homomorphism, and the normal subgroup plus cosets are isomorphic to the image of the isomorphism, which is what the fundamental theorem on homomorphisms says.

Therefore "there aren't that many group homomorphism", and a normal subgroup it is a concrete and natural way to uniquely represent that homomorphism.

The best way to think about the, is to always think first: what is the homomorphism? And then work out everything else from there.

Simple group

Does not have any non-trivial normal subgroup.

And therefore, going back to our intuition that due to the fundamental theorem on homomorphisms there is one normal group per homomorphism, a simple group is one that has no non-trivial homomorphisms.