Why do we have $3$ Isomorphism Theorems? This is a bit of a soft question, but I've wondered about it since being introduced to the $3$ isomorphism theorems (I'm aware of the $4^{th}$ as well, but it is not typically presented in the classroom as far as I know).
For setup purposes, say $G$ is a group with $H$ and $K$ normal subgroups of $G$ (and $K \subseteq H$), and $S$ a subgroup of $G$.  It seems natural to ask what groups are isomorphic to $G / H$, $\frac{G / H}{H / K}$, and $SH / H$, the answers to which ultimately yield the isomorphism theorems.
I'm probably wrong here, but it seems like there must be more exotic ways of combining $G$, $H$, $K$, and $S$ (or even other (normal) subgroups) into quotient groups and asking what they are isomorphic to.  My question is, why aren't there more?  Of course we don't want $1729$ isomorphism theorems clogging textbooks, but why these $3$? Perhaps most / all questions of quotient groups can be determined from these $3$ theorems?

Well, actually we only need the first isomorphism theorem, because the rest follows from this one. But for referal reasons the second and third isomorphism theorem has been made to theorems. The other theorems are lessed used and are most of the times presented as lemma's or propositions, because they are used in specific situations. This was the explanation to this question in our class. I hope this will answer your question.
