Fourth isomorphism theorem

Name
This result is termed the lattice isomorphism theorem, the fourth isomorphism theorem, and the correspondence theorem.

Statement with symbols
Let $$G$$ be a group and let $$N$$ be a fact about::normal subgroup of $$G$$. Then, we have a bijection:

Set of subgroups of $$G$$ containing $$N$$ $$ \leftrightarrow$$ Set of subgroups of $$G/N$$

If $$\varphi:G \to G/N$$ is the quotient map, then this bijection is given by:

$$H \mapsto \varphi(H)$$

in the forward direction, and:

$$K \mapsto \varphi^{-1}(K)$$

in the reverse direction. Moreover:


 * 1) Under the bijection, normality is preserved. In other words, a subgroup containing $$N$$ is normal if and only if its image under $$\varphi$$ is normal.
 * 2) The bijection is an isomorphism between the lattice of subgroups of $$G$$ containing $$N$$, and the lattice of subgroups of $$G/N$$. In other words, the bijection preserves partial order: $$A \le B$$ if and only if $$\varphi(A) \le \varphi(B)$$. It also preserves intersections and joins.
 * 3) The bijection preserves index. If $$A,B$$ are subgroups of $$G$$ containing $$N$$, with $$A \le B$$, then $$[B:A] = [\varphi(B):\varphi(A)]$$.

Textbook references

 * , Page 99, Theorem 20, Section 3.3 (few steps of proof given, but full proof not provided)
 * , Page 75, Exercise 8, Section 7 (Restriction of a homomorphism to a subgroup) (starred problem, termed Correspondence Problem)