Fourth isomorphism theorem

This article gives the statement, and possibly proof, of a basic fact in group theory.
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This article is about an isomorphism theorem in group theory.
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Name

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

Statement

Statement with symbols

Let $G$ be a group and let $N$ be a 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:

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.
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: $A\leq B$ if and only if $\varphi (A)\leq \varphi (B)$ . It also preserves intersections and joins.
The bijection preserves index. If $A,B$ are subgroups of $G$ containing $N$ , with $A\leq B$ , then $[B:A]=[\varphi (B):\varphi (A)]$ .

References

Textbook references

Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 99, Theorem 20, Section 3.3 (few steps of proof given, but full proof not provided)
Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 75, Exercise 8, Section 7 (Restriction of a homomorphism to a subgroup) (starred problem, termed Correspondence Problem)