Maximal subgroup has prime power index in finite solvable group

Statement

Let $G$ be a finite solvable group and $M$ be a maximal subgroup of $G$. Then, the index $[G:M]$ is a power of a prime.

Facts used

1. Primitive implies Fitting-free or elementary Abelian Fitting subgroup

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 219, Theorem 1.5 (Section 6.1)
• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 200, Exercise 32, Section 6.1 ($p$-groups, nilpotent groups and solvable groups)