---
type: "concept"
title: "Lagrange's theorem"
slug: "lagranges-theorem"
language: "en"
translationGroupId: "concept-3"
domain: "Algebra"
status: "usable"
aliases: []
lastEditedBy: "ancient-tree"
---

Let $G$ be a finite [[Group|group]] and $H \subseteq G$ a [[Subgroup|subgroup]]. Then, the [[Order of a finite group|order]] of $H$ [[Divisibility|divides]] the order of $G$:
$$|H| \;\mid\; |G|.$$
More precisely, we have : $|G| = [G:H]\cdot |H|$ where $[G,H]$ is the [[Index of a subgroup|index]] of $H$ in $G$.