---
tag: 4DWC4XFS34
kind: example
name: Lagrange's theorem in S3
msc: [20D60]
illustrates: [3C53TD]
uses: [0515A6D2Y1, 0X1JXT47GA]
sources:
  - url: https://en.wikipedia.org/wiki/Dihedral_group_of_order_6
    locator: Subgroups
---

## Example

The [[symmetric group|0515A6D2Y1]] $S_3$ has order $6$.
Its [[subgroups|0X1JXT47GA]] are $\{e\}$, the three subgroups $\{e, (1\,2)\}$, $\{e, (1\,3)\}$ and $\{e, (2\,3)\}$, the subgroup $A_3$ of order $3$, and $S_3$.
Their orders are $1$, $2$, $3$ and $6$. Each of these numbers divides $6$, as [[Lagrange's theorem|3C53TD]] states.

## Remarks

No subgroup has order $4$ or $5$, because these numbers do not divide $6$.