---
tag: 0B8BW4YWTF
kind: concept
name: Lattice of subgroups
aliases: [subgroup-lattice]
msc: ["20E15"]
uses: [0N0PXCTYA7, 0RG9SPQSWM, 0ZTGMDNVTB]
sources:
  - url: https://en.wikipedia.org/wiki/Lattice_of_subgroups
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Lattice_of_subgroups
    locator: Properties
same_as:
  wikidata: Q3538331
  wikipedia: Lattice of subgroups
---

## Summary

The [[subgroups|0X1JXT47GA]] of a [[group|G5S1M9]] $G$, ordered by inclusion, form a complete lattice.
The meet of a family of subgroups is their intersection, and the join is the [[subgroup generated by|0RP7VWQ0YN]] their union.
The least element is the trivial subgroup $\{e\}$ and the greatest element is $G$.

## Remarks

For two subgroups $H$ and $K$, the meet is $H \cap K$ and the join is $\langle H \cup K \rangle$.
The lattice of subgroups of $(\mathbb{Z}, +)$ is the set of the subgroups $n\mathbb{Z}$, and $m\mathbb{Z} \subseteq n\mathbb{Z}$ exactly when $n$ divides $m$.
The lattice of normal subgroups of a group is a modular lattice.

## Explanations

### L1 human

You can draw the subgroups of a small group as a diagram, with a line from each subgroup up to each next bigger subgroup.
This diagram shows the lattice of subgroups.
Two subgroups always have a largest common subgroup below them and a smallest subgroup above them.