Lattice of subgroups

Summary

The subgroups of a group $G$, 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 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.