# Lattice of subgroups

## Contents

## Definition

The lattice of subgroups is a lattice (that is, a special kind of partially ordered set) whose elements are the subgroups are where the partial order is subgroup inclusion. Important points about this lattice:

- The meet operation in the lattice is intersection of subgroups
- The join operation in the lattice is join of subgroups
- The lattice is bounded, with the upper bound being the improper subgroup (or the whole group) and the lower bound being the trivial subgroup.
- The lattice is a complete lattice, that is, the meet and join operation can both be performed for infinitely many elements.

## Importance

### Lattice-theoretic properties of subgroups

`Further information: Lattice-theoretic properties of subgroups`