# Lattice-complemented does not satisfy intermediate subgroup condition

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., lattice-complemented subgroup)notsatisfying a subgroup metaproperty (i.e., intermediate subgroup condition).

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## Statement

It can happen that is a lattice-complemented subgroup of , and , with *not* a lattice-complemented subgroup in .

## Proof

### Example of the symmetric group

`Further information: symmetric group:S4`

Let be the symmetric group on a set of size four.

Consider the subgroups:

.

is a subgroup of order two and is a subgroup of order four containing . Further:

- is a lattice-complemented subgroup in : The symmetric group on the subset is a lattice complement to in .
- is not a lattice-complemented subgroup in : Indeed, is a cyclic group of order four and is a subgroup of order two, and has no lattice complements.