# Subgroups of Small Rank

LiE contains a small database with the types of the maximal proper subgroups of complex reductive simply connected Lie groups $g$, where $g$ is simple and of rank at most $8$. We copied this list into Magma, and it can be accessed using the following functions.

## `LiEMaximalSubgroups() -> SeqEnum`

All maximal subgroups as described above, as a sequence of pairs. Each pair consists of a string denoting the simple group at hand, and a sequence of strings denoting its maximal subgroups.

## `MaximalSubgroups(G): MonStgElt -> SeqEnum[MonStgElt]`

The maximal subgroups of the complex reductive simply connected simple Lie group whose Cartan type is the string $G$, represented as a sequence of strings.

## `RestrictionMatrix(G, H): MonStgElt, MonStgElt -> AlgMatElt`

```magma
Index: RngIntElt                    Default: 
```

The restriction matrix for the maximal proper subgroup of type $H$ of $G$. If more than one maximal subgroup of $G$ is of type $H$, the parameter `Index` must be set to indicate which one is required.

## `Example: Subgroup DB (ex-23802f)`

Using the subgroup database:

```magma
> MaximalSubgroups("E7");
[ A2, A1, A1, A1F4, G2C3, A1G2, A1A1, D6A1, A7, A5A2 ]
> M := RestrictionMatrix("E7", "A1" : Index := 2); M;
[26]
[37]
[50]
[72]
[57]
[40]
[21]
> R := RootDatum("E7" : Isogeny := "SC");
> S := RootDatum("A1" : Isogeny := "SC");
> D := AdjointRepresentationDecomposition(R);
> RepresentationDimension(D);
133
> E := Branch(S, D, M); #E;
8
> RepresentationDimension(E);
133

```