Listing subgroups of a group

Question

I made a program to list all the subgroups of any group and I came up with satisfactory result for $\operatorname{Symmetric Group}[3]$ as

$\left\{\{\text{Cycles}[\{\}]\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{cc} 1 & 2 \\ \end{array} \right)\right]\right\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{cc} 1 & 3 \\ \end{array} \right)\right]\right\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{cc} 2 & 3 \\ \end{array} \right)\right]\right\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{ccc} 1 & 2 & 3 \\ \end{array} \right)\right],\text{Cycles}\left[\left( \begin{array}{ccc} 1 & 3 & 2 \\ \end{array} \right)\right]\right\}\right\}$

It excludes the whole set itself though it can be added seperately. But in case of $SymmetricGroup[4]$ I am getting following

$\left\{\{\text{Cycles}[\{\}]\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{cc} 1 & 2 \\ \end{array} \right)\right]\right\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{cc} 1 & 3 \\ \end{array} \right)\right]\right\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{cc} 1 & 4 \\ \end{array} \right)\right]\right\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{cc} 2 & 3 \\ \end{array} \right)\right]\right\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{cc} 2 & 4 \\ \end{array} \right)\right]\right\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{cc} 3 & 4 \\ \end{array} \right)\right]\right\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array} \right)\right]\right\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{cc} 1 & 3 \\ 2 & 4 \\ \end{array} \right)\right]\right\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \\ \end{array} \right)\right]\right\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{ccc} 1 & 2 & 3 \\ \end{array} \right)\right],\text{Cycles}\left[\left( \begin{array}{ccc} 1 & 3 & 2 \\ \end{array} \right)\right]\right\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{ccc} 1 & 2 & 4 \\ \end{array} \right)\right],\text{Cycles}\left[\left( \begin{array}{ccc} 1 & 4 & 2 \\ \end{array} \right)\right]\right\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{ccc} 1 & 3 & 4 \\ \end{array} \right)\right],\text{Cycles}\left[\left( \begin{array}{ccc} 1 & 4 & 3 \\ \end{array} \right)\right]\right\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left( \begin{array}{ccc} 2 & 3 & 4 \\ \end{array} \right)\right],\text{Cycles}\left[\left( \begin{array}{ccc} 2 & 4 & 3 \\ \end{array} \right)\right]\right\}\right\}$

The matrix form shows double transposition. Can someone please check for me if I am getting appropriate results? I doubt I am!!

Answer 1

I have the impression that you only list the cyclic subgroups.

For $S_3$, the full group $S_3$ ist missing as a subgroup (you are mentioning that in your question).

For $S_4$, several subgroups are missing. In total, there should be $30$ of them. $14$ of them are cyclic, which are exactly the ones you listed. To give you a concrete example, the famous Klein Four subgroup $$\{\operatorname{id},(12)(34),(13)(24),(14)(23)\}$$ is not contained in your list.

Answer 2

By using GAP, you can make them all so easier and visible.

  gap> LoadPackage("sonata");;
  gap> s4:=SymmetricGroup(IsPermGroup,4);;
  gap> g:=Subgroups(s4);; 
  gap> List([1..Size(g)],k->g[k]);