Sylow Subgroups of the Rubik's Cube Group

The Rubik's Cube Group has order $$2^{27} 3^{14} 5^3 7^2 11$$. What is known about the Sylow subgroups of this group? Do they have an intuitive meaning with regards to the symmetry of the cube?

• This problem is amenable to the free software GAP: see here (39.13-1) and here. – Shaun Apr 3 '19 at 18:28
• Here is the gap tag. If you're happy with computing the subgroups, then I suggest you add it with an edit. – Shaun Apr 3 '19 at 18:59
• There are even questions having both gap and rubiks-cube tags, e.g. this – Alexander Konovalov Apr 3 '19 at 20:44