Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
996 conjugacy classes of subgroups; only the 509 normal subgroups are shown.
This group has 996 conjugacy classes of subgroups — too many to draw. Only its 509 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D163 |
| commutator, socle, cosocle | C22×C163 | C25 |
| Fitting | #543×C163 | C2 |
| prime radical | C24×C163 | C23 |
| hypercenter | #543 | D163 |
| hypocenter | C163 | #3458 |
| LCS | G ▷ C22×C163 ▷ C163 | |
| UCS | C1 ◁ C22 ◁ #543 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23782294 | C2 | semidirect |
| 3. #23782217 | C2 | semidirect |
| 7. #19104734 | C22 | semidirect |
| 8. #19104733 | C22 | semidirect |
| 9. #19104725 | C22 | semidirect |
| 12. Q8×Dic163 | C22 | semidirect |
| 13. #19104700 | C22 | semidirect |
| 22. C22×Dic163 | Q8 | semidirect |
| 61. C163 | #3458 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23782294 | 2 |
| #23782291 | 2 |
| #23782217 | 2 |
| C2×#19104691 | 2 |
| #543×C163 | 2 |
| #3458 | 163 |