Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
528 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D163 |
| commutator, socle, cosocle | C23×C163 | C24 |
| Fitting | #400×C163 | C2 |
| prime radical | C23×D163 | C23 |
| hypercenter | #400 | D163 |
| hypocenter | C163 | #2396 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ #400 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19104700 | C2 | semidirect |
| 4. #23781599 | C2 | semidirect |
| 6. #23781585 | C2 | semidirect |
| 8. #23781516 | C2 | semidirect |
| 9. #23781516 | C2 | semidirect |
| 11. #400×C163 | C2 | semidirect |
| 47. C2×C4×D163 | Q8 | semidirect |
| 128. C163 | #2396 | semidirect |
| 132. #400 | D163 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19104700 | 2 |
| C2×#19104693 | 2 |
| #23781600 | 2 |
| #23781599 | 2 |
| #23781589 | 2 |
| #23781585 | 2 |
| #23781578 | 2 |
| #23781516 | 2 |
| #23781512 | 2 |
| #400×C163 | 2 |
| #2396 | 163 |