Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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 | #398×C163 | C2 |
| prime radical | #23781575 | C2 |
| hypercenter | #398 | D163 |
| hypocenter | C163 | #2392 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ #398 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19104697 | C2 | semidirect |
| 2. C2×#19104697 | C2 | semidirect |
| 3. C2×#19104693 | C2 | semidirect |
| 4. #23781591 | C2 | semidirect |
| 5. #23781587 | C2 | semidirect |
| 6. #23781583 | C2 | semidirect |
| 7. #23781582 | C2 | semidirect |
| 9. #23781516 | C2 | semidirect |
| 10. #23781512 | C2 | semidirect |
| 11. #398×C163 | C2 | semidirect |
| 54. C2×C4×D163 | D4 | semidirect |
| 66. C4×Dic163 | D4 | semidirect |
| 148. C163 | #2392 | semidirect |
| 153. #398 | D163 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19104697 | 2 |
| C2×#19104693 | 2 |
| #23781591 | 2 |
| #23781587 | 2 |
| #23781583 | 2 |
| #23781582 | 2 |
| #23781575 | 2 |
| #23781516 | 2 |
| #23781512 | 2 |
| #398×C163 | 2 |
| #2392 | 163 |