Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
800 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 181 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 | #395×C163 | C2 |
| prime radical | C23×D163 | C23 |
| hypercenter | #395 | D163 |
| hypocenter | C163 | #2434 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ #395 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#19104734 | C2 | semidirect |
| 4. #23781586 | C2 | semidirect |
| 5. #23781585 | C2 | semidirect |
| 6. #23781584 | C2 | semidirect |
| 7. #23781569 | C2 | semidirect |
| 11. #395×C163 | C2 | semidirect |
| 65. (C2×D163)⋊C4 | Q8 | semidirect |
| 96. C2×Dic163 | C2×Q8 | semidirect |
| 113. C2×C4×C163 | C2×Q8 | semidirect |
| 123. C2×D163 | C4⋊Q8 | semidirect |
| 140. C163 | #2434 | semidirect |
| 148. #395 | D163 | semidirect |
| 254. C2×C4 | Q8×D163 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×D163 | 2 |
| C2×#19104734 | 2 |
| C2×#19104713 | 2 |
| #23781586 | 2 |
| #23781585 | 2 |
| #23781584 | 2 |
| #23781569 | 2 |
| #23781539 | 2 |
| #23781536 | 2 |
| #23781512 | 2 |
| #395×C163 | 2 |
| #2434 | 163 |