Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
564 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 169 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 | #381×C163 | C2 |
| prime radical | C24×C163 | C23 |
| hypercenter | #381 | D163 |
| hypocenter | C163 | #2855 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ #381 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #23781569 | C2 | semidirect |
| 4. #23781562 | C2 | semidirect |
| 5. #23781552 | C2 | semidirect |
| 6. #23781548 | C2 | semidirect |
| 62. C22×C4×C163 | Q8 | semidirect |
| 129. C163 | #2855 | semidirect |
| 182. C22×C4 | Dic326 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19104712 | 2 |
| C22×C4×Dic163 | 2 |
| #23781569 | 2 |
| #23781562 | 2 |
| #23781552 | 2 |
| #23781548 | 2 |
| #23781540 | 2 |
| #23781537 | 2 |
| #23781528 | 2 |
| #23781523 | 2 |
| #381×C163 | 2 |
| #2855 | 163 |