Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
840 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 177 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D163 |
| commutator, socle, cosocle | C23×C163 | C24 |
| Frattini | C23 | C23×D163 |
| Fitting | C2×C4×D4×C163 | C2 |
| hypercenter | C2×C4×D4 | D163 |
| hypocenter | C163 | #2397 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C4×D652 | C2 | semidirect |
| 2. C2×C4×C22⋊D163 | C2 | semidirect |
| 3. #23781601 | C2 | semidirect |
| 4. #23781600 | C2 | semidirect |
| 5. #23781574 | C2 | semidirect |
| 6. #23781573 | C2 | semidirect |
| 7. #23781511 | C2 | semidirect |
| 8. C2×C4×D4×C163 | C2 | semidirect |
| 43. C4×Dic163 | D4 | semidirect |
| 55. C42×C163 | D4 | semidirect |
| 111. C163 | #2397 | semidirect |
| 115. C2×C4×D4 | D163 | semidirect |
| 167. C42 | C22⋊D163 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D652 | 2 |
| C2×C4×C22⋊D163 | 2 |
| #23781601 | 2 |
| #23781600 | 2 |
| #23781574 | 2 |
| #23781573 | 2 |
| #23781511 | 2 |
| C2×C4×D4×C163 | 2 |
| #2397 | 163 |