Finite Group DatabasebetaFinite Group DatabasebetaOrder 20864.
952 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 952 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 | C2×#121×C163 | C2 |
| hypercenter | C2×#121 | D163 |
| hypocenter | C163 | #2431 |
| LCS | G ▷ C23×C163 ▷ C163 | |
| UCS | C1 ◁ C23 ◁ C2×#121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#14579699 | C2 | semidirect |
| 2. C2×#19104731 | C2 | semidirect |
| 3. C2×#19104722 | C2 | semidirect |
| 4. #23781573 | C2 | semidirect |
| 5. #23781570 | C2 | semidirect |
| 6. #23781566 | C2 | semidirect |
| 7. #23781541 | C2 | semidirect |
| 8. C4×C22⋊Dic163 | C2 | semidirect |
| 9. C2×#121×C163 | C2 | semidirect |
| 37. C2×Dic326 | D4 | semidirect |
| 40. C2×D652 | D4 | semidirect |
| 67. C22×C4×C163 | D4 | semidirect |
| 90. C2×Dic163 | #41 | semidirect |
| 148. C163 | #2431 | semidirect |
| 153. C2×#121 | D163 | semidirect |
| 211. C22×C4 | C22⋊D163 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#14579699 | 2 |
| C2×#19104731 | 2 |
| C2×#19104722 | 2 |
| #23781573 | 2 |
| #23781570 | 2 |
| #23781566 | 2 |
| #23781541 | 2 |
| C4×C22⋊Dic163 | 2 |
| C2×#121×C163 | 2 |
| #2431 | 163 |