Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
4886 conjugacy classes of subgroups; only the 907 normal subgroups are shown.
This group has 4886 conjugacy classes of subgroups — too many to draw. Only its 907 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C23×D89 |
| commutator, cosocle | C23×C89 | C25 |
| Frattini | C23 | C24×D89 |
| Fitting | C22×#95×C89 | C2 |
| socle | C24×C89 | C24 |
| hypercenter | C22×#95 | D89 |
| hypocenter | C89 | C2×#2656 |
| LCS | G ▷ C23×C89 ▷ C89 | |
| UCS | C1 ◁ C24 ◁ C22×#95 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#15667986 | C2 | semidirect |
| 2. #26026354 | C2 | direct (N × H) |
| 3. C2×#19151535 | C2 | semidirect |
| 4. C2×#19151508 | C2 | semidirect |
| 5. C22×#95×C89 | C2 | semidirect |
| 9. C2×#15667986 | C22 | semidirect |
| 12. #19151535 | C22 | semidirect |
| 13. #19151508 | C22 | semidirect |
| 16. C2×#95×C89 | C22 | semidirect |
| 31. C23×C4×C89 | D4 | semidirect |
| 57. C22×C4×C89 | C2×D4 | semidirect |
| 101. C2×C89 | #2656 | semidirect |
| 110. C22×#95 | D89 | semidirect |
| 111. C89 | C2×#2656 | semidirect |
| 122. C2×#95 | C2×D89 | semidirect |
| 132. C23×C4 | D356 | semidirect |
| 161. C22×C4 | C2×D356 | semidirect |
| 207. C2 | #26026354 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#15667986 | 2 |
| #26026354 | 2 |
| C2×#19151535 | 2 |
| C2×#19151508 | 2 |
| C22×#95×C89 | 2 |
| C2×#2656 | 89 |