Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
8094 conjugacy classes of subgroups; only the 1123 normal subgroups are shown.
This group has 8094 conjugacy classes of subgroups — too many to draw. Only its 1123 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 | C24×D89 |
| commutator, socle, cosocle | C23×C89 | C25 |
| Fitting | C89×#2867 | C2 |
| hypercenter | #2867 | D89 |
| hypocenter | C89 | C2×#2867 |
| LCS | G ▷ C23×C89 ▷ C89 | |
| UCS | C1 ◁ C23 ◁ #2867 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#127×D89 | C2 | semidirect |
| 2. #26028647 | C2 | semidirect |
| 3. #26028646 | C2 | semidirect |
| 4. C43×D89 | C2 | semidirect |
| 5. C89×#2867 | C2 | semidirect |
| 7. C2×#15668009 | C22 | semidirect |
| 8. C2×#15668008 | C22 | semidirect |
| 11. C42×Dic89 | C22 | semidirect |
| 12. C2×#127×C89 | C22 | semidirect |
| 13. C43×C89 | C22 | semidirect |
| 22. C42×D89 | D4 | semidirect |
| 34. C4×Dic89 | C2×D4 | semidirect |
| 38. C42×C89 | C2×D4 | semidirect |
| 49. C4×D89 | #127 | semidirect |
| 55. C4×C89 | C2×#127 | semidirect |
| 56. Dic89 | C2×#127 | semidirect |
| 61. D89 | #2867 | direct (N × H) |
| 65. #2867 | D89 | direct (N × H) |
| 67. C89 | C2×#2867 | semidirect |
| 72. C2×#127 | C2×D89 | semidirect |
| 75. C43 | C2×D89 | semidirect |
| 100. C42 | D4×D89 | semidirect |
| 118. C4 | #127×D89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#127×D89 | 2 |
| #26028647 | 2 |
| #26028646 | 2 |
| C43×D89 | 2 |
| C89×#2867 | 2 |
| C2×#2867 | 89 |