Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
3666 conjugacy classes of subgroups; only the 667 normal subgroups are shown.
This group has 3666 conjugacy classes of subgroups — too many to draw. Only its 667 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 | C2×#383×C89 | C2 |
| socle | C24×C89 | C24 |
| hypercenter | C2×#383 | D89 |
| hypocenter | C89 | C2×#2818 |
| LCS | G ▷ C23×C89 ▷ C89 | |
| UCS | C1 ◁ C24 ◁ C2×#383 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#15668009 | C2 | semidirect |
| 2. C22×#15668006 | C2 | semidirect |
| 3. #26028295 | C2 | direct (N × H) |
| 4. C2×#19151516 | C2 | semidirect |
| 5. C2×#19151508 | C2 | semidirect |
| 6. C2×#383×C89 | C2 | semidirect |
| 10. C2×#15668009 | C22 | semidirect |
| 12. C2×#15668006 | C22 | semidirect |
| 16. #19151516 | C22 | semidirect |
| 17. #19151508 | C22 | semidirect |
| 19. #383×C89 | C22 | semidirect |
| 39. C2×C42×C89 | D4 | semidirect |
| 66. C42×C89 | C2×D4 | semidirect |
| 105. C2×C89 | #2818 | semidirect |
| 115. C2×#383 | D89 | semidirect |
| 116. C89 | C2×#2818 | semidirect |
| 129. #383 | C2×D89 | semidirect |
| 149. C2×C42 | D356 | semidirect |
| 176. C42 | C2×D356 | semidirect |
| 216. C2 | #26028295 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#15668009 | 2 |
| C22×#15668006 | 2 |
| #26028295 | 2 |
| C2×#19151516 | 2 |
| C2×#19151508 | 2 |
| C2×#383×C89 | 2 |
| C2×#2818 | 89 |