Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
32890 conjugacy classes of subgroups; only the 1579 normal subgroups are shown.
This group has 32890 conjugacy classes of subgroups — too many to draw. Only its 1579 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 | C27×C89 | C2 |
| socle | C24×C89 | C24 |
| hypercenter | C27 | D89 |
| hypocenter | C89 | C2×#2846 |
| LCS | G ▷ C23×C89 ▷ C89 | |
| UCS | C1 ◁ C24 ◁ C27 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#15667981 | C2 | semidirect |
| 2. #26028529 | C2 | direct (N × H) |
| 3. C2×#19151454 | C2 | semidirect |
| 4. C27×C89 | C2 | semidirect |
| 5. C23×C22⋊D89 | C22 | semidirect |
| 7. C2×#15667981 | C22 | semidirect |
| 8. C22×C22⋊Dic89 | C22 | semidirect |
| 9. #19151454 | C22 | semidirect |
| 12. C26×C89 | C22 | semidirect |
| 13. C24×D89 | C23 | semidirect |
| 15. C22×C22⋊D89 | C23 | semidirect |
| 16. C23×Dic89 | C23 | semidirect |
| 19. C2×C22⋊Dic89 | C23 | semidirect |
| 23. C25×C89 | D4 | semidirect |
| 27. C23×D89 | C24 | semidirect |
| 31. C22×Dic89 | C24 | semidirect |
| 40. C24×C89 | C2×D4 | semidirect |
| 49. C23×C89 | #120 | semidirect |
| 62. C22×C89 | C2×#120 | semidirect |
| 69. C2×C89 | #2846 | semidirect |
| 72. C27 | D89 | semidirect |
| 76. C89 | C2×#2846 | semidirect |
| 79. C26 | C2×D89 | semidirect |
| 89. C25 | C22⋊D89 | semidirect |
| 107. C24 | C2×C22⋊D89 | semidirect |
| 123. C23 | #15667981 | semidirect |
| 136. C22 | C2×#15667981 | semidirect |
| 141. C2 | #26028529 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#15667981 | 2 |
| #26028529 | 2 |
| C2×#19151454 | 2 |
| C27×C89 | 2 |
| C2×#2846 | 89 |