Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
9486 conjugacy classes of subgroups; only the 6366 normal subgroups are shown.
This group has 9486 conjugacy classes of subgroups — too many to draw. Only its 6366 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C25×C89 | C23 |
| commutator, Frattini, cosocle | C22 | C26×C89 |
| prime radical | C26×C89 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#126×C89 | C2 | direct (N × H) |
| 2. C23×C4⋊C4×C89 | C2 | semidirect |
| 4. C23×C42×C89 | C2 | semidirect |
| 8. C2×#126×C89 | C22 | direct (N × H) |
| 9. C22×C4⋊C4×C89 | C22 | semidirect |
| 11. C22×C42×C89 | C22 | semidirect |
| 19. #126×C89 | C23 | direct (N × H) |
| 20. C2×C4⋊C4×C89 | C23 | semidirect |
| 22. C2×C42×C89 | C23 | semidirect |
| 32. C4⋊C4×C89 | C24 | semidirect |
| 34. C42×C89 | C24 | semidirect |
| 37. C23×C89 | #126 | direct (N × H) |
| 47. C22×C89 | C2×#126 | direct (N × H) |
| 52. C23×#126 | C89 | direct (N × H) |
| 54. C2×C89 | C22×#126 | direct (N × H) |
| 56. C22×#126 | C2×C89 | direct (N × H) |
| 57. C23×C4⋊C4 | C2×C89 | semidirect |
| 59. C23×C42 | C2×C89 | semidirect |
| 60. C89 | C23×#126 | direct (N × H) |
| 64. C2×#126 | C22×C89 | direct (N × H) |
| 65. C22×C4⋊C4 | C22×C89 | semidirect |
| 67. C22×C42 | C22×C89 | semidirect |
| 75. #126 | C23×C89 | direct (N × H) |
| 76. C2×C4⋊C4 | C23×C89 | semidirect |
| 78. C2×C42 | C23×C89 | semidirect |
| 88. C4⋊C4 | C24×C89 | semidirect |
| 90. C42 | C24×C89 | semidirect |
| 93. C23 | #126×C89 | direct (N × H) |
| 103. C22 | C2×#126×C89 | direct (N × H) |
| 109. C2 | C22×#126×C89 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#126×C89 | 2 |
| C23×C4⋊C4×C89 | 2 |
| C23×C22⋊C4×C89 | 2 |
| C23×C42×C89 | 2 |
| C23×#126 | 89 |