Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
3550 conjugacy classes of subgroups; only the 974 normal subgroups are shown.
This group has 3550 conjugacy classes of subgroups — too many to draw. Only its 974 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C89 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C89 |
| prime radical | C26×C89 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#2840 | C2 | direct (N × H) |
| 2. C2×#393×C89 | C2 | semidirect |
| 5. C23×Q8×C89 | C22 | semidirect |
| 7. #393×C89 | C22 | semidirect |
| 9. C2×#95×C89 | C22 | semidirect |
| 11. C25×C89 | Q8 | semidirect |
| 15. C22×Q8×C89 | C23 | semidirect |
| 19. #95×C89 | C23 | semidirect |
| 25. C24×C89 | C2×Q8 | semidirect |
| 55. C2×#2840 | C89 | direct (N × H) |
| 57. C2×C89 | #2840 | direct (N × H) |
| 60. #2840 | C2×C89 | direct (N × H) |
| 61. C2×#393 | C2×C89 | semidirect |
| 63. C89 | C2×#2840 | direct (N × H) |
| 65. C23×Q8 | C22×C89 | semidirect |
| 67. #393 | C22×C89 | semidirect |
| 69. C2×#95 | C22×C89 | semidirect |
| 71. C25 | Q8×C89 | semidirect |
| 75. C22×Q8 | C23×C89 | semidirect |
| 79. #95 | C23×C89 | semidirect |
| 85. C24 | C2×Q8×C89 | semidirect |
| 116. C2 | C89×#2840 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2840 | 2 |
| C2×#393×C89 | 2 |
| C2×#379×C89 | 2 |
| C2×#2840 | 89 |