Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
6640 conjugacy classes of subgroups; only the 5670 normal subgroups are shown.
This group has 6640 conjugacy classes of subgroups — too many to draw. Only its 5670 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42×C89 | C24 |
| commutator, Frattini, cosocle | C22 | C26×C89 |
| socle | C22×C89 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#3531 | C2 | semidirect |
| 2. C89×#3529 | C2 | semidirect |
| 3. C89×#3444 | C2 | semidirect |
| 4. #564×C89 | C22 | semidirect |
| 5. #563×C89 | C22 | semidirect |
| 6. #562×C89 | C22 | semidirect |
| 7. H(3,𝔽4)×C89 | C22 | semidirect |
| 8. #560×C89 | C22 | semidirect |
| 9. #533×C89 | C22 | semidirect |
| 11. #529×C89 | C22 | semidirect |
| 12. #525×C89 | C22 | semidirect |
| 13. C4×#41×C89 | C22 | semidirect |
| 14. C22×C42×C89 | C22 | semidirect |
| 57. #62694 | C89 | direct (N × H) |
| 60. #3531 | C2×C89 | semidirect |
| 61. #3529 | C2×C89 | semidirect |
| 62. #3444 | C2×C89 | semidirect |
| 63. C89 | #62694 | direct (N × H) |
| 64. #564 | C22×C89 | semidirect |
| 65. #563 | C22×C89 | semidirect |
| 66. #562 | C22×C89 | semidirect |
| 67. H(3,𝔽4) | C22×C89 | semidirect |
| 68. #560 | C22×C89 | semidirect |
| 69. #533 | C22×C89 | semidirect |
| 71. #529 | C22×C89 | semidirect |
| 72. #525 | C22×C89 | semidirect |
| 73. C4×#41 | C22×C89 | semidirect |
| 74. C22×C42 | C22×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3531 | 2 |
| C89×#3529 | 2 |
| C89×#3444 | 2 |
| #62694 | 89 |