Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
1186 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1186 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | #1781 |
| commutator | C24 | C22×C4×C89 |
| Frattini, cosocle | C22×D4 | C23×C89 |
| prime radical | C89×#3484 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C23×C89 ◁ C23×D4×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#2121 | C2 | semidirect |
| 3. C89×#1783 | C2 | semidirect |
| 4. C89×#1781 | C2 | semidirect |
| 5. C2×H(5,𝔽2)×C89 | C4 | semidirect |
| 14. C2×#120×C89 | C4 | semidirect |
| 15. #518×C89 | C4 | semidirect |
| 17. C2×#100×C89 | C22 | semidirect |
| 19. C2×#99×C89 | C22 | semidirect |
| 87. C24×C89 | C22⋊C4 | semidirect |
| 130. C22×C4×C89 | C22⋊C4 | semidirect |
| 146. C22⋊C4×C89 | C22⋊C4 | semidirect |
| 256. #12705 | C89 | direct (N × H) |
| 268. #2121 | C2×C89 | semidirect |
| 269. #1783 | C2×C89 | semidirect |
| 270. #1781 | C2×C89 | semidirect |
| 271. C89 | #12705 | direct (N × H) |
| 272. C2×H(5,𝔽2) | C4×C89 | semidirect |
| 281. C2×#120 | C4×C89 | semidirect |
| 282. #518 | C4×C89 | semidirect |
| 284. C2×#100 | C22×C89 | semidirect |
| 286. C2×#99 | C22×C89 | semidirect |
| 354. C24 | C22⋊C4×C89 | semidirect |
| 397. C22×C4 | C22⋊C4×C89 | semidirect |
| 413. C22⋊C4 | C22⋊C4×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3484 | 2 |
| C89×#2121 | 2 |
| C89×#1783 | 2 |
| C89×#1781 | 2 |
| #12705 | 89 |