Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
2508 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 2508 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C2×#99 |
| commutator | C24 | C22×C4×C89 |
| Frattini, cosocle | C22×D4 | C23×C89 |
| prime radical | C22×#120×C89 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C24×C89 ◁ C23×D4×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×AΣL(1,𝔽16)×C89 | C2 | semidirect |
| 3. C89×#1783 | C2 | semidirect |
| 4. C89×#1781 | C2 | semidirect |
| 5. C26×C89 | C4 | semidirect |
| 11. C2×#120×C89 | C4 | semidirect |
| 14. C22×C22⋊C4×C89 | C4 | semidirect |
| 16. C2×#100×C89 | C22 | semidirect |
| 18. C2×#99×C89 | C22 | semidirect |
| 95. C24×C89 | C22⋊C4 | semidirect |
| 135. C22⋊C4×C89 | C22⋊C4 | semidirect |
| 243. C22×C89 | AΣL(1,𝔽16) | semidirect |
| 251. #12687 | C89 | direct (N × H) |
| 263. C2×AΣL(1,𝔽16) | C2×C89 | semidirect |
| 264. #1783 | C2×C89 | semidirect |
| 265. #1781 | C2×C89 | semidirect |
| 266. C89 | #12687 | direct (N × H) |
| 267. C26 | C4×C89 | semidirect |
| 273. C2×#120 | C4×C89 | semidirect |
| 276. C22×C22⋊C4 | C4×C89 | semidirect |
| 278. C2×#100 | C22×C89 | semidirect |
| 280. C2×#99 | C22×C89 | semidirect |
| 357. C24 | C22⋊C4×C89 | semidirect |
| 397. C22⋊C4 | C22⋊C4×C89 | semidirect |
| 505. C22 | AΣL(1,𝔽16)×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#120×C89 | 2 |
| C2×AΣL(1,𝔽16)×C89 | 2 |
| C89×#1783 | 2 |
| C89×#1781 | 2 |
| #12687 | 89 |