Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
892 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 892 conjugacy classes of subgroups — too many to draw. Only its 98 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 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C23×C89 |
| prime radical | C2×B(4,𝔽2)×C89 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C89 ◁ C22×D4×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#458×C89 | C2 | semidirect |
| 3. C89×#2697 | C2 | semidirect |
| 4. C2×#352×C89 | C2 | semidirect |
| 5. C2×AΣL(1,𝔽16)×C89 | C2 | semidirect |
| 6. C89×#1792 | C2 | semidirect |
| 7. C89×#1393 | C2 | semidirect |
| 47. C23×C4×C89 | D4 | semidirect |
| 64. C2×C4⋊C4×C89 | D4 | semidirect |
| 78. C2×C22⋊C4×C89 | D4 | semidirect |
| 79. C2×C42×C89 | D4 | semidirect |
| 90. #99×C89 | D4 | semidirect |
| 324. #13344 | C89 | direct (N × H) |
| 336. C2×#458 | C2×C89 | semidirect |
| 338. #2697 | C2×C89 | semidirect |
| 339. C2×#352 | C2×C89 | semidirect |
| 340. C2×AΣL(1,𝔽16) | C2×C89 | semidirect |
| 341. #1792 | C2×C89 | semidirect |
| 342. #1393 | C2×C89 | semidirect |
| 343. C89 | #13344 | direct (N × H) |
| 383. C23×C4 | D4×C89 | semidirect |
| 400. C2×C4⋊C4 | D4×C89 | semidirect |
| 414. C2×C22⋊C4 | D4×C89 | semidirect |
| 415. C2×C42 | D4×C89 | semidirect |
| 426. #99 | D4×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#458×C89 | 2 |
| C2×B(4,𝔽2)×C89 | 2 |
| C89×#2697 | 2 |
| C2×#352×C89 | 2 |
| C2×AΣL(1,𝔽16)×C89 | 2 |
| C89×#1792 | 2 |
| C89×#1393 | 2 |
| #13344 | 89 |