Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
1008 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 102 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 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23×C89 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C4×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#3006 | C2 | semidirect |
| 2. C89×#2539 | C2 | semidirect |
| 3. C89×#2404 | C2 | semidirect |
| 4. C89×#1789 | C2 | semidirect |
| 5. C89×#1510 | C2 | semidirect |
| 6. C89×#1462 | C2 | semidirect |
| 7. C89×#1284 | C2 | semidirect |
| 39. C22×Q8×C89 | D4 | semidirect |
| 44. C22×D4×C89 | D4 | semidirect |
| 45. C22×D4×C89 | D4 | semidirect |
| 62. C2×C4⋊C4×C89 | D4 | semidirect |
| 76. C2×C42×C89 | D4 | semidirect |
| 83. C22⋊C8×C89 | D4 | semidirect |
| 87. #95×C89 | D4 | semidirect |
| 284. #9938 | C89 | direct (N × H) |
| 294. #3006 | C2×C89 | semidirect |
| 295. #2539 | C2×C89 | semidirect |
| 296. #2404 | C2×C89 | semidirect |
| 297. #1789 | C2×C89 | semidirect |
| 298. #1510 | C2×C89 | semidirect |
| 299. #1462 | C2×C89 | semidirect |
| 300. #1284 | C2×C89 | semidirect |
| 301. C89 | #9938 | direct (N × H) |
| 333. C22×Q8 | D4×C89 | semidirect |
| 338. C22×D4 | D4×C89 | semidirect |
| 339. C22×D4 | D4×C89 | semidirect |
| 356. C2×C4⋊C4 | D4×C89 | semidirect |
| 370. C2×C42 | D4×C89 | semidirect |
| 377. C22⋊C8 | D4×C89 | semidirect |
| 381. #95 | D4×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3006 | 2 |
| C89×#2539 | 2 |
| C89×#2404 | 2 |
| C89×#1789 | 2 |
| C89×#1510 | 2 |
| C89×#1462 | 2 |
| C89×#1284 | 2 |
| #9938 | 89 |