Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
1184 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 130 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 | #386 |
| commutator | C24 | C22×C4×C89 |
| Frattini, cosocle | C25 | C23×C89 |
| prime radical | C89×#2403 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C25×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#2309 | C2 | semidirect |
| 3. C89×#1781 | C2 | semidirect |
| 4. C89×#1781 | C2 | semidirect |
| 5. C89×#1304 | C2 | semidirect |
| 9. C2×#120×C89 | C4 | semidirect |
| 16. #386×C89 | C4 | semidirect |
| 53. C2×C22⋊C4×C89 | D4 | semidirect |
| 193. C2×C4×C89 | #99 | semidirect |
| 234. #9497 | C89 | direct (N × H) |
| 246. #2309 | C2×C89 | semidirect |
| 247. #1781 | C2×C89 | semidirect |
| 248. #1781 | C2×C89 | semidirect |
| 249. #1304 | C2×C89 | semidirect |
| 250. C89 | #9497 | direct (N × H) |
| 254. C2×#120 | C4×C89 | semidirect |
| 261. #386 | C4×C89 | semidirect |
| 298. C2×C22⋊C4 | D4×C89 | semidirect |
| 438. C2×C4 | #99×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2403 | 2 |
| C89×#2309 | 2 |
| C89×#1781 | 2 |
| C89×#1304 | 2 |
| #9497 | 89 |