Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
390 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 90 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 | #393 |
| commutator, Frattini, cosocle | C2×C42 | C23×C89 |
| prime radical | #532×C89 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#2606 | C2 | semidirect |
| 5. C89×#1300 | C2 | semidirect |
| 6. C89×#1288 | C2 | semidirect |
| 7. C89×#1285 | C2 | semidirect |
| 174. #10380 | C89 | direct (N × H) |
| 181. #2606 | C2×C89 | semidirect |
| 185. #1300 | C2×C89 | semidirect |
| 186. #1288 | C2×C89 | semidirect |
| 187. #1285 | C2×C89 | semidirect |
| 188. C89 | #10380 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2606 | 2 |
| C89×#1980 | 2 |
| C89×#1514 | 2 |
| C89×#1505 | 2 |
| C89×#1300 | 2 |
| C89×#1288 | 2 |
| C89×#1285 | 2 |
| #10380 | 89 |