Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
654 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 654 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 |
| prime radical | C25×C89 | D4 |
| 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×#3095 | C2 | semidirect |
| 4. C89×#1826 | C2 | semidirect |
| 5. C89×#1511 | C2 | semidirect |
| 7. C89×#1284 | C2 | semidirect |
| 222. #9944 | C89 | direct (N × H) |
| 230. #3095 | C2×C89 | semidirect |
| 233. #1826 | C2×C89 | semidirect |
| 234. #1511 | C2×C89 | semidirect |
| 236. #1284 | C2×C89 | semidirect |
| 237. C89 | #9944 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3095 | 2 |
| C89×#2561 | 2 |
| C89×#2482 | 2 |
| C89×#1826 | 2 |
| C89×#1511 | 2 |
| C89×#1462 | 2 |
| C89×#1284 | 2 |
| #9944 | 89 |