Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
474 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 474 conjugacy classes of subgroups — too many to draw. Only its 94 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 | #394 |
| commutator, Frattini, cosocle | C23×C4 | C23×C89 |
| prime radical | C2×#121×C89 | C22 |
| 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 |
|---|---|---|
| 2. C89×#1901 | C2 | semidirect |
| 4. C89×#1851 | C2 | semidirect |
| 5. C89×#1508 | C2 | semidirect |
| 7. C89×#1293 | C2 | semidirect |
| 213. #10558 | C89 | direct (N × H) |
| 223. #1901 | C2×C89 | semidirect |
| 225. #1851 | C2×C89 | semidirect |
| 226. #1508 | C2×C89 | semidirect |
| 228. #1293 | C2×C89 | semidirect |
| 229. C89 | #10558 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3187 | 2 |
| C89×#1901 | 2 |
| C89×#1900 | 2 |
| C89×#1851 | 2 |
| C89×#1508 | 2 |
| C89×#1507 | 2 |
| C89×#1293 | 2 |
| #10558 | 89 |