Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
1866 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 1866 conjugacy classes of subgroups — too many to draw. Only its 118 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 | C2×#99 |
| commutator | C24 | C22×C4×C89 |
| Frattini, cosocle | C2×C22⋊C4 | C23×C89 |
| prime radical | C26×C89 | C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C24×C89 ◁ C22×C22⋊C4×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#1462 | C2 | semidirect |
| 3. C89×#1316 | C2 | semidirect |
| 13. #323×C89 | C22 | semidirect |
| 63. C24×C89 | M16 | semidirect |
| 165. #11499 | C89 | direct (N × H) |
| 175. #1462 | C2×C89 | semidirect |
| 176. #1316 | C2×C89 | semidirect |
| 177. C89 | #11499 | direct (N × H) |
| 187. #323 | C22×C89 | semidirect |
| 237. C24 | M16×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#379×C89 | 2 |
| C89×#1462 | 2 |
| C89×#1316 | 2 |
| #11499 | 89 |