Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
1062 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C89 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| prime radical | C89×#2847 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C24×C4×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#2848 | C2 | semidirect |
| 3. C89×#2077 | C2 | semidirect |
| 4. C89×#2035 | C2 | semidirect |
| 5. C89×#2034 | C2 | semidirect |
| 6. C89×#2000 | C2 | semidirect |
| 7. C89×#1779 | C2 | semidirect |
| 9. C22×SD16×C89 | C22 | semidirect |
| 13. C2×#106×C89 | C22 | semidirect |
| 14. C2×#103×C89 | C22 | semidirect |
| 15. C2×#102×C89 | C22 | semidirect |
| 23. #340×C89 | C22 | semidirect |
| 24. #340×C89 | C22 | semidirect |
| 58. C24×C89 | SD16 | semidirect |
| 154. #32708 | C89 | direct (N × H) |
| 160. #2848 | C2×C89 | semidirect |
| 162. #2077 | C2×C89 | semidirect |
| 163. #2035 | C2×C89 | semidirect |
| 164. #2034 | C2×C89 | semidirect |
| 165. #2000 | C2×C89 | semidirect |
| 166. #1779 | C2×C89 | semidirect |
| 167. C89 | #32708 | direct (N × H) |
| 169. C22×SD16 | C22×C89 | semidirect |
| 173. C2×#106 | C22×C89 | semidirect |
| 174. C2×#103 | C22×C89 | semidirect |
| 175. C2×#102 | C22×C89 | semidirect |
| 183. #340 | C22×C89 | semidirect |
| 184. #340 | C22×C89 | semidirect |
| 218. C24 | SD16×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2848 | 2 |
| C89×#2847 | 2 |
| C89×#2077 | 2 |
| C89×#2035 | 2 |
| C89×#2034 | 2 |
| C89×#2000 | 2 |
| C89×#1779 | 2 |
| #32708 | 89 |