Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
8094 conjugacy classes of subgroups; only the 1243 normal subgroups are shown.
This group has 8094 conjugacy classes of subgroups — too many to draw. Only its 1243 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C23×D89 |
| commutator, cosocle | C23×C89 | C25 |
| Frattini | C23 | C24×D89 |
| Fitting | C22×#127×C89 | C2 |
| socle | C24×C89 | C24 |
| hypercenter | C22×#127 | D89 |
| hypocenter | C89 | C2×#2867 |
| LCS | G ▷ C23×C89 ▷ C89 | |
| UCS | C1 ◁ C24 ◁ C22×#127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#15668009 | C2 | semidirect |
| 2. C22×#15668008 | C2 | semidirect |
| 3. #26028646 | C2 | direct (N × H) |
| 4. C2×C42×Dic89 | C2 | semidirect |
| 5. C22×#127×C89 | C2 | semidirect |
| 9. C2×#15668009 | C22 | semidirect |
| 11. C2×#15668008 | C22 | semidirect |
| 13. C42×Dic89 | C22 | semidirect |
| 16. C2×#127×C89 | C22 | semidirect |
| 27. C2×C4×Dic89 | D4 | semidirect |
| 35. C2×C42×C89 | D4 | semidirect |
| 47. C4×Dic89 | C2×D4 | semidirect |
| 58. C42×C89 | C2×D4 | semidirect |
| 71. C2×C4×C89 | #127 | semidirect |
| 75. C2×Dic89 | #127 | semidirect |
| 88. C4×C89 | C2×#127 | semidirect |
| 89. Dic89 | C2×#127 | semidirect |
| 92. C2×C89 | #2867 | semidirect |
| 98. C22×#127 | D89 | semidirect |
| 101. C89 | C2×#2867 | semidirect |
| 110. C2×#127 | C2×D89 | semidirect |
| 131. C2×C42 | C22⋊D89 | semidirect |
| 154. C42 | C2×C22⋊D89 | semidirect |
| 171. C2×C4 | #15668008 | semidirect |
| 185. C4 | C2×#15668008 | semidirect |
| 188. C2 | #26028646 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#15668009 | 2 |
| C22×#15668008 | 2 |
| #26028646 | 2 |
| C2×C42×Dic89 | 2 |
| C22×#127×C89 | 2 |
| C2×#2867 | 89 |