Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
4122 conjugacy classes of subgroups; only the 883 normal subgroups are shown.
This group has 4122 conjugacy classes of subgroups — too many to draw. Only its 883 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D89 |
| commutator, socle, cosocle | C23×C89 | C25 |
| Fitting | C42×Q8×C89 | C2 |
| hypercenter | C42×Q8 | D89 |
| hypocenter | C89 | #36592 |
| LCS | G ▷ C23×C89 ▷ C89 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19152270 | C2 | semidirect |
| 2. #26028647 | C2 | semidirect |
| 3. #26025194 | C2 | semidirect |
| 4. C4×#15668009 | C2 | semidirect |
| 5. C42×Q8×C89 | C2 | semidirect |
| 32. C4×Q8×C89 | D4 | semidirect |
| 57. C42×C89 | #41 | semidirect |
| 62. Q8×C89 | #127 | semidirect |
| 82. C4×C89 | #550 | semidirect |
| 93. C42×Q8 | D89 | semidirect |
| 94. C89 | #36592 | semidirect |
| 118. C4×Q8 | D356 | semidirect |
| 146. C42 | #13083575 | semidirect |
| 151. Q8 | #15668009 | semidirect |
| 172. C4 | #19152270 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19152270 | 2 |
| #26028647 | 2 |
| #26025194 | 2 |
| C4×#15668009 | 2 |
| C42×Q8×C89 | 2 |
| #36592 | 89 |