Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #19151433 |
| commutator, cosocle | C32×C89 | C23 |
| Frattini | C32 | C22×D89 |
| Fitting | C2×D32×C89 | C2 |
| socle | C22×C89 | D32 |
| hypercenter | C2×D32 | D89 |
| hypocenter | C89 | C2×D64 |
| LCS | G ▷ C32×C89 ▷ C16×C89 ▷ C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ C2×D32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D2848 | C2 | semidirect |
| 2. #26018464 | C2 | direct (N × H) |
| 3. C2×#19151317 | C2 | semidirect |
| 4. C2×D32×C89 | C2 | semidirect |
| 7. D2848 | C22 | semidirect |
| 8. #19151317 | C22 | semidirect |
| 11. D32×C89 | C22 | semidirect |
| 49. C2×C89 | D64 | semidirect |
| 53. C2×D32 | D89 | semidirect |
| 56. C89 | C2×D64 | semidirect |
| 62. D32 | C2×D89 | semidirect |
| 101. C2 | #26018464 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D2848 | 2 |
| #26018464 | 2 |
| C2×#19151317 | 2 |
| C2×D32×C89 | 2 |
| C2×D64 | 89 |