Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
734 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 734 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23×C89 |
| socle | C22×C89 | #392 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C2×C42×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#3100 | C2 | semidirect |
| 2. C89×#2397 | C2 | semidirect |
| 3. C89×#1963 | C2 | semidirect |
| 4. C89×#1486 | C2 | semidirect |
| 5. C89×#1274 | C2 | semidirect |
| 27. C4⋊Q8×C89 | D4 | semidirect |
| 28. #127×C89 | D4 | semidirect |
| 51. C4×C8×C89 | D4 | semidirect |
| 87. C42×C89 | D8 | semidirect |
| 144. #10012 | C89 | direct (N × H) |
| 150. #3100 | C2×C89 | semidirect |
| 151. #2397 | C2×C89 | semidirect |
| 152. #1963 | C2×C89 | semidirect |
| 153. #1486 | C2×C89 | semidirect |
| 154. #1274 | C2×C89 | semidirect |
| 155. C89 | #10012 | direct (N × H) |
| 177. C4⋊Q8 | D4×C89 | semidirect |
| 178. #127 | D4×C89 | semidirect |
| 201. C4×C8 | D4×C89 | semidirect |
| 237. C42 | D8×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3100 | 2 |
| C89×#2397 | 2 |
| C89×#1963 | 2 |
| C89×#1486 | 2 |
| C89×#1274 | 2 |
| #10012 | 89 |