Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
558 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 558 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, socle | C22×C89 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C89 |
| prime radical | C89×#3150 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C22×C4×C89 ◁ C2×#97×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#510×C89 | C2 | semidirect |
| 3. C2×#360×C89 | C2 | semidirect |
| 4. C89×#1971 | C2 | semidirect |
| 5. C89×#1377 | C2 | semidirect |
| 35. C2×Q16×C89 | D4 | semidirect |
| 52. M32×C89 | D4 | semidirect |
| 148. #13342 | C89 | direct (N × H) |
| 154. C2×#510 | C2×C89 | semidirect |
| 156. C2×#360 | C2×C89 | semidirect |
| 157. #1971 | C2×C89 | semidirect |
| 158. #1377 | C2×C89 | semidirect |
| 159. C89 | #13342 | direct (N × H) |
| 189. C2×Q16 | D4×C89 | semidirect |
| 206. M32 | D4×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#510×C89 | 2 |
| C89×#3150 | 2 |
| C2×#360×C89 | 2 |
| C89×#1971 | 2 |
| C89×#1377 | 2 |
| #13342 | 89 |