Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
302 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 78 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 | C4×C8 | C23×C89 |
| prime radical | C89×#3403 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C42×C89 ◁ #446×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#2243 | C2 | semidirect |
| 3. C89×#2226 | C2 | semidirect |
| 4. C89×#1694 | C2 | semidirect |
| 5. C89×#1360 | C2 | semidirect |
| 6. C89×#1338 | C2 | semidirect |
| 7. C89×#1333 | C2 | semidirect |
| 143. #12390 | C89 | direct (N × H) |
| 150. #2243 | C2×C89 | semidirect |
| 151. #2226 | C2×C89 | semidirect |
| 152. #1694 | C2×C89 | semidirect |
| 153. #1360 | C2×C89 | semidirect |
| 154. #1338 | C2×C89 | semidirect |
| 155. #1333 | C2×C89 | semidirect |
| 156. C89 | #12390 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3403 | 2 |
| C89×#2243 | 2 |
| C89×#2226 | 2 |
| C89×#1694 | 2 |
| C89×#1360 | 2 |
| C89×#1338 | 2 |
| C89×#1333 | 2 |
| #12390 | 89 |