Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
726 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 726 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 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23×C89 |
| prime radical | C89×#3186 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C4×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#2854 | C2 | semidirect |
| 3. C89×#1817 | C2 | semidirect |
| 4. C89×#1519 | C2 | semidirect |
| 5. C89×#1289 | C2 | semidirect |
| 73. C22×C4×C89 | D8 | semidirect |
| 163. #10321 | C89 | direct (N × H) |
| 173. #2854 | C2×C89 | semidirect |
| 174. #1817 | C2×C89 | semidirect |
| 175. #1519 | C2×C89 | semidirect |
| 176. #1289 | C2×C89 | semidirect |
| 177. C89 | #10321 | direct (N × H) |
| 245. C22×C4 | D8×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3186 | 2 |
| C89×#2854 | 2 |
| C89×#1817 | 2 |
| C89×#1519 | 2 |
| C89×#1289 | 2 |
| #10321 | 89 |