Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
382 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8×C89 | #99 |
| commutator | C23 | C22×C8×C89 |
| Frattini, cosocle | C2×M16 | C23×C89 |
| socle | C2×C89 | #1459 |
| prime radical | #525×C89 | C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C89 ◁ C2×C8×C89 ◁ C2×#131×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#2115 | C2 | semidirect |
| 3. C89×#1328 | C2 | semidirect |
| 4. C89×#1320 | C2 | semidirect |
| 126. #11483 | C89 | direct (N × H) |
| 133. #2115 | C2×C89 | semidirect |
| 134. #1328 | C2×C89 | semidirect |
| 135. #1320 | C2×C89 | semidirect |
| 136. C89 | #11483 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2929 | 2 |
| C89×#2115 | 2 |
| C89×#1328 | 2 |
| C89×#1320 | 2 |
| #11483 | 89 |