Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
458 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 458 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 | C2×C4×C89 | #120 |
| commutator | C22×C4 | C22×C4×C89 |
| Frattini, cosocle | C23×C4 | C23×C89 |
| socle | C22×C89 | #385 |
| prime radical | C22×D4×C89 | Q8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C23×C4×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C89×#1855 | C2 | semidirect |
| 5. C89×#1298 | C2 | semidirect |
| 117. C2×C4×C89 | #108 | semidirect |
| 148. #8860 | C89 | direct (N × H) |
| 158. #1855 | C2×C89 | semidirect |
| 160. #1298 | C2×C89 | semidirect |
| 161. C89 | #8860 | direct (N × H) |
| 273. C2×C4 | #108×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2975 | 2 |
| C89×#1870 | 2 |
| C89×#1855 | 2 |
| C89×#1459 | 2 |
| C89×#1298 | 2 |
| #8860 | 89 |