Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
136 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 58 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 | #357 |
| commutator | C16 | C2×C8×C89 |
| Frattini, cosocle | C4×C16 | C22×C89 |
| prime radical | C2×D16×C89 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C42×C89 ◁ C4×C8×C89 ◁ C4×C16×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#1397 | C2 | semidirect |
| 3. C89×#1372 | C2 | semidirect |
| 14. D16×C89 | C8 | semidirect |
| 56. #7385 | C89 | direct (N × H) |
| 62. #1397 | C2×C89 | semidirect |
| 63. #1372 | C2×C89 | semidirect |
| 64. C89 | #7385 | direct (N × H) |
| 75. D16 | C8×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×D16×C89 | 2 |
| C89×#1397 | 2 |
| C89×#1372 | 2 |
| #7385 | 89 |