Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
356 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | #1316 |
| commutator | C22×C4 | C2×C8×C89 |
| Frattini, cosocle | #381 | C22×C89 |
| prime radical | C22×#41×C89 | C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C23×C89 ◁ C24×C89 ◁ #381×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#1317 | C2 | semidirect |
| 11. C22×Q8×C89 | C8 | semidirect |
| 13. C22×D4×C89 | C8 | semidirect |
| 97. #7339 | C89 | direct (N × H) |
| 105. #1317 | C2×C89 | semidirect |
| 106. C89 | #7339 | direct (N × H) |
| 115. C22×Q8 | C8×C89 | semidirect |
| 117. C22×D4 | C8×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#1782 | 2 |
| C89×#1317 | 2 |
| #7339 | 89 |