Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
780 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 98 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 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C23×C89 |
| prime radical | C2×H(5,𝔽2)×C89 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C89 ◁ C22×D4×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C89×#2717 | C2 | semidirect |
| 4. C2×#354×C89 | C2 | semidirect |
| 5. C2×#352×C89 | C2 | semidirect |
| 7. C89×#1390 | C2 | semidirect |
| 295. #13349 | C89 | direct (N × H) |
| 308. #2717 | C2×C89 | semidirect |
| 309. C2×#354 | C2×C89 | semidirect |
| 310. C2×#352 | C2×C89 | semidirect |
| 312. #1390 | C2×C89 | semidirect |
| 313. C89 | #13349 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#458×C89 | 2 |
| C2×#455×C89 | 2 |
| C89×#2717 | 2 |
| C2×#354×C89 | 2 |
| C2×#352×C89 | 2 |
| C89×#1790 | 2 |
| C89×#1390 | 2 |
| #13349 | 89 |