Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
458 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 74 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 | C2×C4⋊C4 | C23×C89 |
| prime radical | C89×#1598 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C89 ◁ C2×C4⋊C4×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#2609 | C2 | semidirect |
| 2. C89×#2040 | C2 | semidirect |
| 3. C89×#1610 | C2 | semidirect |
| 5. C89×#1346 | C2 | semidirect |
| 6. C89×#1345 | C2 | semidirect |
| 7. C89×#1324 | C2 | semidirect |
| 114. C42×C89 | D8 | semidirect |
| 189. #12154 | C89 | direct (N × H) |
| 197. #2609 | C2×C89 | semidirect |
| 198. #2040 | C2×C89 | semidirect |
| 199. #1610 | C2×C89 | semidirect |
| 201. #1346 | C2×C89 | semidirect |
| 202. #1345 | C2×C89 | semidirect |
| 203. #1324 | C2×C89 | semidirect |
| 204. C89 | #12154 | direct (N × H) |
| 311. C42 | D8×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2609 | 2 |
| C89×#2040 | 2 |
| C89×#1610 | 2 |
| C89×#1598 | 2 |
| C89×#1346 | 2 |
| C89×#1345 | 2 |
| C89×#1324 | 2 |
| #12154 | 89 |