Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
1556 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1556 conjugacy classes of subgroups — too many to draw. Only its 142 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 | C2×D4 | C22×C4×C89 |
| Frattini, cosocle | C22×D4 | C23×C89 |
| prime radical | C2×D42×C89 | C2 |
| LCS | G ▷ C2×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 |
|---|---|---|
| 2. C2×#353×C89 | C2 | semidirect |
| 3. C2×AΣL(1,𝔽16)×C89 | C2 | semidirect |
| 4. C89×#1889 | C2 | semidirect |
| 5. C89×#1887 | C2 | semidirect |
| 9. D42×C89 | C4 | semidirect |
| 14. #518×C89 | C22 | semidirect |
| 17. C2×#100×C89 | C22 | semidirect |
| 18. C2×#99×C89 | C22 | semidirect |
| 74. C24×C89 | C22⋊C4 | semidirect |
| 93. C2×D4×C89 | C22⋊C4 | semidirect |
| 118. C22⋊C4×C89 | C22⋊C4 | semidirect |
| 121. C42×C89 | C22⋊C4 | semidirect |
| 202. #12759 | C89 | direct (N × H) |
| 212. C2×#353 | C2×C89 | semidirect |
| 213. C2×AΣL(1,𝔽16) | C2×C89 | semidirect |
| 214. #1889 | C2×C89 | semidirect |
| 215. #1887 | C2×C89 | semidirect |
| 216. C89 | #12759 | direct (N × H) |
| 220. D42 | C4×C89 | semidirect |
| 225. #518 | C22×C89 | semidirect |
| 228. C2×#100 | C22×C89 | semidirect |
| 229. C2×#99 | C22×C89 | semidirect |
| 285. C24 | C22⋊C4×C89 | semidirect |
| 304. C2×D4 | C22⋊C4×C89 | semidirect |
| 329. C22⋊C4 | C22⋊C4×C89 | semidirect |
| 332. C42 | C22⋊C4×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D42×C89 | 2 |
| C2×#353×C89 | 2 |
| C2×AΣL(1,𝔽16)×C89 | 2 |
| C89×#1889 | 2 |
| C89×#1887 | 2 |
| #12759 | 89 |