Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
1112 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C89 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C89 |
| socle | C2×C89 | C2×B(4,𝔽2) |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C2×C4×C89 ◁ C2×#41×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#3529 | C2 | semidirect |
| 2. C89×#3018 | C2 | semidirect |
| 3. C89×#2203 | C2 | semidirect |
| 4. C89×#2202 | C2 | semidirect |
| 5. C89×#2132 | C2 | semidirect |
| 6. #585×C89 | C22 | semidirect |
| 8. #563×C89 | C22 | semidirect |
| 10. H(3,𝔽4)×C89 | C22 | semidirect |
| 15. #456×C89 | C22 | semidirect |
| 20. #413×C89 | C22 | semidirect |
| 21. #356×C89 | C22 | semidirect |
| 22. #355×C89 | C22 | semidirect |
| 163. #33564 | C89 | direct (N × H) |
| 169. #3529 | C2×C89 | semidirect |
| 170. #3018 | C2×C89 | semidirect |
| 171. #2203 | C2×C89 | semidirect |
| 172. #2202 | C2×C89 | semidirect |
| 173. #2132 | C2×C89 | semidirect |
| 174. C89 | #33564 | direct (N × H) |
| 175. #585 | C22×C89 | semidirect |
| 177. #563 | C22×C89 | semidirect |
| 179. H(3,𝔽4) | C22×C89 | semidirect |
| 184. #456 | C22×C89 | semidirect |
| 189. #413 | C22×C89 | semidirect |
| 190. #356 | C22×C89 | semidirect |
| 191. #355 | C22×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3529 | 2 |
| C89×#3018 | 2 |
| C89×#2203 | 2 |
| C89×#2202 | 2 |
| C89×#2132 | 2 |
| #33564 | 89 |