Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
384 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 78 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 | #1619 |
| commutator, Frattini, cosocle | C4×C8 | C23×C89 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C2×C4×C89 ◁ C4×C8×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#3291 | C2 | semidirect |
| 2. C89×#2213 | C2 | semidirect |
| 3. C89×#1721 | C2 | semidirect |
| 4. C89×#1359 | C2 | semidirect |
| 5. C89×#1336 | C2 | semidirect |
| 9. #493×C89 | C22 | semidirect |
| 16. #364×C89 | C22 | semidirect |
| 18. #322×C89 | C22 | semidirect |
| 59. Q16×C89 | D8 | semidirect |
| 69. D8×C89 | D8 | semidirect |
| 115. #12314 | C89 | direct (N × H) |
| 121. #3291 | C2×C89 | semidirect |
| 122. #2213 | C2×C89 | semidirect |
| 123. #1721 | C2×C89 | semidirect |
| 124. #1359 | C2×C89 | semidirect |
| 125. #1336 | C2×C89 | semidirect |
| 126. C89 | #12314 | direct (N × H) |
| 130. #493 | C22×C89 | semidirect |
| 137. #364 | C22×C89 | semidirect |
| 139. #322 | C22×C89 | semidirect |
| 180. Q16 | D8×C89 | semidirect |
| 190. D8 | D8×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3291 | 2 |
| C89×#2213 | 2 |
| C89×#1721 | 2 |
| C89×#1359 | 2 |
| C89×#1336 | 2 |
| #12314 | 89 |