Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
160 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 70 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 | #2226 |
| commutator, Frattini, cosocle | C2×C16 | C23×C89 |
| prime radical | #508×C89 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C2×C4×C89 ◁ C4×C8×C89 ◁ C4×C16×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#2254 | C2 | semidirect |
| 3. C89×#1426 | C2 | semidirect |
| 4. C89×#1422 | C2 | semidirect |
| 18. Q32×C89 | Q8 | semidirect |
| 20. D16×C89 | Q8 | semidirect |
| 65. #13710 | C89 | direct (N × H) |
| 69. #2254 | C2×C89 | semidirect |
| 71. #1426 | C2×C89 | semidirect |
| 72. #1422 | C2×C89 | semidirect |
| 74. C89 | #13710 | direct (N × H) |
| 87. Q32 | Q8×C89 | semidirect |
| 89. D16 | Q8×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2254 | 2 |
| C89×#2178 | 2 |
| C89×#1426 | 2 |
| C89×#1422 | 2 |
| C89×#1418 | 2 |
| #13710 | 89 |