Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
184 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 86 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 | C2×D16 |
| commutator | C16 | C22×C4×C89 |
| Frattini, cosocle | C2×C16 | C23×C89 |
| socle | C2×C89 | C4×D16 |
| prime radical | C89×#2262 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C2×C8×C89 ◁ C4×C8×C89 ◁ C4×C16×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#2178 | C2 | semidirect |
| 3. C89×#1425 | C2 | semidirect |
| 4. C89×#1417 | C2 | semidirect |
| 5. C89×#1397 | C2 | semidirect |
| 8. Q64×C89 | C4 | semidirect |
| 9. SD64×C89 | C4 | semidirect |
| 10. D32×C89 | C4 | semidirect |
| 58. #13656 | C89 | direct (N × H) |
| 63. #2178 | C2×C89 | semidirect |
| 64. #1425 | C2×C89 | semidirect |
| 65. #1417 | C2×C89 | semidirect |
| 66. #1397 | C2×C89 | semidirect |
| 67. C89 | #13656 | direct (N × H) |
| 70. Q64 | C4×C89 | semidirect |
| 71. SD64 | C4×C89 | semidirect |
| 72. D32 | C4×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2262 | 2 |
| C89×#2178 | 2 |
| C89×#1425 | 2 |
| C89×#1417 | 2 |
| C89×#1397 | 2 |
| #13656 | 89 |