Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
872 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 872 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C89 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23×C89 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C2×C42×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#2679 | C2 | semidirect |
| 2. C89×#2011 | C2 | semidirect |
| 3. C89×#1270 | C2 | semidirect |
| 5. C2×#127×C89 | C22 | semidirect |
| 10. #336×C89 | C22 | semidirect |
| 44. C2×D4×C89 | D8 | semidirect |
| 97. #7734 | C89 | direct (N × H) |
| 103. #2679 | C2×C89 | semidirect |
| 104. #2011 | C2×C89 | semidirect |
| 105. #1270 | C2×C89 | semidirect |
| 106. C89 | #7734 | direct (N × H) |
| 108. C2×#127 | C22×C89 | semidirect |
| 113. #336 | C22×C89 | semidirect |
| 147. C2×D4 | D8×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2679 | 2 |
| C89×#2011 | 2 |
| C89×#1270 | 2 |
| #7734 | 89 |