Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
374 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C89 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23×C89 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C2×C42×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 35. #97×C89 | Q8 | semidirect |
| 36. #97×C89 | Q8 | semidirect |
| 108. #10584 | C89 | direct (N × H) |
| 119. C89 | #10584 | direct (N × H) |
| 149. #97 | Q8×C89 | semidirect |
| 150. #97 | Q8×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2844 | 2 |
| C89×#1847 | 2 |
| C89×#1836 | 2 |
| C89×#1295 | 2 |
| #10584 | 89 |