Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
522 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 522 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C23×C89 |
| prime radical | C89×#2006 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C89 ◁ C2×C4⋊C4×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#2683 | C2 | semidirect |
| 3. C89×#1604 | C2 | semidirect |
| 4. C89×#1354 | C2 | semidirect |
| 5. C89×#1318 | C2 | semidirect |
| 61. C2×D4×C89 | SD16 | semidirect |
| 147. #12126 | C89 | direct (N × H) |
| 155. #2683 | C2×C89 | semidirect |
| 157. #1604 | C2×C89 | semidirect |
| 158. #1354 | C2×C89 | semidirect |
| 159. #1318 | C2×C89 | semidirect |
| 160. C89 | #12126 | direct (N × H) |
| 216. C2×D4 | SD16×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2683 | 2 |
| C89×#2006 | 2 |
| C89×#1604 | 2 |
| C89×#1354 | 2 |
| C89×#1318 | 2 |
| #12126 | 89 |