Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
506 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 506 conjugacy classes of subgroups — too many to draw. Only its 218 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 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C89 |
| prime radical | C2×D8×C89 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C4×Q8×C89 ◁ C8×Q8×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#3395 | C2 | semidirect |
| 3. C89×#2241 | C2 | semidirect |
| 4. C89×#2225 | C2 | semidirect |
| 6. Q8×C16×C89 | C2 | semidirect |
| 75. Q8×C89 | SD32 | semidirect |
| 99. #33926 | C89 | direct (N × H) |
| 104. #3395 | C2×C89 | semidirect |
| 106. #2241 | C2×C89 | semidirect |
| 107. #2225 | C2×C89 | semidirect |
| 109. Q8×C16 | C2×C89 | semidirect |
| 111. C89 | #33926 | direct (N × H) |
| 179. Q8 | SD32×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3395 | 2 |
| Q8×D8×C89 | 2 |
| C89×#2241 | 2 |
| C89×#2225 | 2 |
| C89×#2208 | 2 |
| Q8×C16×C89 | 2 |
| C4×SD32×C89 | 2 |
| #33926 | 89 |