Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
274 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 274 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 | C2×C89 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C23×C89 |
| prime radical | C89×#3417 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C2×C4×C89 ◁ #131×C89 ◁ #504×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#2264 | C2 | semidirect |
| 3. C2×Q64×C89 | C2 | semidirect |
| 4. C89×#2222 | C2 | semidirect |
| 5. C89×#1420 | C2 | semidirect |
| 6. C89×#1417 | C2 | semidirect |
| 7. C89×#1401 | C2 | semidirect |
| 112. #13672 | C89 | direct (N × H) |
| 118. #2264 | C2×C89 | semidirect |
| 119. C2×Q64 | C2×C89 | semidirect |
| 120. #2222 | C2×C89 | semidirect |
| 121. #1420 | C2×C89 | semidirect |
| 122. #1417 | C2×C89 | semidirect |
| 123. #1401 | C2×C89 | semidirect |
| 124. C89 | #13672 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3417 | 2 |
| C89×#2264 | 2 |
| C2×Q64×C89 | 2 |
| C89×#2222 | 2 |
| C89×#1420 | 2 |
| C89×#1417 | 2 |
| C89×#1401 | 2 |
| #13672 | 89 |