Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
4778 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.
This group has 4778 conjugacy classes of subgroups — too many to draw. Only its 1114 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 | C22×#120 |
| commutator, Frattini, cosocle | C23 | C25×C89 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C22×Q8×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×2−1+6 | C2 | semidirect |
| 2. C89×#3487 | C2 | semidirect |
| 3. C89×#3027 | C2 | semidirect |
| 4. C89×#3025 | C2 | semidirect |
| 5. C89×#2884 | C2 | semidirect |
| 10. C22×#41×C89 | C22 | semidirect |
| 13. #535×C89 | C22 | semidirect |
| 14. #525×C89 | C22 | semidirect |
| 15. #519×C89 | C22 | semidirect |
| 16. #458×C89 | C22 | semidirect |
| 17. B(4,𝔽2)×C89 | C22 | semidirect |
| 18. #410×C89 | C22 | semidirect |
| 28. C2×#41×C89 | D4 | semidirect |
| 31. C22×Q8×C89 | D4 | semidirect |
| 105. #60402 | C89 | direct (N × H) |
| 110. 2−1+6 | C2×C89 | semidirect |
| 111. #3487 | C2×C89 | semidirect |
| 112. #3027 | C2×C89 | semidirect |
| 113. #3025 | C2×C89 | semidirect |
| 114. #2884 | C2×C89 | semidirect |
| 115. C89 | #60402 | direct (N × H) |
| 120. C22×#41 | C22×C89 | semidirect |
| 123. #535 | C22×C89 | semidirect |
| 124. #525 | C22×C89 | semidirect |
| 125. #519 | C22×C89 | semidirect |
| 126. #458 | C22×C89 | semidirect |
| 127. B(4,𝔽2) | C22×C89 | semidirect |
| 128. #410 | C22×C89 | semidirect |
| 138. C2×#41 | D4×C89 | semidirect |
| 141. C22×Q8 | D4×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×2−1+6 | 2 |
| C89×#3487 | 2 |
| C89×#3027 | 2 |
| C89×#3025 | 2 |
| C89×#2884 | 2 |
| #60402 | 89 |