Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
994 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 254 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 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C89 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C89×#3184 | C2 | semidirect |
| 2. C89×#3151 | C2 | semidirect |
| 3. C89×#1721 | C2 | semidirect |
| 4. C89×#1715 | C2 | semidirect |
| 5. C89×#1712 | C2 | semidirect |
| 6. C89×#1705 | C2 | semidirect |
| 7. C89×#1455 | C2 | semidirect |
| 11. #498×C89 | C22 | semidirect |
| 15. #493×C89 | C22 | semidirect |
| 18. #486×C89 | C22 | semidirect |
| 22. #322×C89 | C22 | semidirect |
| 23. C82×C89 | C22 | semidirect |
| 89. C8×C89 | AGL(1,ℤ/8) | semidirect |
| 100. #19745 | C89 | direct (N × H) |
| 106. #3184 | C2×C89 | semidirect |
| 107. #3151 | C2×C89 | semidirect |
| 108. #1721 | C2×C89 | semidirect |
| 109. #1715 | C2×C89 | semidirect |
| 110. #1712 | C2×C89 | semidirect |
| 111. #1705 | C2×C89 | semidirect |
| 112. #1455 | C2×C89 | semidirect |
| 113. C89 | #19745 | direct (N × H) |
| 117. #498 | C22×C89 | semidirect |
| 121. #493 | C22×C89 | semidirect |
| 124. #486 | C22×C89 | semidirect |
| 128. #322 | C22×C89 | semidirect |
| 129. C82 | C22×C89 | semidirect |
| 195. C8 | AGL(1,ℤ/8)×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#3184 | 2 |
| C89×#3151 | 2 |
| C89×#1721 | 2 |
| C89×#1715 | 2 |
| C89×#1712 | 2 |
| C89×#1705 | 2 |
| C89×#1455 | 2 |
| #19745 | 89 |