Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
3798 conjugacy classes of subgroups; only the 1486 normal subgroups are shown.
This group has 3798 conjugacy classes of subgroups — too many to draw. Only its 1486 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C89 | C24 |
| commutator | C22 | C24×C4×C89 |
| Frattini, cosocle | C23 | C25×C89 |
| prime radical | C22×#127×C89 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C89×#2380 | C2 | direct (N × H) |
| 3. C2×#386×C89 | C2 | semidirect |
| 4. C2×#383×C89 | C2 | semidirect |
| 7. C2×#127×C89 | C4 | semidirect |
| 11. #386×C89 | C22 | semidirect |
| 12. #383×C89 | C22 | semidirect |
| 17. C22×D4×C89 | C2×C4 | semidirect |
| 23. #127×C89 | C2×C4 | semidirect |
| 26. C2×C42×C89 | C2×C4 | semidirect |
| 37. C2×D4×C89 | C22×C4 | semidirect |
| 46. C42×C89 | C22×C4 | semidirect |
| 73. C2×#2380 | C89 | direct (N × H) |
| 75. C2×C89 | #2380 | direct (N × H) |
| 79. #2380 | C2×C89 | direct (N × H) |
| 80. C2×#386 | C2×C89 | semidirect |
| 81. C2×#383 | C2×C89 | semidirect |
| 82. C89 | C2×#2380 | direct (N × H) |
| 85. C2×#127 | C4×C89 | semidirect |
| 89. #386 | C22×C89 | semidirect |
| 90. #383 | C22×C89 | semidirect |
| 95. C22×D4 | C2×C4×C89 | semidirect |
| 101. #127 | C2×C4×C89 | semidirect |
| 104. C2×C42 | C2×C4×C89 | semidirect |
| 115. C2×D4 | C22×C4×C89 | semidirect |
| 124. C42 | C22×C4×C89 | semidirect |
| 152. C2 | C89×#2380 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#127×C89 | 2 |
| C89×#2380 | 2 |
| C2×#386×C89 | 2 |
| C2×#383×C89 | 2 |
| C2×#2380 | 89 |