Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1102 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C53 |
| prime radical | C2×#120×C53 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C4×C53 | C2 | semidirect |
| 2. C2×#123×C53 | C2 | semidirect |
| 4. C53×#386 | C2 | semidirect |
| 5. C53×#385 | C2 | semidirect |
| 6. C53×#381 | C2 | semidirect |
| 7. C53×#379 | C2 | semidirect |
| 17. C2×C4⋊C4×C53 | C22 | semidirect |
| 22. C2×C22⋊C4×C53 | C22 | semidirect |
| 23. #95×C53 | C22 | semidirect |
| 39. C22×C4×C53 | D4 | semidirect |
| 87. C22×C53 | #123 | semidirect |
| 95. #2849 | C53 | direct (N × H) |
| 102. C24×C4 | C2×C53 | semidirect |
| 103. C2×#123 | C2×C53 | semidirect |
| 105. #386 | C2×C53 | semidirect |
| 106. #385 | C2×C53 | semidirect |
| 107. #381 | C2×C53 | semidirect |
| 108. #379 | C2×C53 | semidirect |
| 109. C53 | #2849 | direct (N × H) |
| 119. C2×C4⋊C4 | C22×C53 | semidirect |
| 124. C2×C22⋊C4 | C22×C53 | semidirect |
| 125. #95 | C22×C53 | semidirect |
| 141. C22×C4 | D4×C53 | semidirect |
| 189. C22 | #123×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C4×C53 | 2 |
| C2×#123×C53 | 2 |
| C2×#120×C53 | 2 |
| C53×#386 | 2 |
| C53×#385 | 2 |
| C53×#381 | 2 |
| C53×#379 | 2 |
| #2849 | 53 |