Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
582 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 582 conjugacy classes of subgroups — too many to draw. Only its 216 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 | C22×D4×C53 | C22 |
| 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. C2×#126×C53 | C2 | semidirect |
| 2. C22×C42×C53 | C2 | semidirect |
| 3. C53×#388 | C2 | semidirect |
| 4. C53×#387 | C2 | semidirect |
| 6. C53×#385 | C2 | semidirect |
| 7. C53×#383 | C2 | semidirect |
| 40. C42×C53 | D4 | semidirect |
| 68. C22×C53 | #126 | semidirect |
| 77. #2852 | C53 | direct (N × H) |
| 83. C2×#126 | C2×C53 | semidirect |
| 84. C22×C42 | C2×C53 | semidirect |
| 85. #388 | C2×C53 | semidirect |
| 86. #387 | C2×C53 | semidirect |
| 88. #385 | C2×C53 | semidirect |
| 89. #383 | C2×C53 | semidirect |
| 90. C53 | #2852 | direct (N × H) |
| 123. C42 | D4×C53 | semidirect |
| 151. C22 | #126×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#126×C53 | 2 |
| C22×C42×C53 | 2 |
| C53×#388 | 2 |
| C53×#387 | 2 |
| C53×#386 | 2 |
| C53×#385 | 2 |
| C53×#383 | 2 |
| #2852 | 53 |