Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
740 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 740 conjugacy classes of subgroups — too many to draw. Only its 296 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 | C22 | C23×C4×C53 |
| Frattini, cosocle | C23 | C24×C53 |
| prime radical | C2×#127×C53 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×D4×C53 | C2 | semidirect |
| 3. C53×#390 | C2 | semidirect |
| 4. C53×#386 | C2 | semidirect |
| 5. C53×#382 | C2 | semidirect |
| 6. C53×#376 | C2 | semidirect |
| 10. #127×C53 | C4 | semidirect |
| 24. C2×D4×C53 | C2×C4 | semidirect |
| 25. C2×D4×C53 | C2×C4 | semidirect |
| 37. C42×C53 | C2×C4 | semidirect |
| 38. C42×C53 | D4 | semidirect |
| 73. #2328 | C53 | direct (N × H) |
| 80. C2×C4×D4 | C2×C53 | semidirect |
| 81. #390 | C2×C53 | semidirect |
| 82. #386 | C2×C53 | semidirect |
| 83. #382 | C2×C53 | semidirect |
| 84. #376 | C2×C53 | semidirect |
| 85. C53 | #2328 | direct (N × H) |
| 89. #127 | C4×C53 | semidirect |
| 103. C2×D4 | C2×C4×C53 | semidirect |
| 104. C2×D4 | C2×C4×C53 | semidirect |
| 116. C42 | C2×C4×C53 | semidirect |
| 117. C42 | D4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#127×C53 | 2 |
| C2×C4×D4×C53 | 2 |
| C53×#390 | 2 |
| C53×#386 | 2 |
| C53×#382 | 2 |
| C53×#376 | 2 |
| #2328 | 53 |