Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
924 conjugacy classes of subgroups; only the 288 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 288 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 |
| 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×#127×C53 | C2 | semidirect |
| 2. C2×#121×C53 | C2 | semidirect |
| 3. C22×C42×C53 | C2 | semidirect |
| 4. C53×#390 | C2 | semidirect |
| 5. C53×#384 | C2 | semidirect |
| 24. C22×C4×C53 | D4 | semidirect |
| 31. C42×C53 | D4 | semidirect |
| 54. C22×C53 | #127 | semidirect |
| 60. C4×C53 | #121 | semidirect |
| 61. #2856 | C53 | direct (N × H) |
| 67. C2×#127 | C2×C53 | semidirect |
| 68. C2×#121 | C2×C53 | semidirect |
| 69. C22×C42 | C2×C53 | semidirect |
| 70. #390 | C2×C53 | semidirect |
| 71. #384 | C2×C53 | semidirect |
| 72. C53 | #2856 | direct (N × H) |
| 91. C22×C4 | D4×C53 | semidirect |
| 98. C42 | D4×C53 | semidirect |
| 121. C22 | #127×C53 | semidirect |
| 127. C4 | #121×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×#121×C53 | 2 |
| C22×C42×C53 | 2 |
| C53×#390 | 2 |
| C53×#384 | 2 |
| #2856 | 53 |