Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
198 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C53 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ C4×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×SD32×C53 | C2 | semidirect |
| 2. C2×D16×C53 | C2 | semidirect |
| 3. C53×#495 | C2 | semidirect |
| 4. C53×#493 | C2 | semidirect |
| 5. C53×#346 | C2 | semidirect |
| 31. C16×C53 | D4 | semidirect |
| 55. #2250 | C53 | direct (N × H) |
| 60. C2×SD32 | C2×C53 | semidirect |
| 61. C2×D16 | C2×C53 | semidirect |
| 62. #495 | C2×C53 | semidirect |
| 63. #493 | C2×C53 | semidirect |
| 64. #346 | C2×C53 | semidirect |
| 65. C53 | #2250 | direct (N × H) |
| 91. C16 | D4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×SD32×C53 | 2 |
| C2×D16×C53 | 2 |
| C53×#495 | 2 |
| C53×#493 | 2 |
| C53×#346 | 2 |
| #2250 | 53 |