Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
370 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 168 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 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C53 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ #117×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#540 | C2 | semidirect |
| 2. C53×#478 | C2 | semidirect |
| 3. C53×#469 | C2 | semidirect |
| 4. C53×#466 | C2 | semidirect |
| 5. C53×#459 | C2 | semidirect |
| 6. C53×#424 | C2 | semidirect |
| 67. #3259 | C53 | direct (N × H) |
| 72. #540 | C2×C53 | semidirect |
| 73. #478 | C2×C53 | semidirect |
| 74. #469 | C2×C53 | semidirect |
| 75. #466 | C2×C53 | semidirect |
| 76. #459 | C2×C53 | semidirect |
| 77. #424 | C2×C53 | semidirect |
| 78. C53 | #3259 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#540 | 2 |
| C53×#478 | 2 |
| C53×#469 | 2 |
| C53×#466 | 2 |
| C53×#459 | 2 |
| C53×#424 | 2 |
| #3259 | 53 |