Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
460 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 192 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 ◁ C4×Q8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#550 | C2 | semidirect |
| 2. C53×#493 | C2 | semidirect |
| 3. C53×#459 | C2 | semidirect |
| 4. C8×Q8×C53 | C2 | semidirect |
| 5. C4×D8×C53 | C2 | semidirect |
| 32. Q8×C53 | D8 | semidirect |
| 43. C8×C53 | #41 | semidirect |
| 53. #3391 | C53 | direct (N × H) |
| 58. #550 | C2×C53 | semidirect |
| 59. #493 | C2×C53 | semidirect |
| 60. #459 | C2×C53 | semidirect |
| 61. C8×Q8 | C2×C53 | semidirect |
| 62. C4×D8 | C2×C53 | semidirect |
| 63. C53 | #3391 | direct (N × H) |
| 90. Q8 | D8×C53 | semidirect |
| 101. C8 | #41×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#550 | 2 |
| C53×#493 | 2 |
| C53×#459 | 2 |
| C8×Q8×C53 | 2 |
| C4×D8×C53 | 2 |
| #3391 | 53 |