Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
456 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 184 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 |
| prime radical | C22×D4×C53 | C22 |
| 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×#126×C53 | C2 | semidirect |
| 2. C53×#398 | C2 | semidirect |
| 3. C53×#396 | C2 | semidirect |
| 5. C53×#384 | C2 | semidirect |
| 29. C42×C53 | D4 | semidirect |
| 54. #2823 | C53 | direct (N × H) |
| 59. C2×#126 | C2×C53 | semidirect |
| 60. #398 | C2×C53 | semidirect |
| 61. #396 | C2×C53 | semidirect |
| 63. #384 | C2×C53 | semidirect |
| 64. C53 | #2823 | direct (N × H) |
| 88. C42 | D4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#126×C53 | 2 |
| C53×#398 | 2 |
| C53×#396 | 2 |
| C53×#394 | 2 |
| C53×#384 | 2 |
| #2823 | 53 |