Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
788 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 788 conjugacy classes of subgroups — too many to draw. Only its 248 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×#121×C53 | C2 | semidirect |
| 2. C2×C4×D4×C53 | C2 | semidirect |
| 3. C53×#394 | C2 | semidirect |
| 4. C53×#384 | C2 | semidirect |
| 5. C53×#380 | C2 | semidirect |
| 23. C2×D4×C53 | D4 | semidirect |
| 36. C42×C53 | D4 | semidirect |
| 74. C4×C53 | #121 | semidirect |
| 75. #2537 | C53 | direct (N × H) |
| 82. C2×#121 | C2×C53 | semidirect |
| 83. C2×C4×D4 | C2×C53 | semidirect |
| 84. #394 | C2×C53 | semidirect |
| 85. #384 | C2×C53 | semidirect |
| 86. #380 | C2×C53 | semidirect |
| 87. C53 | #2537 | direct (N × H) |
| 105. C2×D4 | D4×C53 | semidirect |
| 118. C42 | D4×C53 | semidirect |
| 156. C4 | #121×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#121×C53 | 2 |
| C2×C4×D4×C53 | 2 |
| C53×#394 | 2 |
| C53×#384 | 2 |
| C53×#380 | 2 |
| #2537 | 53 |