Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
1224 conjugacy classes of subgroups; only the 288 normal subgroups are shown.
This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 288 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. C24×C4×C53 | C2 | semidirect |
| 2. C2×#121×C53 | C2 | semidirect |
| 3. C2×#120×C53 | C2 | semidirect |
| 4. C53×#386 | C2 | semidirect |
| 5. C53×#380 | C2 | semidirect |
| 8. C22×D4×C53 | C22 | semidirect |
| 14. C2×C4⋊C4×C53 | C22 | semidirect |
| 16. C2×C22⋊C4×C53 | C22 | semidirect |
| 17. #95×C53 | C22 | semidirect |
| 20. C24×C53 | D4 | semidirect |
| 31. C22×C4×C53 | D4 | semidirect |
| 68. C22×C53 | #121 | semidirect |
| 74. C4×C53 | #120 | semidirect |
| 75. #2847 | C53 | direct (N × H) |
| 81. C24×C4 | C2×C53 | semidirect |
| 82. C2×#121 | C2×C53 | semidirect |
| 83. C2×#120 | C2×C53 | semidirect |
| 84. #386 | C2×C53 | semidirect |
| 85. #380 | C2×C53 | semidirect |
| 86. C53 | #2847 | direct (N × H) |
| 89. C22×D4 | C22×C53 | semidirect |
| 95. C2×C4⋊C4 | C22×C53 | semidirect |
| 97. C2×C22⋊C4 | C22×C53 | semidirect |
| 98. #95 | C22×C53 | semidirect |
| 101. C24 | D4×C53 | semidirect |
| 112. C22×C4 | D4×C53 | semidirect |
| 149. C22 | #121×C53 | semidirect |
| 155. C4 | #120×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C4×C53 | 2 |
| C2×#121×C53 | 2 |
| C2×#120×C53 | 2 |
| C53×#386 | 2 |
| C53×#380 | 2 |
| #2847 | 53 |