Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
480 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 232 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 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C53 |
| prime radical | C22×D4×C53 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C4⋊Q8×C53 | C2 | semidirect |
| 3. C53×#474 | C2 | direct (N × H) |
| 4. C2×#107×C53 | C2 | semidirect |
| 5. C2×C4⋊C8×C53 | C2 | semidirect |
| 12. C4⋊Q8×C53 | C22 | semidirect |
| 20. #107×C53 | C22 | semidirect |
| 21. C4⋊C8×C53 | C22 | semidirect |
| 28. C2×D4×C53 | Q8 | semidirect |
| 55. D4×C53 | C2×Q8 | semidirect |
| 85. C2×#474 | C53 | direct (N × H) |
| 87. C2×C53 | #474 | direct (N × H) |
| 91. C2×C4⋊Q8 | C2×C53 | semidirect |
| 93. #474 | C2×C53 | direct (N × H) |
| 94. C2×#107 | C2×C53 | semidirect |
| 95. C2×C4⋊C8 | C2×C53 | semidirect |
| 97. C53 | C2×#474 | direct (N × H) |
| 103. C4⋊Q8 | C22×C53 | semidirect |
| 111. #107 | C22×C53 | semidirect |
| 112. C4⋊C8 | C22×C53 | semidirect |
| 119. C2×D4 | Q8×C53 | semidirect |
| 146. D4 | C2×Q8×C53 | semidirect |
| 177. C2 | C53×#474 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊Q8×C53 | 2 |
| C2×C4×D4×C53 | 2 |
| C53×#474 | 2 |
| C2×#107×C53 | 2 |
| C2×C4⋊C8×C53 | 2 |
| C2×#102×C53 | 2 |
| C2×#474 | 53 |