Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
728 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 728 conjugacy classes of subgroups — too many to draw. Only its 216 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×#127×C53 | C2 | semidirect |
| 2. C2×#121×C53 | C2 | semidirect |
| 3. C2×#117×C53 | C2 | semidirect |
| 4. C53×#394 | C2 | semidirect |
| 5. C53×#392 | C2 | semidirect |
| 6. C53×#390 | C2 | semidirect |
| 7. C53×#382 | C2 | semidirect |
| 34. C22×C4×C53 | D4 | semidirect |
| 46. C42×C53 | D4 | semidirect |
| 91. #2619 | C53 | direct (N × H) |
| 98. C2×#127 | C2×C53 | semidirect |
| 99. C2×#121 | C2×C53 | semidirect |
| 100. C2×#117 | C2×C53 | semidirect |
| 101. #394 | C2×C53 | semidirect |
| 102. #392 | C2×C53 | semidirect |
| 103. #390 | C2×C53 | semidirect |
| 104. #382 | C2×C53 | semidirect |
| 105. C53 | #2619 | direct (N × H) |
| 132. C22×C4 | D4×C53 | semidirect |
| 144. C42 | D4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#127×C53 | 2 |
| C2×#121×C53 | 2 |
| C2×#117×C53 | 2 |
| C53×#394 | 2 |
| C53×#392 | 2 |
| C53×#390 | 2 |
| C53×#382 | 2 |
| #2619 | 53 |