Finite Group DatabasebetaFinite Group DatabasebetaOrder 6784.
370 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C53 |
| prime radical | C2×D4×C53 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C4×Q8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#557 | C2 | semidirect |
| 3. C53×#487 | C2 | semidirect |
| 4. C53×#475 | C2 | semidirect |
| 6. C8×Q8×C53 | C2 | semidirect |
| 47. Q8×C53 | SD16 | semidirect |
| 70. #3359 | C53 | direct (N × H) |
| 75. #557 | C2×C53 | semidirect |
| 77. #487 | C2×C53 | semidirect |
| 78. #475 | C2×C53 | semidirect |
| 80. C8×Q8 | C2×C53 | semidirect |
| 82. C53 | #3359 | direct (N × H) |
| 122. Q8 | SD16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#557 | 2 |
| D4×Q8×C53 | 2 |
| C53×#487 | 2 |
| C53×#475 | 2 |
| C53×#461 | 2 |
| C8×Q8×C53 | 2 |
| C4×SD16×C53 | 2 |
| #3359 | 53 |