Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
134 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1404 |
| commutator | #125 | C2×C4×C53 |
| Frattini, cosocle | #490 | C22×C53 |
| prime radical | #127×C53 | D4 |
| LCS | G ▷ #125 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C53 ◁ C2×C4×C53 ◁ C42×C53 ◁ C53×#490 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C53×#564 | C4 | semidirect |
| 10. C53×#330 | C4 | semidirect |
| 52. #7524 | C53 | direct (N × H) |
| 68. #564 | C4×C53 | semidirect |
| 74. #330 | C4×C53 | semidirect |
| 75. C53 | #7524 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#1658 | 2 |
| C53×#1412 | 2 |
| C53×#1355 | 2 |
| #7524 | 53 |