Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
138 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 138 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 | #1409 |
| commutator | C4×C8 | C2×C4×C53 |
| Frattini, cosocle | #492 | C22×C53 |
| prime radical | #127×C53 | D4 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C53 ◁ C2×C4×C53 ◁ C42×C53 ◁ C53×#492 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C53×#498 | C4 | semidirect |
| 10. C82×C53 | C4 | semidirect |
| 48. #7536 | C53 | direct (N × H) |
| 63. #498 | C4×C53 | semidirect |
| 69. C82 | C4×C53 | semidirect |
| 70. C53 | #7536 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#1708 | 2 |
| C53×#1410 | 2 |
| C53×#1409 | 2 |
| #7536 | 53 |