Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
294 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #354 |
| commutator | C42 | C42×C53 |
| Frattini, cosocle | #532 | C22×C53 |
| socle | C2×C53 | #1393 |
| prime radical | C53×#532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C53×#532 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C53×#423 | C4 | semidirect |
| 7. C43×C53 | C4 | semidirect |
| 8. C53×#354 | C4 | semidirect |
| 68. #7500 | C53 | direct (N × H) |
| 89. #423 | C4×C53 | semidirect |
| 92. C43 | C4×C53 | semidirect |
| 93. #354 | C4×C53 | semidirect |
| 95. C53 | #7500 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2126 | 2 |
| C53×#1962 | 2 |
| #7500 | 53 |