Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
216 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 126 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 | C2×#106 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C4×C8 | C23×C53 |
| prime radical | C23×C53 | #106 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×#1368 | C2 | semidirect |
| 52. #12021 | C53 | direct (N × H) |
| 63. #1368 | C2×C53 | semidirect |
| 75. C53 | #12021 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×M32×C53 | 2 |
| C53×#1568 | 2 |
| C53×#1368 | 2 |
| #12021 | 53 |