Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
222 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 46 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 | #353 |
| commutator | C42 | C2×C8×C53 |
| Frattini, cosocle | #423 | C22×C53 |
| socle | C2×C53 | #1318 |
| prime radical | C53×#532 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C53×#423 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#1327 | C2 | semidirect |
| 61. #7357 | C53 | direct (N × H) |
| 75. #1327 | C2×C53 | semidirect |
| 85. C53 | #7357 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#1962 | 2 |
| C53×#1327 | 2 |
| #7357 | 53 |