Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
340 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 94 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 | #354 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | #124 | C23×C53 |
| prime radical | C22×D4×C53 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C53 ◁ C22×C4×C53 ◁ C2×#124×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C53×#1414 | C2 | direct (N × H) |
| 5. C2×#125×C53 | C4 | semidirect |
| 9. C53×#488 | C4 | semidirect |
| 15. C2×C4×C8×C53 | C4 | semidirect |
| 26. #125×C53 | C2×C4 | semidirect |
| 46. C4×C8×C53 | C2×C4 | semidirect |
| 111. C2×#1414 | C53 | direct (N × H) |
| 133. #1414 | C2×C53 | direct (N × H) |
| 138. C2×C53 | #1414 | direct (N × H) |
| 140. C2×#125 | C4×C53 | semidirect |
| 144. #488 | C4×C53 | semidirect |
| 150. C2×C4×C8 | C4×C53 | semidirect |
| 155. C53 | C2×#1414 | direct (N × H) |
| 162. #125 | C2×C4×C53 | semidirect |
| 182. C4×C8 | C2×C4×C53 | semidirect |
| 269. C2 | C53×#1414 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#488 | 2 |
| C2×C53×#355 | 2 |
| C2×C53×#354 | 2 |
| C53×#1414 | 2 |
| C2×#1414 | 53 |