Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
298 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 298 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 | C2×C4×C53 | D16 |
| commutator | C16 | C22×C4×C53 |
| Frattini, cosocle | C2×C16 | C23×C53 |
| socle | C22×C53 | C2×D16 |
| prime radical | C2×C53×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C22×C8×C53 ◁ C22×C16×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C32×C53 | C2 | semidirect |
| 3. C53×#2158 | C2 | semidirect |
| 4. C53×#1416 | C2 | semidirect |
| 5. C53×#1415 | C2 | semidirect |
| 8. C53×#508 | C4 | semidirect |
| 77. #13615 | C53 | direct (N × H) |
| 89. C22×C32 | C2×C53 | semidirect |
| 90. #2158 | C2×C53 | semidirect |
| 91. #1416 | C2×C53 | semidirect |
| 92. #1415 | C2×C53 | semidirect |
| 99. #508 | C4×C53 | semidirect |
| 109. C53 | #13615 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#508 | 2 |
| C22×C32×C53 | 2 |
| C53×#2158 | 2 |
| C53×#1416 | 2 |
| C53×#1415 | 2 |
| #13615 | 53 |