Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
530 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 530 conjugacy classes of subgroups — too many to draw. Only its 174 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 | #390 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C2×#127×C53 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#1848 | C2 | semidirect |
| 3. C53×#1546 | C2 | semidirect |
| 5. C53×#1450 | C2 | semidirect |
| 100. #9768 | C53 | direct (N × H) |
| 114. #1848 | C2×C53 | semidirect |
| 115. #1546 | C2×C53 | semidirect |
| 117. #1450 | C2×C53 | semidirect |
| 136. C53 | #9768 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#492 | 2 |
| C53×#1848 | 2 |
| C53×#1546 | 2 |
| C53×#1545 | 2 |
| C53×#1450 | 2 |
| #9768 | 53 |