Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
364 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 110 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 | #486 |
| commutator, Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #2023 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C2×#97×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#3162 | C2 | semidirect |
| 4. C53×#1389 | C2 | semidirect |
| 5. C53×#1375 | C2 | semidirect |
| 20. C2×Q16×C53 | Q8 | semidirect |
| 22. C2×D8×C53 | Q8 | semidirect |
| 100. #13373 | C53 | direct (N × H) |
| 114. #3162 | C2×C53 | semidirect |
| 117. #1389 | C2×C53 | semidirect |
| 118. #1375 | C2×C53 | semidirect |
| 135. C53 | #13373 | direct (N × H) |
| 139. C2×Q16 | Q8×C53 | semidirect |
| 141. C2×D8 | Q8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3162 | 2 |
| C53×#2145 | 2 |
| C53×#1975 | 2 |
| C53×#1389 | 2 |
| C53×#1375 | 2 |
| #13373 | 53 |