Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
416 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 114 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 | #124 |
| commutator, Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | #393 |
| prime radical | C23×C53 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 21. C4⋊Q8×C53 | Q8 | semidirect |
| 101. #10371 | C53 | direct (N × H) |
| 139. C53 | #10371 | direct (N × H) |
| 142. C4⋊Q8 | Q8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2399 | 2 |
| C53×#1995 | 2 |
| C53×#1503 | 2 |
| C53×#1303 | 2 |
| C53×#1274 | 2 |
| #10371 | 53 |