Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
250 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 250 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 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C53 |
| prime radical | C53×#2246 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ C4×C8×C53 ◁ C4×C16×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C53×SD64 | C2 | semidirect |
| 2. C53×#2247 | C2 | semidirect |
| 4. C4×C32×C53 | C2 | semidirect |
| 27. C32×C53 | D4 | semidirect |
| 55. #13727 | C53 | direct (N × H) |
| 60. C4×C53 | SD64 | semidirect |
| 63. C2×SD64 | C2×C53 | semidirect |
| 64. #2247 | C2×C53 | semidirect |
| 66. C4×C32 | C2×C53 | semidirect |
| 79. C53 | #13727 | direct (N × H) |
| 93. C32 | D4×C53 | semidirect |
| 125. C4 | C53×SD64 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×SD64 | 2 |
| C53×#2247 | 2 |
| C53×#2246 | 2 |
| C4×C32×C53 | 2 |
| #13727 | 53 |