Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
180 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 78 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 | C2×D16×C53 | C22 |
| 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. C53×#2253 | C2 | semidirect |
| 3. C53×#1416 | C2 | semidirect |
| 5. C4×C32×C53 | C2 | semidirect |
| 56. #13721 | C53 | direct (N × H) |
| 65. #2253 | C2×C53 | semidirect |
| 67. #1416 | C2×C53 | semidirect |
| 69. C4×C32 | C2×C53 | semidirect |
| 82. C53 | #13721 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2253 | 2 |
| C53×#2249 | 2 |
| C53×#1416 | 2 |
| C53×#1415 | 2 |
| C4×C32×C53 | 2 |
| #13721 | 53 |