Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
556 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 106 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 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C53×#3205 | C2 |
| LCS | G ▷ C2×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 |
|---|---|---|
| 1. C53×#3254 | C2 | semidirect |
| 3. C53×#3138 | C2 | semidirect |
| 4. C53×#1965 | C2 | semidirect |
| 5. C53×#1525 | C2 | semidirect |
| 6. C53×#1485 | C2 | semidirect |
| 7. C53×#1299 | C2 | semidirect |
| 165. #10112 | C53 | direct (N × H) |
| 187. #3254 | C2×C53 | semidirect |
| 189. #3138 | C2×C53 | semidirect |
| 190. #1965 | C2×C53 | semidirect |
| 191. #1525 | C2×C53 | semidirect |
| 192. #1485 | C2×C53 | semidirect |
| 193. #1299 | C2×C53 | semidirect |
| 223. C53 | #10112 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3254 | 2 |
| C53×#3205 | 2 |
| C53×#3138 | 2 |
| C53×#1965 | 2 |
| C53×#1525 | 2 |
| C53×#1485 | 2 |
| C53×#1299 | 2 |
| #10112 | 53 |