Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
622 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 622 conjugacy classes of subgroups — too many to draw. Only its 102 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 |
| 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×#3213 | C2 | semidirect |
| 2. C53×#3205 | C2 | semidirect |
| 3. C2×C53×#496 | C2 | semidirect |
| 4. C53×#1555 | C2 | semidirect |
| 5. C53×#1530 | C2 | semidirect |
| 6. C53×#1469 | C2 | semidirect |
| 7. C53×#1302 | C2 | semidirect |
| 171. #10128 | C53 | direct (N × H) |
| 195. #3213 | C2×C53 | semidirect |
| 196. #3205 | C2×C53 | semidirect |
| 197. C2×#496 | C2×C53 | semidirect |
| 198. #1555 | C2×C53 | semidirect |
| 199. #1530 | C2×C53 | semidirect |
| 200. #1469 | C2×C53 | semidirect |
| 201. #1302 | C2×C53 | semidirect |
| 232. C53 | #10128 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3213 | 2 |
| C53×#3205 | 2 |
| C2×C53×#496 | 2 |
| C53×#1555 | 2 |
| C53×#1530 | 2 |
| C53×#1469 | 2 |
| C53×#1302 | 2 |
| #10128 | 53 |