Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
678 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 678 conjugacy classes of subgroups — too many to draw. Only its 142 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 | #379 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C53×#3512 | C2 |
| LCS | G ▷ C22×C4 ▷ 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 |
|---|---|---|
| 2. C53×#1543 | C2 | semidirect |
| 3. C53×#1536 | C2 | semidirect |
| 4. C53×#1488 | C2 | semidirect |
| 5. C53×#1470 | C2 | semidirect |
| 137. #8476 | C53 | direct (N × H) |
| 155. #1543 | C2×C53 | semidirect |
| 156. #1536 | C2×C53 | semidirect |
| 157. #1488 | C2×C53 | semidirect |
| 158. #1470 | C2×C53 | semidirect |
| 183. C53 | #8476 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3512 | 2 |
| C53×#1543 | 2 |
| C53×#1536 | 2 |
| C53×#1488 | 2 |
| C53×#1470 | 2 |
| #8476 | 53 |