Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
610 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 610 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 | C23×C4 | C23×C53 |
| prime radical | C53×#3192 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#3094 | C2 | semidirect |
| 3. C53×#2624 | C2 | semidirect |
| 4. C53×#1964 | C2 | semidirect |
| 5. C53×#1826 | C2 | semidirect |
| 6. C53×#1507 | C2 | semidirect |
| 7. C53×#1305 | C2 | semidirect |
| 194. #9932 | C53 | direct (N × H) |
| 228. #3094 | C2×C53 | semidirect |
| 229. #2624 | C2×C53 | semidirect |
| 230. #1964 | C2×C53 | semidirect |
| 231. #1826 | C2×C53 | semidirect |
| 232. #1507 | C2×C53 | semidirect |
| 233. #1305 | C2×C53 | semidirect |
| 268. C53 | #9932 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3192 | 2 |
| C53×#3094 | 2 |
| C53×#2624 | 2 |
| C53×#1964 | 2 |
| C53×#1826 | 2 |
| C53×#1507 | 2 |
| C53×#1305 | 2 |
| #9932 | 53 |