Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
336 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 336 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 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23×C53 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ C53×#446 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#3403 | C2 | semidirect |
| 2. C53×#2250 | C2 | semidirect |
| 3. C53×#2206 | C2 | semidirect |
| 4. C53×#1688 | C2 | semidirect |
| 5. C53×#1360 | C2 | semidirect |
| 6. C53×#1338 | C2 | semidirect |
| 7. C53×#1333 | C2 | semidirect |
| 137. #12336 | C53 | direct (N × H) |
| 156. #3403 | C2×C53 | semidirect |
| 157. #2250 | C2×C53 | semidirect |
| 158. #2206 | C2×C53 | semidirect |
| 159. #1688 | C2×C53 | semidirect |
| 160. #1360 | C2×C53 | semidirect |
| 161. #1338 | C2×C53 | semidirect |
| 162. #1333 | C2×C53 | semidirect |
| 188. C53 | #12336 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3403 | 2 |
| C53×#2250 | 2 |
| C53×#2206 | 2 |
| C53×#1688 | 2 |
| C53×#1360 | 2 |
| C53×#1338 | 2 |
| C53×#1333 | 2 |
| #12336 | 53 |