Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
552 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #493 |
| commutator, Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #2011 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C2×#97×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#3414 | C2 | semidirect |
| 2. C53×#3150 | C2 | semidirect |
| 3. C53×#2143 | C2 | semidirect |
| 4. C53×#1975 | C2 | semidirect |
| 5. C53×#1375 | C2 | semidirect |
| 35. C2×Q16×C53 | D4 | semidirect |
| 41. C2×D8×C53 | D4 | semidirect |
| 57. C2×C16×C53 | D4 | semidirect |
| 142. #13340 | C53 | direct (N × H) |
| 158. #3414 | C2×C53 | semidirect |
| 159. #3150 | C2×C53 | semidirect |
| 160. #2143 | C2×C53 | semidirect |
| 161. #1975 | C2×C53 | semidirect |
| 162. #1375 | C2×C53 | semidirect |
| 186. C53 | #13340 | direct (N × H) |
| 199. C2×Q16 | D4×C53 | semidirect |
| 205. C2×D8 | D4×C53 | semidirect |
| 221. C2×C16 | D4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3414 | 2 |
| C53×#3150 | 2 |
| C53×#2143 | 2 |
| C53×#1975 | 2 |
| C53×#1375 | 2 |
| #13340 | 53 |