Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1118 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #120 |
| commutator | C23 | C22×C8×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C22×C53 | #379 |
| prime radical | C2×H(5,𝔽2)×C53 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C23×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#1869 | C2 | semidirect |
| 3. C53×#1459 | C2 | semidirect |
| 5. C23×C8×C53 | C22 | semidirect |
| 13. C2×C22⋊C8×C53 | C22 | semidirect |
| 18. H(5,𝔽2)×C53 | C8 | semidirect |
| 102. C23×C53 | C22⋊C8 | semidirect |
| 125. C2×C4×C53 | C22⋊C8 | semidirect |
| 130. #8316 | C53 | direct (N × H) |
| 151. #1869 | C2×C53 | semidirect |
| 152. #1459 | C2×C53 | semidirect |
| 160. C23×C8 | C22×C53 | semidirect |
| 168. C2×C22⋊C8 | C22×C53 | semidirect |
| 172. C53 | #8316 | direct (N × H) |
| 174. H(5,𝔽2) | C8×C53 | semidirect |
| 258. C23 | C22⋊C8×C53 | semidirect |
| 281. C2×C4 | C22⋊C8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×H(5,𝔽2)×C53 | 2 |
| C53×#1869 | 2 |
| C53×#1459 | 2 |
| #8316 | 53 |