Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
910 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 910 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, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| LCS | G ▷ C42 ▷ 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 |
|---|---|---|
| 1. C53×#3568 | C2 | semidirect |
| 2. C53×#3217 | C2 | semidirect |
| 3. C53×#1657 | C2 | semidirect |
| 4. C53×#1655 | C2 | semidirect |
| 5. C53×#1535 | C2 | semidirect |
| 6. C53×#563 | C22 | semidirect |
| 7. C53×H(3,𝔽4) | C22 | semidirect |
| 12. C2×#127×C53 | C22 | semidirect |
| 15. C53×#490 | C22 | semidirect |
| 20. C53×#432 | C22 | semidirect |
| 21. C53×#330 | C22 | semidirect |
| 22. C53×#329 | C22 | semidirect |
| 124. #19094 | C53 | direct (N × H) |
| 138. #3568 | C2×C53 | semidirect |
| 139. #3217 | C2×C53 | semidirect |
| 140. #1657 | C2×C53 | semidirect |
| 141. #1655 | C2×C53 | semidirect |
| 142. #1535 | C2×C53 | semidirect |
| 147. #563 | C22×C53 | semidirect |
| 148. H(3,𝔽4) | C22×C53 | semidirect |
| 153. C2×#127 | C22×C53 | semidirect |
| 156. #490 | C22×C53 | semidirect |
| 161. #432 | C22×C53 | semidirect |
| 162. #330 | C22×C53 | semidirect |
| 163. #329 | C22×C53 | semidirect |
| 164. C53 | #19094 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3568 | 2 |
| C53×#3217 | 2 |
| C53×#1657 | 2 |
| C53×#1655 | 2 |
| C53×#1535 | 2 |
| #19094 | 53 |