Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1162 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 1162 conjugacy classes of subgroups — too many to draw. Only its 182 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×#102 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C23×D8×C53 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C4×C53 ◁ C23×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C53×#361 | C2 | semidirect |
| 3. C53×#2112 | C2 | semidirect |
| 4. C53×#1810 | C2 | semidirect |
| 12. C2×M32×C53 | C22 | semidirect |
| 13. C2×#108×C53 | C22 | semidirect |
| 107. #12736 | C53 | direct (N × H) |
| 116. C22×C53 | #361 | semidirect |
| 124. C2×#361 | C2×C53 | semidirect |
| 125. #2112 | C2×C53 | semidirect |
| 126. #1810 | C2×C53 | semidirect |
| 139. C2×M32 | C22×C53 | semidirect |
| 140. C2×#108 | C22×C53 | semidirect |
| 145. C53 | #12736 | direct (N × H) |
| 243. C22 | C53×#361 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D8×C53 | 2 |
| C2×C53×#361 | 2 |
| C53×#2112 | 2 |
| C53×#1810 | 2 |
| #12736 | 53 |