Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
730 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 730 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C22×#41×C53 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C22×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#1728 | C2 | semidirect |
| 3. C53×#1272 | C2 | semidirect |
| 4. C53×#1270 | C2 | semidirect |
| 51. C2×Q8×C53 | M16 | semidirect |
| 58. C2×D4×C53 | M16 | semidirect |
| 134. #7631 | C53 | direct (N × H) |
| 160. #1728 | C2×C53 | semidirect |
| 161. #1272 | C2×C53 | semidirect |
| 162. #1270 | C2×C53 | semidirect |
| 182. C53 | #7631 | direct (N × H) |
| 217. C2×Q8 | M16×C53 | semidirect |
| 224. C2×D4 | M16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2297 | 2 |
| C53×#1728 | 2 |
| C53×#1272 | 2 |
| C53×#1270 | 2 |
| #7631 | 53 |