Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
556 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 126 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 | #386 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C53×#3151 | C2 |
| 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 |
|---|---|---|
| 2. C53×#2979 | C2 | semidirect |
| 3. C53×#1560 | C2 | semidirect |
| 4. C53×#1545 | C2 | semidirect |
| 5. C53×#1299 | C2 | semidirect |
| 116. #9437 | C53 | direct (N × H) |
| 133. #2979 | C2×C53 | semidirect |
| 134. #1560 | C2×C53 | semidirect |
| 135. #1545 | C2×C53 | semidirect |
| 136. #1299 | C2×C53 | semidirect |
| 158. C53 | #9437 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3151 | 2 |
| C53×#2979 | 2 |
| C53×#1560 | 2 |
| C53×#1545 | 2 |
| C53×#1299 | 2 |
| #9437 | 53 |