Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
420 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 98 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 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C22×D4×C53 | D4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×#2084 | C2 | semidirect |
| 4. C53×#2083 | C2 | semidirect |
| 5. C53×#1271 | C2 | semidirect |
| 6. C53×#1271 | C2 | semidirect |
| 155. #7793 | C53 | direct (N × H) |
| 184. #2084 | C2×C53 | semidirect |
| 185. #2083 | C2×C53 | semidirect |
| 186. #1271 | C2×C53 | semidirect |
| 187. #1271 | C2×C53 | semidirect |
| 216. C53 | #7793 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2727 | 2 |
| C53×#2093 | 2 |
| C53×#2084 | 2 |
| C53×#2083 | 2 |
| C53×#1271 | 2 |
| C53×#1270 | 2 |
| #7793 | 53 |