Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
400 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #331 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C4×C8 | C23×C53 |
| socle | C2×C53 | C2×#331 |
| prime radical | C53×#3147 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C2×C4×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C53×#364 | C2 | semidirect |
| 3. C53×#1358 | C2 | semidirect |
| 4. C53×#1357 | C2 | semidirect |
| 83. #11920 | C53 | direct (N × H) |
| 96. C2×#364 | C2×C53 | semidirect |
| 97. #1358 | C2×C53 | semidirect |
| 98. #1357 | C2×C53 | semidirect |
| 113. C53 | #11920 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3147 | 2 |
| C2×C53×#364 | 2 |
| C53×#1358 | 2 |
| C53×#1357 | 2 |
| #11920 | 53 |