Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
240 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×D8 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C4×C8 | C23×C53 |
| socle | C22×C53 | C4×D8 |
| prime radical | C2×D16×C53 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C4×C8×C53 ◁ C82×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#1589 | C2 | semidirect |
| 3. C8×D8×C53 | C2 | semidirect |
| 4. C53×#1371 | C2 | semidirect |
| 5. C53×#1331 | C2 | semidirect |
| 6. C53×#1331 | C2 | semidirect |
| 7. C53×#1312 | C2 | semidirect |
| 51. D8×C53 | M16 | semidirect |
| 89. C8×C53 | D16 | semidirect |
| 92. #12069 | C53 | direct (N × H) |
| 108. #1589 | C2×C53 | semidirect |
| 109. C8×D8 | C2×C53 | semidirect |
| 110. #1371 | C2×C53 | semidirect |
| 111. #1331 | C2×C53 | semidirect |
| 112. #1331 | C2×C53 | semidirect |
| 113. #1312 | C2×C53 | semidirect |
| 134. C53 | #12069 | direct (N × H) |
| 163. D8 | M16×C53 | semidirect |
| 201. C8 | D16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×D16×C53 | 2 |
| C53×#1589 | 2 |
| C8×D8×C53 | 2 |
| C53×#1371 | 2 |
| C53×#1331 | 2 |
| C53×#1312 | 2 |
| #12069 | 53 |