Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
372 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1619 |
| commutator, Frattini, cosocle | C4×C8 | C23×C53 |
| prime radical | C53×#3292 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C4×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2214 | C2 | semidirect |
| 3. C53×#1709 | C2 | semidirect |
| 4. C53×#1357 | C2 | semidirect |
| 5. C53×#1335 | C2 | semidirect |
| 8. C53×#494 | C22 | semidirect |
| 14. C53×#364 | C22 | semidirect |
| 16. C82×C53 | C22 | semidirect |
| 79. #12312 | C53 | direct (N × H) |
| 90. #2214 | C2×C53 | semidirect |
| 91. #1709 | C2×C53 | semidirect |
| 92. #1357 | C2×C53 | semidirect |
| 93. #1335 | C2×C53 | semidirect |
| 100. #494 | C22×C53 | semidirect |
| 106. #364 | C22×C53 | semidirect |
| 108. C82 | C22×C53 | semidirect |
| 109. C53 | #12312 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3292 | 2 |
| C53×#2214 | 2 |
| C53×#1709 | 2 |
| C53×#1357 | 2 |
| C53×#1335 | 2 |
| #12312 | 53 |