Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
316 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 90 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 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23×C53 |
| prime radical | C2×D8×C53 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ C8×Q8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2252 | C2 | semidirect |
| 4. C53×#1674 | C2 | semidirect |
| 5. C53×#1364 | C2 | semidirect |
| 6. C53×Q8⋊C16 | C2 | semidirect |
| 63. D8×C53 | Q16 | semidirect |
| 108. #12404 | C53 | direct (N × H) |
| 124. #2252 | C2×C53 | semidirect |
| 126. #1674 | C2×C53 | semidirect |
| 127. #1364 | C2×C53 | semidirect |
| 128. Q8⋊C16 | C2×C53 | semidirect |
| 151. C53 | #12404 | direct (N × H) |
| 190. D8 | Q16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×D8×C53 | 2 |
| C53×#2252 | 2 |
| C53×#2224 | 2 |
| C53×#1674 | 2 |
| C53×#1364 | 2 |
| C53×Q8⋊C16 | 2 |
| C53×#1331 | 2 |
| #12404 | 53 |