Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
506 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 506 conjugacy classes of subgroups — too many to draw. Only its 218 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 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C53 |
| prime radical | C2×D8×C53 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C4×Q8×C53 ◁ C8×Q8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#3395 | C2 | semidirect |
| 3. C53×#2241 | C2 | semidirect |
| 4. C53×#2225 | C2 | semidirect |
| 6. Q8×C16×C53 | C2 | semidirect |
| 75. Q8×C53 | SD32 | semidirect |
| 88. #33926 | C53 | direct (N × H) |
| 100. #3395 | C2×C53 | semidirect |
| 102. #2241 | C2×C53 | semidirect |
| 103. #2225 | C2×C53 | semidirect |
| 105. Q8×C16 | C2×C53 | semidirect |
| 127. C53 | #33926 | direct (N × H) |
| 179. Q8 | SD32×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3395 | 2 |
| Q8×D8×C53 | 2 |
| C53×#2241 | 2 |
| C53×#2225 | 2 |
| C53×#2208 | 2 |
| Q8×C16×C53 | 2 |
| C4×SD32×C53 | 2 |
| #33926 | 53 |