Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
284 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 74 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 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C23×C53 |
| prime radical | C53×#3415 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ #131×C53 ◁ C53×#504 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2264 | C2 | semidirect |
| 3. C2×C53×SD64 | C2 | semidirect |
| 4. C53×#2222 | C2 | semidirect |
| 5. C53×#1419 | C2 | semidirect |
| 6. C53×#1417 | C2 | semidirect |
| 7. C53×#1401 | C2 | semidirect |
| 13. C2×Q32×C53 | C22 | semidirect |
| 21. C53×M64 | C22 | semidirect |
| 22. C2×C32×C53 | C22 | semidirect |
| 23. C53×#368 | C22 | semidirect |
| 103. #13671 | C53 | direct (N × H) |
| 116. #2264 | C2×C53 | semidirect |
| 117. C2×SD64 | C2×C53 | semidirect |
| 118. #2222 | C2×C53 | semidirect |
| 119. #1419 | C2×C53 | semidirect |
| 120. #1417 | C2×C53 | semidirect |
| 121. #1401 | C2×C53 | semidirect |
| 132. C2×Q32 | C22×C53 | semidirect |
| 140. M64 | C22×C53 | semidirect |
| 141. C2×C32 | C22×C53 | semidirect |
| 142. #368 | C22×C53 | semidirect |
| 144. C53 | #13671 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3415 | 2 |
| C53×#2264 | 2 |
| C2×C53×SD64 | 2 |
| C53×#2222 | 2 |
| C53×#1419 | 2 |
| C53×#1417 | 2 |
| C53×#1401 | 2 |
| #13671 | 53 |