Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
316 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 94 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 | #1505 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C2×M16 | C23×C53 |
| prime radical | C53×#3176 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C53×#405 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2166 | C2 | semidirect |
| 3. C53×#1747 | C2 | semidirect |
| 4. C53×#1341 | C2 | semidirect |
| 5. C53×#1339 | C2 | semidirect |
| 7. C53×#499 | C4 | semidirect |
| 8. C53×#493 | C4 | semidirect |
| 12. C53×#442 | C4 | semidirect |
| 97. #11770 | C53 | direct (N × H) |
| 112. #2166 | C2×C53 | semidirect |
| 113. #1747 | C2×C53 | semidirect |
| 114. #1341 | C2×C53 | semidirect |
| 115. #1339 | C2×C53 | semidirect |
| 121. #499 | C4×C53 | semidirect |
| 122. #493 | C4×C53 | semidirect |
| 126. #442 | C4×C53 | semidirect |
| 136. C53 | #11770 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3176 | 2 |
| C53×#2166 | 2 |
| C53×#1747 | 2 |
| C53×#1341 | 2 |
| C53×#1339 | 2 |
| #11770 | 53 |