Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
984 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 984 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C25×C53 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C24×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2076 | C2 | semidirect |
| 3. C53×#2036 | C2 | semidirect |
| 4. C53×#2001 | C2 | semidirect |
| 7. C22×Q16×C53 | C22 | semidirect |
| 9. C2×#107×C53 | C22 | semidirect |
| 10. C2×#103×C53 | C22 | semidirect |
| 16. C53×#340 | C22 | semidirect |
| 17. C53×#340 | C22 | semidirect |
| 40. C24×C53 | Q16 | semidirect |
| 89. #32710 | C53 | direct (N × H) |
| 105. #2076 | C2×C53 | semidirect |
| 106. #2036 | C2×C53 | semidirect |
| 107. #2001 | C2×C53 | semidirect |
| 114. C22×Q16 | C22×C53 | semidirect |
| 116. C2×#107 | C22×C53 | semidirect |
| 117. C2×#103 | C22×C53 | semidirect |
| 123. #340 | C22×C53 | semidirect |
| 124. #340 | C22×C53 | semidirect |
| 125. C53 | #32710 | direct (N × H) |
| 148. C24 | Q16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2848 | 2 |
| C53×#2076 | 2 |
| C53×#2036 | 2 |
| C53×#2001 | 2 |
| C53×#1779 | 2 |
| #32710 | 53 |