Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
218 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C53 | D4 |
| commutator | C4 | C22×C16×C53 |
| Frattini, cosocle | C4×C8 | C23×C53 |
| socle, prime radical | C22×C53 | C8×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C53 ◁ C4×C16×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 31. Q16×C53 | C16 | direct (N × H) |
| 45. C16×C53 | Q16 | direct (N × H) |
| 60. C16×Q16 | C53 | direct (N × H) |
| 87. C53 | C16×Q16 | direct (N × H) |
| 103. Q16 | C16×C53 | direct (N × H) |
| 117. C16 | Q16×C53 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×C16×C53 | 2 |
| C8×Q16×C53 | 2 |
| C53×#1370 | 2 |
| C53×Q8⋊C16 | 2 |
| C8×C16×C53 | 2 |
| C16×Q16 | 53 |