Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
414 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 414 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 | C22×C53 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C23×C53 |
| prime radical | C22×D4×C53 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C4⋊C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2028 | C2 | semidirect |
| 3. C53×#1611 | C2 | semidirect |
| 5. C53×#1346 | C2 | semidirect |
| 7. C53×#1319 | C2 | semidirect |
| 70. C2×Q8×C53 | SD16 | semidirect |
| 154. #12210 | C53 | direct (N × H) |
| 179. #2028 | C2×C53 | semidirect |
| 180. #1611 | C2×C53 | semidirect |
| 182. #1346 | C2×C53 | semidirect |
| 184. #1319 | C2×C53 | semidirect |
| 210. C53 | #12210 | direct (N × H) |
| 254. C2×Q8 | SD16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2728 | 2 |
| C53×#2028 | 2 |
| C53×#1611 | 2 |
| C53×#1599 | 2 |
| C53×#1346 | 2 |
| C53×#1345 | 2 |
| C53×#1319 | 2 |
| #12210 | 53 |