Finite Group DatabasebetaFinite Group DatabasebetaOrder 17152.
1136 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 222 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×C67 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C67 |
| prime radical | C2×2−1+4×C67 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C67 ◁ C23×C67 ◁ C22×Q8×C67 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q82×C67 | C2 | semidirect |
| 3. C67×#2205 | C2 | direct (N × H) |
| 4. C2×#356×C67 | C2 | semidirect |
| 8. Q82×C67 | C22 | semidirect |
| 16. #356×C67 | C22 | semidirect |
| 123. C2×#2205 | C67 | direct (N × H) |
| 127. C2×C67 | #2205 | direct (N × H) |
| 129. C2×Q82 | C2×C67 | semidirect |
| 131. #2205 | C2×C67 | direct (N × H) |
| 132. C2×#356 | C2×C67 | semidirect |
| 133. C67 | C2×#2205 | direct (N × H) |
| 137. Q82 | C22×C67 | semidirect |
| 145. #356 | C22×C67 | semidirect |
| 255. C2 | C67×#2205 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q82×C67 | 2 |
| C2×#456×C67 | 2 |
| C67×#2205 | 2 |
| C2×#356×C67 | 2 |
| C2×#2205 | 67 |