Finite Group Databasebeta

← C2×C3×C4×C5×C7 #11719829

Subgroups — C2×C3×C4×C5×C7 #11719829

Order 840.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×C7C211yesmaximal, characteristic
2C3×C4×C5×C7C212yesmaximal, direct product factor
3C2×C4×C5×C7C311yesmaximal, characteristic, direct product factor, Hall
4C2×C3×C5×C7C412yesdirect product factor
5C2×C3×C5×C7C2211yescharacteristic
6C2×C3×C4×C7C511yesmaximal, characteristic, direct product factor, Hall
7C22×C5×C7C2×C311yescharacteristic
8C4×C5×C7C2×C312yesdirect product factor
9C2×C3×C4×C5C711yesmaximal, characteristic, direct product factor, Hall
10C3×C5×C7C2×C411yescharacteristic, direct product factor, Hall
11C22×C3×C7C2×C511yescharacteristic
12C3×C4×C7C2×C512yesdirect product factor
13C2×C5×C7C3×C412yesdirect product factor
14C2×C5×C7C22×C311yescharacteristic
15C22×C3×C5C2×C711yescharacteristic
16C3×C4×C5C2×C712yesdirect product factor
17C2×C4×C7C3×C511yescharacteristic, direct product factor, Hall
18C2×C3×C7C4×C512yesdirect product factor
19C2×C3×C7C22×C511yescharacteristic
20C2×C4×C5C3×C711yescharacteristic, direct product factor, Hall
21C5×C7C2×C3×C411yescharacteristic, direct product factor, Hall
22C2×C3×C5C4×C712yesdirect product factor
23C2×C3×C5C22×C711yescharacteristic
24C22×C7C2×C3×C511yescharacteristic
25C4×C7C2×C3×C512yesdirect product factor
26C2×C3×C4C5×C711yescharacteristic, direct product factor, Hall
27C3×C7C2×C4×C511yescharacteristic, direct product factor, Hall
28C22×C5C2×C3×C711yescharacteristic
29C4×C5C2×C3×C712yesdirect product factor
30C3×C5C2×C4×C711yescharacteristic, direct product factor, Hall
31C2×C7C3×C4×C512yesdirect product factor
32C2×C7C22×C3×C511yescharacteristic
33C22×C3C2×C5×C711yescharacteristic
34C3×C4C2×C5×C712yesdirect product factor
35C2×C5C3×C4×C712yesdirect product factor
36C2×C5C22×C3×C711yescharacteristic
37C2×C4C3×C5×C711yescharacteristic, direct product factor, Sylow
38C7C2×C3×C4×C511yescharacteristic, direct product factor, Sylow
39C2×C3C4×C5×C712yesdirect product factor
40C2×C3C22×C5×C711yescharacteristic
41C5C2×C3×C4×C711yescharacteristic, direct product factor, Sylow
42C22C2×C3×C5×C711yescharacteristic
43C4C2×C3×C5×C712yesdirect product factor
44C3C2×C4×C5×C711yescharacteristic, direct product factor, Sylow
45C2C3×C4×C5×C712yesdirect product factor
46C2C22×C3×C5×C711yescharacteristic
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2C22×C3×C5×C7
socle, prime radicalC22×C3×C5×C7C2
LCSG ▷ C1
UCSC1 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×C4×C5×C7C2direct (N × H)
3. C2×C4×C5×C7C3direct (N × H)
4. C2×C3×C5×C7C4direct (N × H)
6. C2×C3×C4×C7C5direct (N × H)
8. C4×C5×C7C2×C3direct (N × H)
9. C2×C3×C4×C5C7direct (N × H)
10. C3×C5×C7C2×C4direct (N × H)
12. C3×C4×C7C2×C5direct (N × H)
13. C2×C5×C7C3×C4direct (N × H)
16. C3×C4×C5C2×C7direct (N × H)
17. C2×C4×C7C3×C5direct (N × H)
18. C2×C3×C7C4×C5direct (N × H)
20. C2×C4×C5C3×C7direct (N × H)
21. C5×C7C2×C3×C4direct (N × H)
22. C2×C3×C5C4×C7direct (N × H)
25. C4×C7C2×C3×C5direct (N × H)
26. C2×C3×C4C5×C7direct (N × H)
27. C3×C7C2×C4×C5direct (N × H)
29. C4×C5C2×C3×C7direct (N × H)
30. C3×C5C2×C4×C7direct (N × H)
31. C2×C7C3×C4×C5direct (N × H)
34. C3×C4C2×C5×C7direct (N × H)
35. C2×C5C3×C4×C7direct (N × H)
37. C2×C4C3×C5×C7direct (N × H)
38. C7C2×C3×C4×C5direct (N × H)
39. C2×C3C4×C5×C7direct (N × H)
41. C5C2×C3×C4×C7direct (N × H)
43. C4C2×C3×C5×C7direct (N × H)
44. C3C2×C4×C5×C7direct (N × H)
45. C2C3×C4×C5×C7direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3×C5×C72
C3×C4×C5×C72
C2×C4×C5×C73
C2×C3×C4×C75
C2×C3×C4×C57