Finite Group Databasebeta

← C2×C4×C5×D3×C7 #13132861

Subgroups — C2×C4×C5×D3×C7 #13132861

Order 1680.

216 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×D3×C7C211yesmaximal, characteristic
2C2×C3×C4×C5×C7C211yesmaximal, characteristic, complemented
3C2×C5×C7×Dic3C211yesmaximal, characteristic, complemented
4C4×C5×D3×C7C214yesmaximal, direct product factor
5C22×C4×C5×C7—31nomaximal, Hall
6C22×C3×C5×C7C2211yescharacteristic
7C2×C5×D3×C7C414yesdirect product factor
8C2×C5×D3×C7C2212yes—
9C3×C4×C5×C7C2212yescomplemented
10C5×C7×Dic3C2212yescomplemented
11C2×C4×D3×C7C511yesmaximal, characteristic, direct product factor, Hall
12C23×C5×C7—31no—
13C2×C4×C5×C7—31no—
14C2×C4×C5×C7—34no—
15C2×C4×C5×C7D311yescharacteristic, direct product factor
16C2×C4×C5×D3C711yesmaximal, characteristic, direct product factor, Hall
17C2×C3×C5×C7C2×C412yescomplemented
18C2×C3×C5×C7C2311yescharacteristic
19C5×D3×C7C2×C414yesdirect product factor
20C22×D3×C7C2×C511yescharacteristic
21C2×C3×C4×C7C2×C511yescharacteristic, complemented
22C2×C7×Dic3C2×C511yescharacteristic, complemented
23C4×D3×C7C2×C514yesdirect product factor
24C22×C5×C7—34no—
25C22×C5×C7—32no—
26C22×C5×C7C2×D311yescharacteristic
27C4×C5×C7—32no—
28C4×C5×C7C2×D312yesdirect product factor
29C22×C5×D3C2×C711yescharacteristic
30C2×C3×C4×C5C2×C711yescharacteristic, complemented
31C2×C5×Dic3C2×C711yescharacteristic, complemented
32C4×C5×D3C2×C714yesdirect product factor
33C22×C4×C7—31noHall
34C3×C5×C7C22×C411yescharacteristic, complemented, Hall
35C22×C3×C7C22×C511yescharacteristic
36C2×D3×C7C4×C514yesdirect product factor
37C2×D3×C7C22×C512yes—
38C3×C4×C7C22×C512yescomplemented
39C7×Dic3C22×C512yescomplemented
40C22×C4×C5—31noHall
41C2×C5×C7—34no—
42C2×C5×C7C4×D312yesdirect product factor
43C2×C5×C7C22×D311yescharacteristic
44C22×C3×C5C22×C711yescharacteristic
45C2×C5×D3C4×C714yesdirect product factor
46C2×C5×D3C22×C712yes—
47C3×C4×C5C22×C712yescomplemented
48C5×Dic3C22×C712yescomplemented
49C23×C7—31no—
50C2×C4×C7—31no—
51C2×C4×C7—34no—
52C2×C4×C7C5×D311yescharacteristic, direct product factor
53C2×C4×D3C5×C711yescharacteristic, direct product factor, Hall
54C2×C3×C7C2×C4×C512yescomplemented
55C2×C3×C7C23×C511yescharacteristic
56D3×C7C2×C4×C514yesdirect product factor
57C23×C5—31no—
58C2×C4×C5—31no—
59C2×C4×C5—34no—
60C2×C4×C5D3×C711yescharacteristic, direct product factor
61C5×C7C2×C4×D311yescharacteristic, direct product factor, Hall
62C2×C3×C5C2×C4×C712yescomplemented
63C2×C3×C5C23×C711yescharacteristic
64C5×D3C2×C4×C714yesdirect product factor
65C22×C7—34no—
66C22×C7—32no—
67C22×C7C2×C5×D311yescharacteristic
68C4×C7—32no—
69C4×C7C2×C5×D312yesdirect product factor
70C22×D3C2×C5×C711yescharacteristic
71C2×C3×C4C2×C5×C711yescharacteristic, complemented
72C2×Dic3C2×C5×C711yescharacteristic, complemented
73C4×D3C2×C5×C714yesdirect product factor
74C3×C7C22×C4×C511yescharacteristic, complemented, Hall
75C22×C5—34no—
76C22×C5—32no—
77C22×C5C2×D3×C711yescharacteristic
78C4×C5—32no—
79C4×C5C2×D3×C712yesdirect product factor
80C22×C4—31noSylow
81C3×C5C22×C4×C711yescharacteristic, complemented, Hall
82C2×C7—34no—
83C2×C7C4×C5×D312yesdirect product factor
84C2×C7C22×C5×D311yescharacteristic
85C22×C3C22×C5×C711yescharacteristic
86C2×D3C4×C5×C714yesdirect product factor
87C2×D3C22×C5×C712yes—
88C3×C4C22×C5×C712yescomplemented
89Dic3C22×C5×C712yescomplemented
90C2×C5—34no—
91C2×C5C4×D3×C712yesdirect product factor
92C2×C5C22×D3×C711yescharacteristic
93C23—31no—
94C2×C4—31no—
95C2×C4—34no—
96C2×C4C5×D3×C711yescharacteristic, direct product factor
97C7C2×C4×C5×D311yescharacteristic, direct product factor, Sylow
98C2×C3C2×C4×C5×C712yescomplemented
99C2×C3C23×C5×C711yescharacteristic
100D3C2×C4×C5×C714yesdirect product factor
101C5C2×C4×D3×C711yescharacteristic, direct product factor, Sylow
102C22—34no—
103C22—32no—
104C22C2×C5×D3×C711yescharacteristic
105C4—32no—
106C4C2×C5×D3×C712yesdirect product factor
107C3C22×C4×C5×C711yescharacteristic, complemented, Sylow
108C2—34no—
109C2C4×C5×D3×C712yesdirect product factor
110C2C22×C5×D3×C711yescharacteristic
subgroupG/H
center, hypercenterC2×C4×C5×C7D3
commutator, hypocenterC3C22×C4×C5×C7
FrattiniC2C22×C5×D3×C7
FittingC2×C3×C4×C5×C7C2
socleC22×C3×C5×C7C22
prime radicalC22×C5×D3×C7C2
cosocleC2×C3C23×C5×C7
LCSG ▷ C3
UCSC1 ◁ C2×C4×C5×C7

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×C5×C7C2semidirect
3. C2×C5×C7×Dic3C2semidirect
4. C4×C5×D3×C7C2direct (N × H)
7. C2×C5×D3×C7C4direct (N × H)
9. C3×C4×C5×C7C22semidirect
10. C5×C7×Dic3C22semidirect
11. C2×C4×D3×C7C5direct (N × H)
15. C2×C4×C5×C7D3direct (N × H)
16. C2×C4×C5×D3C7direct (N × H)
17. C2×C3×C5×C7C2×C4semidirect
19. C5×D3×C7C2×C4direct (N × H)
21. C2×C3×C4×C7C2×C5semidirect
22. C2×C7×Dic3C2×C5semidirect
23. C4×D3×C7C2×C5direct (N × H)
28. C4×C5×C7C2×D3direct (N × H)
30. C2×C3×C4×C5C2×C7semidirect
31. C2×C5×Dic3C2×C7semidirect
32. C4×C5×D3C2×C7direct (N × H)
34. C3×C5×C7C22×C4semidirect
36. C2×D3×C7C4×C5direct (N × H)
38. C3×C4×C7C22×C5semidirect
39. C7×Dic3C22×C5semidirect
42. C2×C5×C7C4×D3direct (N × H)
45. C2×C5×D3C4×C7direct (N × H)
47. C3×C4×C5C22×C7semidirect
48. C5×Dic3C22×C7semidirect
52. C2×C4×C7C5×D3direct (N × H)
53. C2×C4×D3C5×C7direct (N × H)
54. C2×C3×C7C2×C4×C5semidirect
56. D3×C7C2×C4×C5direct (N × H)
60. C2×C4×C5D3×C7direct (N × H)
61. C5×C7C2×C4×D3direct (N × H)
62. C2×C3×C5C2×C4×C7semidirect
64. C5×D3C2×C4×C7direct (N × H)
69. C4×C7C2×C5×D3direct (N × H)
71. C2×C3×C4C2×C5×C7semidirect
72. C2×Dic3C2×C5×C7semidirect
73. C4×D3C2×C5×C7direct (N × H)
74. C3×C7C22×C4×C5semidirect
79. C4×C5C2×D3×C7direct (N × H)
81. C3×C5C22×C4×C7semidirect
83. C2×C7C4×C5×D3direct (N × H)
86. C2×D3C4×C5×C7direct (N × H)
88. C3×C4C22×C5×C7semidirect
89. Dic3C22×C5×C7semidirect
91. C2×C5C4×D3×C7direct (N × H)
96. C2×C4C5×D3×C7direct (N × H)
97. C7C2×C4×C5×D3direct (N × H)
98. C2×C3C2×C4×C5×C7semidirect
100. D3C2×C4×C5×C7direct (N × H)
101. C5C2×C4×D3×C7direct (N × H)
106. C4C2×C5×D3×C7direct (N × H)
107. C3C22×C4×C5×C7semidirect
109. C2C4×C5×D3×C7direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×D3×C72
C2×C3×C4×C5×C72
C2×C5×C7×Dic32
C4×C5×D3×C72
C22×C4×C5×C73
C2×C4×D3×C75
C2×C4×C5×D37