Finite Group Databasebeta

← C2×C3×C4×C7×D5 #13132841

Subgroups — C2×C3×C4×C7×D5 #13132841

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×C3×C7×D5C211yesmaximal, characteristic
2C2×C3×C4×C5×C7C211yesmaximal, characteristic, complemented
3C2×C3×C7×Dic5C211yesmaximal, characteristic, complemented
4C3×C4×C7×D5C214yesmaximal, direct product factor
5C2×C4×C7×D5C311yesmaximal, characteristic, direct product factor, Hall
6C22×C3×C5×C7C2211yescharacteristic
7C2×C3×C7×D5C414yesdirect product factor
8C2×C3×C7×D5C2212yes—
9C3×C4×C5×C7C2212yescomplemented
10C3×C7×Dic5C2212yescomplemented
11C22×C3×C4×C7—51nomaximal, Hall
12C22×C7×D5C2×C311yescharacteristic
13C2×C4×C5×C7C2×C311yescharacteristic, complemented
14C2×C7×Dic5C2×C311yescharacteristic, complemented
15C4×C7×D5C2×C314yesdirect product factor
16C2×C3×C4×D5C711yesmaximal, characteristic, direct product factor, Hall
17C2×C3×C5×C7C2×C412yescomplemented
18C2×C3×C5×C7C2311yescharacteristic
19C3×C7×D5C2×C414yesdirect product factor
20C23×C3×C7—51no—
21C2×C3×C4×C7—51no—
22C2×C3×C4×C7—54no—
23C2×C3×C4×C7D511yescharacteristic, direct product factor
24C22×C5×C7C22×C311yescharacteristic
25C2×C7×D5C3×C414yesdirect product factor
26C2×C7×D5C22×C312yes—
27C4×C5×C7C22×C312yescomplemented
28C7×Dic5C22×C312yescomplemented
29C22×C3×D5C2×C711yescharacteristic
30C2×C3×C4×C5C2×C711yescharacteristic, complemented
31C2×C3×Dic5C2×C711yescharacteristic, complemented
32C3×C4×D5C2×C714yesdirect product factor
33C22×C4×C7—51noHall
34C3×C5×C7C22×C411yescharacteristic, complemented, Hall
35C22×C3×C7—54no—
36C22×C3×C7—52no—
37C22×C3×C7C2×D511yescharacteristic
38C3×C4×C7—52no—
39C3×C4×C7C2×D512yesdirect product factor
40C2×C4×D5C3×C711yescharacteristic, direct product factor, Hall
41C2×C5×C7C2×C3×C412yescomplemented
42C2×C5×C7C23×C311yescharacteristic
43C7×D5C2×C3×C414yesdirect product factor
44C22×C3×C5C22×C711yescharacteristic
45C2×C3×D5C4×C714yesdirect product factor
46C2×C3×D5C22×C712yes—
47C3×C4×C5C22×C712yescomplemented
48C3×Dic5C22×C712yescomplemented
49C23×C7—51no—
50C2×C4×C7—51no—
51C2×C4×C7—54no—
52C2×C4×C7C3×D511yescharacteristic, direct product factor
53C22×C3×C4—51noHall
54C2×C3×C7—54no—
55C2×C3×C7C4×D512yesdirect product factor
56C2×C3×C7C22×D511yescharacteristic
57C22×D5C2×C3×C711yescharacteristic
58C2×C4×C5C2×C3×C711yescharacteristic, complemented
59C2×Dic5C2×C3×C711yescharacteristic, complemented
60C4×D5C2×C3×C714yesdirect product factor
61C5×C7C22×C3×C411yescharacteristic, complemented, Hall
62C2×C3×C5C2×C4×C712yescomplemented
63C2×C3×C5C23×C711yescharacteristic
64C3×D5C2×C4×C714yesdirect product factor
65C22×C7—54no—
66C22×C7—52no—
67C22×C7C2×C3×D511yescharacteristic
68C4×C7—52no—
69C4×C7C2×C3×D512yesdirect product factor
70C23×C3—51no—
71C2×C3×C4—51no—
72C2×C3×C4—54no—
73C2×C3×C4C7×D511yescharacteristic, direct product factor
74C3×C7C2×C4×D511yescharacteristic, direct product factor, Hall
75C22×C5C22×C3×C711yescharacteristic
76C2×D5C3×C4×C714yesdirect product factor
77C2×D5C22×C3×C712yes—
78C4×C5C22×C3×C712yescomplemented
79Dic5C22×C3×C712yescomplemented
80C22×C4—51noSylow
81C3×C5C22×C4×C711yescharacteristic, complemented, Hall
82C2×C7—54no—
83C2×C7C3×C4×D512yesdirect product factor
84C2×C7C22×C3×D511yescharacteristic
85C22×C3—54no—
86C22×C3—52no—
87C22×C3C2×C7×D511yescharacteristic
88C3×C4—52no—
89C3×C4C2×C7×D512yesdirect product factor
90C2×C5C2×C3×C4×C712yescomplemented
91C2×C5C23×C3×C711yescharacteristic
92D5C2×C3×C4×C714yesdirect product factor
93C23—51no—
94C2×C4—51no—
95C2×C4—54no—
96C2×C4C3×C7×D511yescharacteristic, direct product factor
97C7C2×C3×C4×D511yescharacteristic, direct product factor, Sylow
98C2×C3—54no—
99C2×C3C4×C7×D512yesdirect product factor
100C2×C3C22×C7×D511yescharacteristic
101C5C22×C3×C4×C711yescharacteristic, complemented, Sylow
102C22—54no—
103C22—52no—
104C22C2×C3×C7×D511yescharacteristic
105C4—52no—
106C4C2×C3×C7×D512yesdirect product factor
107C3C2×C4×C7×D511yescharacteristic, direct product factor, Sylow
108C2—54no—
109C2C3×C4×C7×D512yesdirect product factor
110C2C22×C3×C7×D511yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4×C7D5
commutator, hypocenterC5C22×C3×C4×C7
FrattiniC2C22×C3×C7×D5
FittingC2×C3×C4×C5×C7C2
socleC22×C3×C5×C7C22
prime radicalC22×C3×C7×D5C2
cosocleC2×C5C23×C3×C7
LCSG ▷ C5
UCSC1 ◁ C2×C3×C4×C7

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×C5×C7C2semidirect
3. C2×C3×C7×Dic5C2semidirect
4. C3×C4×C7×D5C2direct (N × H)
5. C2×C4×C7×D5C3direct (N × H)
7. C2×C3×C7×D5C4direct (N × H)
9. C3×C4×C5×C7C22semidirect
10. C3×C7×Dic5C22semidirect
13. C2×C4×C5×C7C2×C3semidirect
14. C2×C7×Dic5C2×C3semidirect
15. C4×C7×D5C2×C3direct (N × H)
16. C2×C3×C4×D5C7direct (N × H)
17. C2×C3×C5×C7C2×C4semidirect
19. C3×C7×D5C2×C4direct (N × H)
23. C2×C3×C4×C7D5direct (N × H)
25. C2×C7×D5C3×C4direct (N × H)
27. C4×C5×C7C22×C3semidirect
28. C7×Dic5C22×C3semidirect
30. C2×C3×C4×C5C2×C7semidirect
31. C2×C3×Dic5C2×C7semidirect
32. C3×C4×D5C2×C7direct (N × H)
34. C3×C5×C7C22×C4semidirect
39. C3×C4×C7C2×D5direct (N × H)
40. C2×C4×D5C3×C7direct (N × H)
41. C2×C5×C7C2×C3×C4semidirect
43. C7×D5C2×C3×C4direct (N × H)
45. C2×C3×D5C4×C7direct (N × H)
47. C3×C4×C5C22×C7semidirect
48. C3×Dic5C22×C7semidirect
52. C2×C4×C7C3×D5direct (N × H)
55. C2×C3×C7C4×D5direct (N × H)
58. C2×C4×C5C2×C3×C7semidirect
59. C2×Dic5C2×C3×C7semidirect
60. C4×D5C2×C3×C7direct (N × H)
61. C5×C7C22×C3×C4semidirect
62. C2×C3×C5C2×C4×C7semidirect
64. C3×D5C2×C4×C7direct (N × H)
69. C4×C7C2×C3×D5direct (N × H)
73. C2×C3×C4C7×D5direct (N × H)
74. C3×C7C2×C4×D5direct (N × H)
76. C2×D5C3×C4×C7direct (N × H)
78. C4×C5C22×C3×C7semidirect
79. Dic5C22×C3×C7semidirect
81. C3×C5C22×C4×C7semidirect
83. C2×C7C3×C4×D5direct (N × H)
89. C3×C4C2×C7×D5direct (N × H)
90. C2×C5C2×C3×C4×C7semidirect
92. D5C2×C3×C4×C7direct (N × H)
96. C2×C4C3×C7×D5direct (N × H)
97. C7C2×C3×C4×D5direct (N × H)
99. C2×C3C4×C7×D5direct (N × H)
101. C5C22×C3×C4×C7semidirect
106. C4C2×C3×C7×D5direct (N × H)
107. C3C2×C4×C7×D5direct (N × H)
109. C2C3×C4×C7×D5direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C7×D52
C2×C3×C4×C5×C72
C2×C3×C7×Dic52
C3×C4×C7×D52
C2×C4×C7×D53
C22×C3×C4×C75
C2×C3×C4×D57