Finite Group Databasebeta

← C2×C4×C7×D15 #13132881

Subgroups — C2×C4×C7×D15 #13132881

Order 1680.

216 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 102 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×C7×D15C211yesmaximal, characteristic
2C2×C3×C4×C5×C7C211yesmaximal, characteristic, complemented
3C2×C7×Dic15C211yesmaximal, characteristic, complemented
4C4×C7×D15C214yesmaximal, direct product factor
5C2×C4×C7×D5—31nomaximal, Hall
6C22×C3×C5×C7C2211yescharacteristic
7C2×C7×D15C414yesdirect product factor
8C2×C7×D15C2212yes—
9C3×C4×C5×C7C2212yescomplemented
10C7×Dic15C2212yescomplemented
11C2×C4×D3×C7—51nomaximal, Hall
12C22×C7×D5—31no—
13C2×C4×C5×C7D311yescharacteristic, complemented
14C2×C7×Dic5—31no—
15C4×C7×D5—34no—
16C2×C4×D15C711yesmaximal, characteristic, direct product factor, Hall
17C2×C3×C5×C7C2×C412yescomplemented
18C2×C3×C5×C7C2311yescharacteristic
19C7×D15C2×C414yesdirect product factor
20C22×D3×C7—51no—
21C2×C3×C4×C7D511yescharacteristic, complemented
22C2×C7×Dic3—51no—
23C4×D3×C7—54no—
24C22×C5×C7C2×D311yescharacteristic
25C2×C7×D5—34no—
26C2×C7×D5—32no—
27C4×C5×C7C2×D312yescomplemented
28C7×Dic5—32no—
29C22×D15C2×C711yescharacteristic
30C2×C3×C4×C5C2×C711yescharacteristic, complemented
31C2×Dic15C2×C711yescharacteristic, complemented
32C4×D15C2×C714yesdirect product factor
33C22×C4×C7—151noHall
34C3×C5×C7C22×C411yescharacteristic, complemented, Hall
35C22×C3×C7C2×D511yescharacteristic
36C2×D3×C7—54no—
37C2×D3×C7—52no—
38C3×C4×C7C2×D512yescomplemented
39C7×Dic3—52no—
40C2×C4×D5—31noHall
41C2×C5×C7C4×D312yescomplemented
42C2×C5×C7C22×D311yescharacteristic
43C7×D5—34no—
44C22×C3×C5C22×C711yescharacteristic
45C2×D15C4×C714yesdirect product factor
46C2×D15C22×C712yes—
47C3×C4×C5C22×C712yescomplemented
48Dic15C22×C712yescomplemented
49C23×C7—151no—
50C2×C4×C7—151no—
51C2×C4×C7—154no—
52C2×C4×C7D1511yescharacteristic, direct product factor
53C2×C4×D3—51noHall
54C2×C3×C7C4×D512yescomplemented
55C2×C3×C7C22×D511yescharacteristic
56D3×C7—54no—
57C22×D5—31no—
58C2×C4×C5D3×C711yescharacteristic, complemented
59C2×Dic5—31no—
60C4×D5—34no—
61C5×C7C2×C4×D311yescharacteristic, complemented, Hall
62C2×C3×C5C2×C4×C712yescomplemented
63C2×C3×C5C23×C711yescharacteristic
64D15C2×C4×C714yesdirect product factor
65C22×C7—154no—
66C22×C7—152no—
67C22×C7C2×D1511yescharacteristic
68C4×C7—152no—
69C4×C7C2×D1512yesdirect product factor
70C22×D3—51no—
71C2×C3×C4C7×D511yescharacteristic, complemented
72C2×Dic3—51no—
73C4×D3—54no—
74C3×C7C2×C4×D511yescharacteristic, complemented, Hall
75C22×C5C2×D3×C711yescharacteristic
76C2×D5—34no—
77C2×D5—32no—
78C4×C5C2×D3×C712yescomplemented
79Dic5—32no—
80C22×C4—151noSylow
81C3×C5C22×C4×C711yescharacteristic, complemented, Hall
82C2×C7—154no—
83C2×C7C4×D1512yesdirect product factor
84C2×C7C22×D1511yescharacteristic
85C22×C3C2×C7×D511yescharacteristic
86C2×D3—54no—
87C2×D3—52no—
88C3×C4C2×C7×D512yescomplemented
89Dic3—52no—
90C2×C5C4×D3×C712yescomplemented
91C2×C5C22×D3×C711yescharacteristic
92D5—34no—
93C23—151no—
94C2×C4—151no—
95C2×C4—154no—
96C2×C4C7×D1511yescharacteristic, direct product factor
97C7C2×C4×D1511yescharacteristic, direct product factor, Sylow
98C2×C3C4×C7×D512yescomplemented
99C2×C3C22×C7×D511yescharacteristic
100D3—54no—
101C5C2×C4×D3×C711yescharacteristic, complemented, Sylow
102C22—154no—
103C22—152no—
104C22C2×C7×D1511yescharacteristic
105C4—152no—
106C4C2×C7×D1512yesdirect product factor
107C3C2×C4×C7×D511yescharacteristic, complemented, Sylow
108C2—154no—
109C2C4×C7×D1512yesdirect product factor
110C2C22×C7×D1511yescharacteristic
subgroupG/H
center, hypercenterC2×C4×C7D15
commutator, hypocenterC3×C5C22×C4×C7
FrattiniC2C22×C7×D15
FittingC2×C3×C4×C5×C7C2
socleC22×C3×C5×C7C22
prime radicalC22×C7×D15C2
cosocleC2×C3×C5C23×C7
LCSG ▷ C3×C5
UCSC1 ◁ C2×C4×C7

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×C5×C7C2semidirect
3. C2×C7×Dic15C2semidirect
4. C4×C7×D15C2direct (N × H)
7. C2×C7×D15C4direct (N × H)
9. C3×C4×C5×C7C22semidirect
10. C7×Dic15C22semidirect
13. C2×C4×C5×C7D3semidirect
16. C2×C4×D15C7direct (N × H)
17. C2×C3×C5×C7C2×C4semidirect
19. C7×D15C2×C4direct (N × H)
21. C2×C3×C4×C7D5semidirect
27. C4×C5×C7C2×D3semidirect
30. C2×C3×C4×C5C2×C7semidirect
31. C2×Dic15C2×C7semidirect
32. C4×D15C2×C7direct (N × H)
34. C3×C5×C7C22×C4semidirect
38. C3×C4×C7C2×D5semidirect
41. C2×C5×C7C4×D3semidirect
45. C2×D15C4×C7direct (N × H)
47. C3×C4×C5C22×C7semidirect
48. Dic15C22×C7semidirect
52. C2×C4×C7D15direct (N × H)
54. C2×C3×C7C4×D5semidirect
58. C2×C4×C5D3×C7semidirect
61. C5×C7C2×C4×D3semidirect
62. C2×C3×C5C2×C4×C7semidirect
64. D15C2×C4×C7direct (N × H)
69. C4×C7C2×D15direct (N × H)
71. C2×C3×C4C7×D5semidirect
74. C3×C7C2×C4×D5semidirect
78. C4×C5C2×D3×C7semidirect
81. C3×C5C22×C4×C7semidirect
83. C2×C7C4×D15direct (N × H)
88. C3×C4C2×C7×D5semidirect
90. C2×C5C4×D3×C7semidirect
96. C2×C4C7×D15direct (N × H)
97. C7C2×C4×D15direct (N × H)
98. C2×C3C4×C7×D5semidirect
101. C5C2×C4×D3×C7semidirect
106. C4C2×C7×D15direct (N × H)
107. C3C2×C4×C7×D5semidirect
109. C2C4×C7×D15direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×D152
C2×C3×C4×C5×C72
C2×C7×Dic152
C4×C7×D152
C2×C4×C7×D53
C2×C4×D3×C75
C2×C4×D157