Finite Group Databasebeta

← C2×C5×D3×Dic7 #13132675

Subgroups — C2×C5×D3×Dic7 #13132675

Order 1680.

216 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 112 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×C5×Dic21C211yesmaximal, characteristic, complemented
3C2×C3×C5×Dic7C211yesmaximal, characteristic, complemented
4C5×D3×Dic7C214yesmaximal, direct product factor
5C22×C5×Dic7—31nomaximal, Hall
6C22×C3×C5×C7C2211yescharacteristic
7C2×C5×D3×C7C414yescomplemented
8C2×C5×D3×C7C2212yes—
9C5×Dic21C2212yescomplemented
10C3×C5×Dic7C2212yescomplemented
11C2×D3×Dic7C511yesmaximal, characteristic, direct product factor, Hall
12C23×C5×C7—31no—
13C2×C5×Dic7—31no—
14C2×C5×Dic7—34no—
15C2×C5×Dic7D311yescharacteristic, direct product factor
16C2×C4×C5×D3—71nomaximal, Hall
17C2×C3×C5×C7C2×C412yescomplemented
18C2×C3×C5×C7C2311yescharacteristic
19C5×D3×C7C2×C414yescomplemented
20C22×D3×C7C2×C511yescharacteristic
21C2×Dic21C2×C511yescharacteristic, complemented
22C2×C3×Dic7C2×C511yescharacteristic, complemented
23D3×Dic7C2×C514yesdirect product factor
24C22×C5×C7—34no—
25C22×C5×C7—32no—
26C22×C5×C7C2×D311yescharacteristic
27C5×Dic7—32no—
28C5×Dic7C2×D312yesdirect product factor
29C22×C5×D3D711yescharacteristic
30C2×C3×C4×C5—71no—
31C2×C5×Dic3—71no—
32C4×C5×D3—74no—
33C22×Dic7—31noHall
34C3×C5×C7C22×C411yescharacteristic, complemented, Hall
35C22×C3×C7C22×C511yescharacteristic
36C2×D3×C7C4×C514yescomplemented
37C2×D3×C7C22×C512yes—
38Dic21C22×C512yescomplemented
39C3×Dic7C22×C512yescomplemented
40C22×C4×C5—211noHall
41C2×C5×C7—34no—
42C2×C5×C7C4×D312yescomplemented
43C2×C5×C7C22×D311yescharacteristic
44C22×C3×C5C2×D711yescharacteristic
45C2×C5×D3Dic714yesdirect product factor
46C2×C5×D3C2×D712yes—
47C3×C4×C5—72no—
48C5×Dic3—72no—
49C23×C7—31no—
50C2×Dic7—31no—
51C2×Dic7—34no—
52C2×Dic7C5×D311yescharacteristic, direct product factor
53C2×C4×D3—71noHall
54C2×C3×C7C2×C4×C512yescomplemented
55C2×C3×C7C23×C511yescharacteristic
56D3×C7C2×C4×C514yescomplemented
57C23×C5—31no—
58C2×C4×C5—71no—
59C2×C4×C5—214no—
60C2×C4×C5—211no—
61C5×C7C2×C4×D311yescharacteristic, complemented, Hall
62C2×C3×C5C2×Dic712yescomplemented
63C2×C3×C5C22×D711yescharacteristic
64C5×D3C2×Dic714yesdirect product factor
65C22×C7—34no—
66C22×C7—32no—
67C22×C7C2×C5×D311yescharacteristic
68Dic7—32no—
69Dic7C2×C5×D312yesdirect product factor
70C22×D3C5×D711yescharacteristic
71C2×C3×C4—71no—
72C2×Dic3—71no—
73C4×D3—74no—
74C3×C7C22×C4×C511yescharacteristic, complemented, Hall
75C22×C5—34no—
76C22×C5—32no—
77C22×C5D3×D711yescharacteristic
78C4×C5—72no—
79C4×C5—212no—
80C22×C4—211noSylow
81C3×C5C22×Dic711yescharacteristic, complemented, Hall
82C2×C7—34no—
83C2×C7C4×C5×D312yescomplemented
84C2×C7C22×C5×D311yescharacteristic
85C22×C3C2×C5×D711yescharacteristic
86C2×D3C5×Dic714yesdirect product factor
87C2×D3C2×C5×D712yes—
88C3×C4—72no—
89Dic3—72no—
90C2×C5—34no—
91C2×C5D3×Dic712yesdirect product factor
92C2×C5C2×D3×D711yescharacteristic
93C23—31no—
94C2×C4—71no—
95C2×C4—214no—
96C2×C4—211no—
97C7C2×C4×C5×D311yescharacteristic, complemented, Sylow
98C2×C3C2×C5×Dic712yescomplemented
99C2×C3C22×C5×D711yescharacteristic
100D3C2×C5×Dic714yesdirect product factor
101C5C2×D3×Dic711yescharacteristic, direct product factor, Sylow
102C22—34no—
103C22—32no—
104C22C5×D3×D711yescharacteristic
105C4—72no—
106C4—212no—
107C3C22×C5×Dic711yescharacteristic, complemented, Sylow
108C2—34no—
109C2C5×D3×Dic712yesdirect product factor
110C2C2×C5×D3×D711yescharacteristic
subgroupG/H
center, hypercenterC22×C5D3×D7
commutator, hypocenterC3×C7C22×C4×C5
FrattiniC2C2×C5×D3×D7
Fitting, socleC22×C3×C5×C7C22
prime radicalC22×C5×D3×C7C2
cosocleC2×C3×C7C23×C5
LCSG ▷ C3×C7
UCSC1 ◁ C22×C5

Complements

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

NH ≅ G/Ntype
2. C2×C5×Dic21C2semidirect
3. C2×C3×C5×Dic7C2semidirect
4. C5×D3×Dic7C2direct (N × H)
7. C2×C5×D3×C7C4semidirect
9. C5×Dic21C22semidirect
10. C3×C5×Dic7C22semidirect
11. C2×D3×Dic7C5direct (N × H)
15. C2×C5×Dic7D3direct (N × H)
17. C2×C3×C5×C7C2×C4semidirect
19. C5×D3×C7C2×C4semidirect
21. C2×Dic21C2×C5semidirect
22. C2×C3×Dic7C2×C5semidirect
23. D3×Dic7C2×C5direct (N × H)
28. C5×Dic7C2×D3direct (N × H)
34. C3×C5×C7C22×C4semidirect
36. C2×D3×C7C4×C5semidirect
38. Dic21C22×C5semidirect
39. C3×Dic7C22×C5semidirect
42. C2×C5×C7C4×D3semidirect
45. C2×C5×D3Dic7direct (N × H)
52. C2×Dic7C5×D3direct (N × H)
54. C2×C3×C7C2×C4×C5semidirect
56. D3×C7C2×C4×C5semidirect
61. C5×C7C2×C4×D3semidirect
62. C2×C3×C5C2×Dic7semidirect
64. C5×D3C2×Dic7direct (N × H)
69. Dic7C2×C5×D3direct (N × H)
74. C3×C7C22×C4×C5semidirect
81. C3×C5C22×Dic7semidirect
83. C2×C7C4×C5×D3semidirect
86. C2×D3C5×Dic7direct (N × H)
91. C2×C5D3×Dic7direct (N × H)
97. C7C2×C4×C5×D3semidirect
98. C2×C3C2×C5×Dic7semidirect
100. D3C2×C5×Dic7direct (N × H)
101. C5C2×D3×Dic7direct (N × H)
107. C3C22×C5×Dic7semidirect
109. C2C5×D3×Dic7direct (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×C5×Dic212
C2×C3×C5×Dic72
C5×D3×Dic72
C22×C5×Dic73
C2×D3×Dic75
C2×C4×C5×D37