Finite Group Databasebeta

← C2×C3×C5×C22⋊D7 #13132839

Subgroups — C2×C3×C5×C22⋊D7 #13132839

Order 1680.

216 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C23×C3×C5×C7C211yesmaximal, characteristic, complemented
2C22×C3×C5×D7C211yesmaximal, characteristic, complemented
3C3×C5×C22⋊D7C214yesmaximal, direct product factor
4C2×C3×C5×Dic7C211yesmaximal, characteristic, complemented
5C2×C5×C22⋊D7C311yesmaximal, characteristic, direct product factor, Hall
6C22×C3×C5×C7—22no—
7C22×C3×C5×C7C2211yescharacteristic
8C22×C3×C5×C7C2212yescomplemented
9C2×C3×C5×D7—22no—
10C2×C3×C5×D7C2212yescomplemented
11C3×C5×Dic7C2212yescomplemented
12C2×C3×C22⋊D7C511yesmaximal, characteristic, direct product factor, Hall
13C23×C5×C7C2×C311yescharacteristic, complemented
14C22×C5×D7C2×C311yescharacteristic, complemented
15C5×C22⋊D7C2×C314yesdirect product factor
16C2×C5×Dic7C2×C311yescharacteristic, complemented
17C2×C3×C5×D4—71nomaximal, Hall
18C2×C3×C5×C7—22no—
19C2×C3×C5×C7D412yescomplemented
20C2×C3×C5×C7C2311yescharacteristic
21C3×C5×D7—22no—
22C23×C3×C7C2×C511yescharacteristic, complemented
23C22×C3×D7C2×C511yescharacteristic, complemented
24C3×C22⋊D7C2×C514yesdirect product factor
25C2×C3×Dic7C2×C511yescharacteristic, complemented
26C22×C5×C7—22no—
27C22×C5×C7C22×C311yescharacteristic
28C22×C5×C7C22×C312yescomplemented
29C2×C5×D7—22no—
30C2×C5×D7C22×C312yescomplemented
31C5×Dic7C22×C312yescomplemented
32C23×C3×C5—71no—
33C23×C3×C5D711yescharacteristic, complemented
34C3×C5×D4—74no—
35C2×C3×C4×C5—71no—
36C2×C22⋊D7C3×C511yescharacteristic, direct product factor, Hall
37C3×C5×C7C2×D411yescharacteristic, complemented, Hall
38C22×C3×C7—22no—
39C22×C3×C7C22×C511yescharacteristic
40C22×C3×C7C22×C512yescomplemented
41C2×C3×D7—22no—
42C2×C3×D7C22×C512yescomplemented
43C3×Dic7C22×C512yescomplemented
44C2×C5×D4—71noHall
45C2×C5×C7—22no—
46C2×C5×C7C3×D412yescomplemented
47C2×C5×C7C23×C311yescharacteristic
48C5×D7—22no—
49C22×C3×C5—22no—
50C22×C3×C5—72no—
51C22×C3×C5—142no—
52C22×C3×C5C2×D711yescharacteristic
53C22×C3×C5C2×D712yescomplemented
54C3×C4×C5—72no—
55C23×C7C2×C3×C511yescharacteristic, complemented
56C22×D7C2×C3×C511yescharacteristic, complemented
57C22⋊D7C2×C3×C514yesdirect product factor
58C2×Dic7C2×C3×C511yescharacteristic, complemented
59C2×C3×D4—71noHall
60C2×C3×C7—22no—
61C2×C3×C7C5×D412yescomplemented
62C2×C3×C7C23×C511yescharacteristic
63C3×D7—22no—
64C23×C5—71no—
65C23×C5C3×D711yescharacteristic, complemented
66C5×D4—74no—
67C2×C4×C5—71no—
68C5×C7C2×C3×D411yescharacteristic, complemented, Hall
69C2×C3×C5—22no—
70C2×C3×C5—142no—
71C2×C3×C5C22⋊D712yesdirect product factor
72C2×C3×C5C22×D711yescharacteristic
73C22×C7—22no—
74C22×C7C22×C3×C511yescharacteristic
75C22×C7C22×C3×C512yescomplemented
76C2×D7—22no—
77C2×D7C22×C3×C512yescomplemented
78Dic7C22×C3×C512yescomplemented
79C23×C3—71no—
80C23×C3C5×D711yescharacteristic, complemented
81C3×D4—74no—
82C2×C3×C4—71no—
83C3×C7C2×C5×D411yescharacteristic, complemented, Hall
84C22×C5—22no—
85C22×C5—72no—
86C22×C5—142no—
87C22×C5C2×C3×D711yescharacteristic
88C22×C5C2×C3×D712yescomplemented
89C4×C5—72no—
90C2×D4—71noSylow
91C3×C5C2×C22⋊D711yescharacteristic, direct product factor, Hall
92C2×C7—22no—
93C2×C7C3×C5×D412yescomplemented
94C2×C7C23×C3×C511yescharacteristic
95D7—22no—
96C22×C3—22no—
97C22×C3—72no—
98C22×C3—142no—
99C22×C3C2×C5×D711yescharacteristic
100C22×C3C2×C5×D712yescomplemented
101C3×C4—72no—
102C2×C5—22no—
103C2×C5—142no—
104C2×C5C3×C22⋊D712yesdirect product factor
105C2×C5C22×C3×D711yescharacteristic
106C23—71no—
107C23C3×C5×D711yescharacteristic, complemented
108D4—74no—
109C2×C4—71no—
110C7C2×C3×C5×D411yescharacteristic, complemented, Sylow
111C2×C3—22no—
112C2×C3—142no—
113C2×C3C5×C22⋊D712yesdirect product factor
114C2×C3C22×C5×D711yescharacteristic
115C5C2×C3×C22⋊D711yescharacteristic, direct product factor, Sylow
116C22—22no—
117C22—72no—
118C22—142no—
119C22C2×C3×C5×D711yescharacteristic
120C22C2×C3×C5×D712yescomplemented
121C4—72no—
122C3C2×C5×C22⋊D711yescharacteristic, direct product factor, Sylow
123C2—22no—
124C2—142no—
125C2C3×C5×C22⋊D712yesdirect product factor
126C2C22×C3×C5×D711yescharacteristic
subgroupG/H
centerC22×C3×C5C2×D7
commutator, cosocleC2×C7C23×C3×C5
FrattiniC2C22×C3×C5×D7
FittingC23×C3×C5×C7C2
socleC22×C3×C5×C7C22
hypercenterC23×C3×C5D7
hypocenterC7C2×C3×C5×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3×C5 ◁ C23×C3×C5

Complements

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

NH ≅ G/Ntype
1. C23×C3×C5×C7C2semidirect
2. C22×C3×C5×D7C2semidirect
3. C3×C5×C22⋊D7C2direct (N × H)
4. C2×C3×C5×Dic7C2semidirect
5. C2×C5×C22⋊D7C3direct (N × H)
8. C22×C3×C5×C7C22semidirect
10. C2×C3×C5×D7C22semidirect
11. C3×C5×Dic7C22semidirect
12. C2×C3×C22⋊D7C5direct (N × H)
13. C23×C5×C7C2×C3semidirect
14. C22×C5×D7C2×C3semidirect
15. C5×C22⋊D7C2×C3direct (N × H)
16. C2×C5×Dic7C2×C3semidirect
19. C2×C3×C5×C7D4semidirect
22. C23×C3×C7C2×C5semidirect
23. C22×C3×D7C2×C5semidirect
24. C3×C22⋊D7C2×C5direct (N × H)
25. C2×C3×Dic7C2×C5semidirect
28. C22×C5×C7C22×C3semidirect
30. C2×C5×D7C22×C3semidirect
31. C5×Dic7C22×C3semidirect
33. C23×C3×C5D7semidirect
36. C2×C22⋊D7C3×C5direct (N × H)
37. C3×C5×C7C2×D4semidirect
40. C22×C3×C7C22×C5semidirect
42. C2×C3×D7C22×C5semidirect
43. C3×Dic7C22×C5semidirect
46. C2×C5×C7C3×D4semidirect
53. C22×C3×C5C2×D7semidirect
55. C23×C7C2×C3×C5semidirect
56. C22×D7C2×C3×C5semidirect
57. C22⋊D7C2×C3×C5direct (N × H)
58. C2×Dic7C2×C3×C5semidirect
61. C2×C3×C7C5×D4semidirect
65. C23×C5C3×D7semidirect
68. C5×C7C2×C3×D4semidirect
71. C2×C3×C5C22⋊D7direct (N × H)
75. C22×C7C22×C3×C5semidirect
77. C2×D7C22×C3×C5semidirect
78. Dic7C22×C3×C5semidirect
80. C23×C3C5×D7semidirect
83. C3×C7C2×C5×D4semidirect
88. C22×C5C2×C3×D7semidirect
91. C3×C5C2×C22⋊D7direct (N × H)
93. C2×C7C3×C5×D4semidirect
100. C22×C3C2×C5×D7semidirect
104. C2×C5C3×C22⋊D7direct (N × H)
107. C23C3×C5×D7semidirect
110. C7C2×C3×C5×D4semidirect
113. C2×C3C5×C22⋊D7direct (N × H)
115. C5C2×C3×C22⋊D7direct (N × H)
120. C22C2×C3×C5×D7semidirect
122. C3C2×C5×C22⋊D7direct (N × H)
125. C2C3×C5×C22⋊D7direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C5×C72
C22×C3×C5×D72
C3×C5×C22⋊D72
C2×C3×C5×Dic72
C2×C5×C22⋊D73
C2×C3×C22⋊D75
C2×C3×C5×D47