Finite Group Databasebeta

← C2×C7×#1220 #13132889

Subgroups — C2×C7×#1220 #13132889

Order 1680.

216 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 86 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×C7×D15C211yesmaximal, characteristic, complemented
3C7×#1220C214yesmaximal, direct product factor
4C2×C7×Dic15C211yesmaximal, characteristic, complemented
5C2×C7×C22⋊D5—31nomaximal, Hall
6C22×C3×C5×C7—22no—
7C22×C3×C5×C7C2211yescharacteristic
8C22×C3×C5×C7C2212yescomplemented
9C2×C7×D15—22no—
10C2×C7×D15C2212yescomplemented
11C7×Dic15C2212yescomplemented
12C2×C7×#67—51nomaximal, Hall
13C23×C5×C7D311yescharacteristic, complemented
14C22×C7×D5—31no—
15C7×C22⋊D5—34no—
16C2×C7×Dic5—31no—
17C2×#1220C711yesmaximal, characteristic, direct product factor, Hall
18C2×C3×C5×C7—22no—
19C2×C3×C5×C7D412yescomplemented
20C2×C3×C5×C7C2311yescharacteristic
21C7×D15—22no—
22C23×C3×C7D511yescharacteristic, complemented
23C22×D3×C7—51no—
24C7×#67—54no—
25C2×C7×Dic3—51no—
26C22×C5×C7—22no—
27C22×C5×C7C2×D311yescharacteristic
28C22×C5×C7C2×D312yescomplemented
29C2×C7×D5—32no—
30C2×C7×D5—62no—
31C7×Dic5—32no—
32C23×C3×C5C2×C711yescharacteristic, complemented
33C22×D15C2×C711yescharacteristic, complemented
34#1220C2×C714yesdirect product factor
35C2×Dic15C2×C711yescharacteristic, complemented
36C2×C7×D4—151noHall
37C3×C5×C7C2×D411yescharacteristic, complemented, Hall
38C22×C3×C7—22no—
39C22×C3×C7C2×D511yescharacteristic
40C22×C3×C7C2×D512yescomplemented
41C2×D3×C7—52no—
42C2×D3×C7—102no—
43C7×Dic3—52no—
44C2×C22⋊D5—31noHall
45C2×C5×C7—22no—
46C2×C5×C7#6712yescomplemented
47C2×C5×C7C22×D311yescharacteristic
48C7×D5—62no—
49C22×C3×C5—22no—
50C22×C3×C5C22×C711yescharacteristic
51C22×C3×C5C22×C712yescomplemented
52C2×D15—22no—
53C2×D15C22×C712yescomplemented
54Dic15C22×C712yescomplemented
55C23×C7—151no—
56C23×C7D1511yescharacteristic, complemented
57C7×D4—154no—
58C2×C4×C7—151no—
59C2×#67—51noHall
60C2×C3×C7—22no—
61C2×C3×C7C22⋊D512yescomplemented
62C2×C3×C7C22×D511yescharacteristic
63D3×C7—102no—
64C23×C5D3×C711yescharacteristic, complemented
65C22×D5—31no—
66C22⋊D5—34no—
67C2×Dic5—31no—
68C5×C7C2×#6711yescharacteristic, complemented, Hall
69C2×C3×C5—22no—
70C2×C3×C5C7×D412yescomplemented
71C2×C3×C5C23×C711yescharacteristic
72D15—22no—
73C22×C7—22no—
74C22×C7—152no—
75C22×C7—302no—
76C22×C7C2×D1511yescharacteristic
77C22×C7C2×D1512yescomplemented
78C4×C7—152no—
79C23×C3C7×D511yescharacteristic, complemented
80C22×D3—51no—
81#67—54no—
82C2×Dic3—51no—
83C3×C7C2×C22⋊D511yescharacteristic, complemented, Hall
84C22×C5—22no—
85C22×C5C2×D3×C711yescharacteristic
86C22×C5C2×D3×C712yescomplemented
87C2×D5—32no—
88C2×D5—62no—
89Dic5—32no—
90C2×D4—151noSylow
91C3×C5C2×C7×D411yescharacteristic, complemented, Hall
92C2×C7—22no—
93C2×C7—302no—
94C2×C7#122012yesdirect product factor
95C2×C7C22×D1511yescharacteristic
96C22×C3—22no—
97C22×C3C2×C7×D511yescharacteristic
98C22×C3C2×C7×D512yescomplemented
99C2×D3—52no—
100C2×D3—102no—
101Dic3—52no—
102C2×C5—22no—
103C2×C5C7×#6712yescomplemented
104C2×C5C22×D3×C711yescharacteristic
105D5—62no—
106C23—151no—
107C23C7×D1511yescharacteristic, complemented
108D4—154no—
109C2×C4—151no—
110C7C2×#122011yescharacteristic, direct product factor, Sylow
111C2×C3—22no—
112C2×C3C7×C22⋊D512yescomplemented
113C2×C3C22×C7×D511yescharacteristic
114D3—102no—
115C5C2×C7×#6711yescharacteristic, complemented, Sylow
116C22—22no—
117C22—152no—
118C22—302no—
119C22C2×C7×D1511yescharacteristic
120C22C2×C7×D1512yescomplemented
121C4—152no—
122C3C2×C7×C22⋊D511yescharacteristic, complemented, Sylow
123C2—22no—
124C2—302no—
125C2C7×#122012yesdirect product factor
126C2C22×C7×D1511yescharacteristic
subgroupG/H
centerC22×C7C2×D15
commutator, cosocleC2×C3×C5C23×C7
FrattiniC2C22×C7×D15
FittingC23×C3×C5×C7C2
socleC22×C3×C5×C7C22
hypercenterC23×C7D15
hypocenterC3×C5C2×C7×D4
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C7 ◁ C23×C7

Complements

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

NH ≅ G/Ntype
1. C23×C3×C5×C7C2semidirect
2. C22×C7×D15C2semidirect
3. C7×#1220C2direct (N × H)
4. C2×C7×Dic15C2semidirect
8. C22×C3×C5×C7C22semidirect
10. C2×C7×D15C22semidirect
11. C7×Dic15C22semidirect
13. C23×C5×C7D3semidirect
17. C2×#1220C7direct (N × H)
19. C2×C3×C5×C7D4semidirect
22. C23×C3×C7D5semidirect
28. C22×C5×C7C2×D3semidirect
32. C23×C3×C5C2×C7semidirect
33. C22×D15C2×C7semidirect
34. #1220C2×C7direct (N × H)
35. C2×Dic15C2×C7semidirect
37. C3×C5×C7C2×D4semidirect
40. C22×C3×C7C2×D5semidirect
46. C2×C5×C7#67semidirect
51. C22×C3×C5C22×C7semidirect
53. C2×D15C22×C7semidirect
54. Dic15C22×C7semidirect
56. C23×C7D15semidirect
61. C2×C3×C7C22⋊D5semidirect
64. C23×C5D3×C7semidirect
68. C5×C7C2×#67semidirect
70. C2×C3×C5C7×D4semidirect
77. C22×C7C2×D15semidirect
79. C23×C3C7×D5semidirect
83. C3×C7C2×C22⋊D5semidirect
86. C22×C5C2×D3×C7semidirect
91. C3×C5C2×C7×D4semidirect
94. C2×C7#1220direct (N × H)
98. C22×C3C2×C7×D5semidirect
103. C2×C5C7×#67semidirect
107. C23C7×D15semidirect
110. C7C2×#1220direct (N × H)
112. C2×C3C7×C22⋊D5semidirect
115. C5C2×C7×#67semidirect
120. C22C2×C7×D15semidirect
122. C3C2×C7×C22⋊D5semidirect
125. C2C7×#1220direct (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×C7×D152
C7×#12202
C2×C7×Dic152
C2×C7×C22⋊D53
C2×C7×#675
C2×#12207