Finite Group Databasebeta

← C2×D3×C7⋊AGL(1,𝔽5) #13132943

Subgroups — C2×D3×C7⋊AGL(1,𝔽5) #13132943

Order 1680.

216 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C2×D3×C7×D5C211yesmaximal, characteristic
2C2×C3⋊(C7⋊AGL(1,𝔽5))C211yesmaximal, characteristic, complemented
3C2×C3×C7⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4D3×C7⋊AGL(1,𝔽5)C214yesmaximal, direct product factor
5C22×C7⋊AGL(1,𝔽5)—31nomaximal, Hall
6C2×C7×D15C411yescharacteristic, complemented
7C2×C5×D3×C7C411yescharacteristic, complemented
8C2×C3×C7×D5C2211yescharacteristic
9D3×C7×D5C412yescomplemented
10D3×C7×D5C2212yes—
11C3⋊(C7⋊AGL(1,𝔽5))C2212yescomplemented
12C3×C7⋊AGL(1,𝔽5)C2212yescomplemented
13C2×D3×Dic7—51nomaximal, Hall
14C22×C7×D5—31no—
15C2×C7⋊AGL(1,𝔽5)—31no—
16C2×C7⋊AGL(1,𝔽5)—34no—
17C2×C7⋊AGL(1,𝔽5)D311yescharacteristic, direct product factor
18C2×D3×AGL(1,𝔽5)—71nomaximal, Hall
19C2×C3×C5×C7C2×C411yescharacteristic, complemented
20C7×D15C2×C412yescomplemented
21C5×D3×C7C2×C412yescomplemented
22C3×C7×D5C2×C411yescharacteristic, complemented
23C3×C7×D5C2311yescharacteristic
24C22×D3×C7—51no—
25C2×Dic21—51no—
26C2×C3×Dic7—51no—
27D3×Dic7—54no—
28C22×C5×C7—31no—
29C2×C7×D5—31no—
30C2×C7×D5—32no—
31C2×C7×D5—32no—
32C2×C7×D5C2×D311yescharacteristic
33C7⋊AGL(1,𝔽5)—32no—
34C7⋊AGL(1,𝔽5)C2×D312yesdirect product factor
35C2×D3×D5D711yescharacteristic
36C2×ΓL(1,𝔽16)—71no—
37C2×C3×AGL(1,𝔽5)—71no—
38D3×AGL(1,𝔽5)—74no—
39C22×Dic7—151noHall
40C3×C5×C7C22×C411yescharacteristic, complemented, Hall
41C22×C3×C7—51no—
42C2×D3×C7—51no—
43C2×D3×C7—52no—
44C2×D3×C7—52no—
45C2×D3×C7AGL(1,𝔽5)11yescharacteristic, complemented
46Dic21—52no—
47C3×Dic7—52no—
48C22×AGL(1,𝔽5)—211noHall
49C2×C5×C7—32no—
50C2×C5×C7C4×D311yescharacteristic, complemented
51C7×D5—32no—
52C7×D5C4×D311yescharacteristic, complemented
53C7×D5C22×D311yescharacteristic
54C2×D15Dic711yescharacteristic, complemented
55C2×C5×D3Dic711yescharacteristic, complemented
56C2×C3×D5C2×D711yescharacteristic
57D3×D5Dic712yescomplemented
58D3×D5C2×D712yes—
59ΓL(1,𝔽16)—72no—
60C3×AGL(1,𝔽5)—72no—
61C23×C7—151no—
62C2×Dic7—51no—
63C2×Dic7—154no—
64C2×Dic7—151no—
65C2×C4×D3—351noHall
66C2×C3×C7—51no—
67C2×C3×C7—51no—
68C2×C3×C7C2×AGL(1,𝔽5)11yescharacteristic, complemented
69D3×C7—52no—
70D3×C7C2×AGL(1,𝔽5)12yescomplemented
71C22×D5—31no—
72C2×AGL(1,𝔽5)—71no—
73C2×AGL(1,𝔽5)—214no—
74C2×AGL(1,𝔽5)—211no—
75C5×C7C2×C4×D311yescharacteristic, complemented, Hall
76C2×C3×C5C2×Dic711yescharacteristic, complemented
77D15C2×Dic712yescomplemented
78C3×D5C2×Dic711yescharacteristic, complemented
79C3×D5C22×D711yescharacteristic
80C5×D3C2×Dic712yescomplemented
81C22×C7—31no—
82C22×C7—51no—
83C22×C7—152no—
84C22×C7—152no—
85C22×C7—151no—
86Dic7—52no—
87Dic7—152no—
88C22×D3—51no—
89C2×C3×C4—351no—
90C2×Dic3—351no—
91C4×D3—354no—
92C3×C7C22×AGL(1,𝔽5)11yescharacteristic, complemented, Hall
93C22×C5—31no—
94C2×D5—31no—
95C2×D5—32no—
96C2×D5—32no—
97C2×D5D3×D711yescharacteristic
98AGL(1,𝔽5)—72no—
99AGL(1,𝔽5)—212no—
100C22×C4—1051noSylow
101C3×C5C22×Dic711yescharacteristic, complemented, Hall
102C2×C7—32no—
103C2×C7—51no—
104C2×C7—51no—
105C2×C7—152no—
106C2×C7D3×AGL(1,𝔽5)11yescharacteristic, complemented
107C22×C3—51no—
108C2×D3—51no—
109C2×D3—52no—
110C2×D3—52no—
111C2×D3C7⋊AGL(1,𝔽5)11yescharacteristic, direct product factor
112C3×C4—352no—
113Dic3—352no—
114C2×C5—32no—
115C2×C5D3×Dic711yescharacteristic, complemented
116D5—32no—
117D5D3×Dic711yescharacteristic, complemented
118D5C2×D3×D711yescharacteristic
119C23—151no—
120C2×C4—351no—
121C2×C4—1054no—
122C2×C4—1051no—
123C7C2×D3×AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
124C2×C3—51no—
125C2×C3—51no—
126C2×C3C2×C7⋊AGL(1,𝔽5)11yescharacteristic, complemented
127D3—52no—
128D3C2×C7⋊AGL(1,𝔽5)12yesdirect product factor
129C5C2×D3×Dic711yescharacteristic, complemented, Sylow
130C22—31no—
131C22—51no—
132C22—152no—
133C22—152no—
134C22—151no—
135C4—352no—
136C4—1052no—
137C3C22×C7⋊AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
138C2—32no—
139C2—51no—
140C2—51no—
141C2—152no—
142C2D3×C7⋊AGL(1,𝔽5)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2D3×C7⋊AGL(1,𝔽5)
commutator, hypocenterC3×C5×C7C22×C4
FrattiniC1G
Fitting, socleC2×C3×C5×C7C2×C4
prime radicalC2×D3×C7×D5C2
cosocleC3×C7×D5C23
LCSG ▷ C3×C5×C7
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊(C7⋊AGL(1,𝔽5))C2semidirect
3. C2×C3×C7⋊AGL(1,𝔽5)C2semidirect
4. D3×C7⋊AGL(1,𝔽5)C2direct (N × H)
6. C2×C7×D15C4semidirect
7. C2×C5×D3×C7C4semidirect
9. D3×C7×D5C4semidirect
11. C3⋊(C7⋊AGL(1,𝔽5))C22semidirect
12. C3×C7⋊AGL(1,𝔽5)C22semidirect
17. C2×C7⋊AGL(1,𝔽5)D3direct (N × H)
19. C2×C3×C5×C7C2×C4semidirect
20. C7×D15C2×C4semidirect
21. C5×D3×C7C2×C4semidirect
22. C3×C7×D5C2×C4semidirect
34. C7⋊AGL(1,𝔽5)C2×D3direct (N × H)
40. C3×C5×C7C22×C4semidirect
45. C2×D3×C7AGL(1,𝔽5)semidirect
50. C2×C5×C7C4×D3semidirect
52. C7×D5C4×D3semidirect
54. C2×D15Dic7semidirect
55. C2×C5×D3Dic7semidirect
57. D3×D5Dic7semidirect
68. C2×C3×C7C2×AGL(1,𝔽5)semidirect
70. D3×C7C2×AGL(1,𝔽5)semidirect
75. C5×C7C2×C4×D3semidirect
76. C2×C3×C5C2×Dic7semidirect
77. D15C2×Dic7semidirect
78. C3×D5C2×Dic7semidirect
80. C5×D3C2×Dic7semidirect
92. C3×C7C22×AGL(1,𝔽5)semidirect
101. C3×C5C22×Dic7semidirect
106. C2×C7D3×AGL(1,𝔽5)semidirect
111. C2×D3C7⋊AGL(1,𝔽5)direct (N × H)
115. C2×C5D3×Dic7semidirect
117. D5D3×Dic7semidirect
123. C7C2×D3×AGL(1,𝔽5)semidirect
126. C2×C3C2×C7⋊AGL(1,𝔽5)semidirect
128. D3C2×C7⋊AGL(1,𝔽5)direct (N × H)
129. C5C2×D3×Dic7semidirect
137. C3C22×C7⋊AGL(1,𝔽5)semidirect
142. C2D3×C7⋊AGL(1,𝔽5)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×C7×D52
C2×C3⋊(C7⋊AGL(1,𝔽5))2
C2×C3×C7⋊AGL(1,𝔽5)2
D3×C7⋊AGL(1,𝔽5)2
C22×C7⋊AGL(1,𝔽5)3
C2×D3×Dic75
C2×D3×AGL(1,𝔽5)7