Finite Group Databasebeta

← C2×C7×#1004 #13062354

Subgroups — C2×C7×#1004 #13062354

Order 1344.

1796 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C7×D4C211yesmaximal, characteristic, complemented
2C22×C7×#67C214yesmaximal, complemented
3C7×#1004C2116yesmaximal, direct product factor
4C2×C7×#237C214yesmaximal, complemented
5C2×D3×C7×D4C214yesmaximal, complemented
6C2×C7×#235C212yesmaximal, complemented
7C2×C7×H(5,𝔽2)—31nomaximal, Hall
8C24×C3×C7C2212yescomplemented
9C23×D3×C7C2214yescomplemented
10C2×C3×C7×D4C2214yescomplemented
11C2×C3×C7×D4C2218yescomplemented
12C22×C3×C4×C7C2211yescharacteristic, complemented
13C2×C7×#67C2214yescomplemented
14C2×C7×#67C2218yescomplemented
15C2×C7×#67C22132yescomplemented
16C22×C7×Dic3C2214yescomplemented
17C7×#237C22132yescomplemented
18D3×C7×D4C22132yescomplemented
19C7×#235C22116yescomplemented
20C2×C7×D12C2212yescomplemented
21C2×C4×D3×C7C2214yescomplemented
22C2×C7×Dic6C2212yescomplemented
23C7×H(5,𝔽2)—316no—
24C2×C7×#41—34no—
25C2×C7×#41—32no—
26C22×C7×D4—34no—
27C22×C7×D4—34no—
28C22×C7×D4D311yescharacteristic, complemented
29C2×#1004C711yesmaximal, characteristic, direct product factor, Hall
30C23×C3×C7—44no—
31C23×C3×C7C2314yes—
32C23×C3×C7C2311yescharacteristic
33C23×C3×C7C2318yescomplemented
34C22×D3×C7—48no—
35C22×D3×C7C2314yes—
36C22×D3×C7C23116yescomplemented
37C3×C7×D4C23116yescomplemented
38C2×C3×C4×C7C2312yes—
39C2×C3×C4×C7C2314yescomplemented
40C7×#67C23116yescomplemented
41C7×#67C23132yescomplemented
42C2×C7×Dic3C2314yes—
43C2×C7×Dic3C23116yescomplemented
44C7×D12C2318yescomplemented
45C4×D3×C7C23116yescomplemented
46C7×Dic6C2318yescomplemented
47C24×C7—34no—
48C24×C7C2×D312yescomplemented
49C7×#41—332no—
50C7×#41—316no—
51C2×C7×Q8—32no—
52C2×C7×D4—38no—
53C2×C7×D4—34no—
54C2×C7×D4—332no—
55C2×C7×D4—332no—
56C2×C7×D4—32no—
57C2×C7×D4C2×D314yescomplemented
58C2×C7×D4C2×D318yescomplemented
59C22×C4×C7—34no—
60C22×C4×C7—34no—
61C22×C4×C7C2×D311yescharacteristic, complemented
62C22×C3×D4C2×C711yescharacteristic, complemented
63C22×#67C2×C714yescomplemented
64#1004C2×C7116yesdirect product factor
65C2×#237C2×C714yescomplemented
66C2×D3×D4C2×C714yescomplemented
67C2×#235C2×C712yescomplemented
68C22×C3×C7—28no—
69C22×C3×C7—22no—
70C22×C3×C7—48no—
71C22×C3×C7C2418yes—
72C22×C3×C7C2412yes—
73C22×C3×C7C2411yescharacteristic
74C2×D3×C7—28no—
75C2×D3×C7—416no—
76C2×D3×C7C2418yes—
77C3×C4×C7C2414yes—
78C7×Dic3C2418yes—
79C2×H(5,𝔽2)—31noSylow
80C23×C7—34no—
81C23×C7—316no—
82C23×C7—44no—
83C23×C7—128no—
84C23×C7C22×D314yes—
85C23×C7C22×D311yescharacteristic
86C23×C7C22×D318yescomplemented
87C7×Q8—38no—
88C7×D4—332no—
89C7×D4—316no—
90C7×D4—38no—
91C7×D4C22×D3116yescomplemented
92C2×C4×C7—316no—
93C2×C4×C7—34no—
94C2×C4×C7—316no—
95C2×C4×C7C22×D312yes—
96C2×C4×C7C22×D314yescomplemented
97C24×C3C22×C712yescomplemented
98C23×D3C22×C714yescomplemented
99C2×C3×D4C22×C714yescomplemented
100C2×C3×D4C22×C718yescomplemented
101C22×C3×C4C22×C711yescharacteristic, complemented
102C2×#67C22×C714yescomplemented
103C2×#67C22×C718yescomplemented
104C2×#67C22×C7132yescomplemented
105C22×Dic3C22×C714yescomplemented
106#237C22×C7132yescomplemented
107D3×D4C22×C7132yescomplemented
108#235C22×C7116yescomplemented
109C2×D12C22×C712yescomplemented
110C2×C4×D3C22×C714yescomplemented
111C2×Dic6C22×C712yescomplemented
112C2×C3×C7—28no—
113C2×C3×C7—22no—
114C2×C3×C7H(5,𝔽2)12yescomplemented
115C2×C3×C7C2511yescharacteristic
116D3×C7—28no—
117H(5,𝔽2)—316no—
118C2×#41—34no—
119C2×#41—32no—
120C22×D4—34no—
121C22×D4—34no—
122C22×D4D3×C711yescharacteristic, complemented
123C22×C7—22no—
124C22×C7—28no—
125C22×C7—38no—
126C22×C7—48no—
127C22×C7—68no—
128C22×C7—1216no—
129C22×C7C23×D318yes—
130C22×C7C23×D312yes—
131C22×C7C23×D311yescharacteristic
132C4×C7—38no—
133C4×C7C23×D314yes—
134C23×C3—44no—
135C23×C3C23×C714yes—
136C23×C3C23×C711yescharacteristic
137C23×C3C23×C718yescomplemented
138C22×D3—48no—
139C22×D3C23×C714yes—
140C22×D3C23×C7116yescomplemented
141C3×D4C23×C7116yescomplemented
142C2×C3×C4C23×C712yes—
143C2×C3×C4C23×C714yescomplemented
144#67C23×C7116yescomplemented
145#67C23×C7132yescomplemented
146C2×Dic3C23×C714yes—
147C2×Dic3C23×C7116yescomplemented
148D12C23×C718yescomplemented
149C4×D3C23×C7116yescomplemented
150Dic6C23×C718yescomplemented
151C3×C7C2×H(5,𝔽2)11yescharacteristic, complemented, Hall
152C24—34no—
153C24C2×D3×C712yescomplemented
154#41—332no—
155#41—316no—
156C2×Q8—32no—
157C2×D4—38no—
158C2×D4—34no—
159C2×D4—332no—
160C2×D4—332no—
161C2×D4—32no—
162C2×D4C2×D3×C714yescomplemented
163C2×D4C2×D3×C718yescomplemented
164C22×C4—34no—
165C22×C4—34no—
166C22×C4C2×D3×C711yescharacteristic, complemented
167C2×C7—22no—
168C2×C7—28no—
169C2×C7—68no—
170C2×C7#100412yesdirect product factor
171C2×C7C24×D311yescharacteristic
172C22×C3—28no—
173C22×C3—22no—
174C22×C3—48no—
175C22×C3C24×C718yes—
176C22×C3C24×C712yes—
177C22×C3C24×C711yescharacteristic
178C2×D3—28no—
179C2×D3—416no—
180C2×D3C24×C718yes—
181C3×C4C24×C714yes—
182Dic3C24×C718yes—
183C23—34no—
184C23—316no—
185C23—44no—
186C23—128no—
187C23C22×D3×C714yes—
188C23C22×D3×C711yescharacteristic
189C23C22×D3×C718yescomplemented
190Q8—38no—
191D4—332no—
192D4—316no—
193D4—38no—
194D4C22×D3×C7116yescomplemented
195C2×C4—34no—
196C2×C4—316no—
197C2×C4—316no—
198C2×C4C22×D3×C712yes—
199C2×C4C22×D3×C714yescomplemented
200C7C2×#100411yescharacteristic, direct product factor, Sylow
201C2×C3—28no—
202C2×C3—22no—
203C2×C3C7×H(5,𝔽2)12yescomplemented
204C2×C3C25×C711yescharacteristic
205D3—28no—
206C22—22no—
207C22—28no—
208C22—38no—
209C22—48no—
210C22—68no—
211C22—1216no—
212C22C23×D3×C718yes—
213C22C23×D3×C712yes—
214C22C23×D3×C711yescharacteristic
215C4—38no—
216C4C23×D3×C714yes—
217C3C2×C7×H(5,𝔽2)11yescharacteristic, complemented, Sylow
218C2—22no—
219C2—28no—
220C2—68no—
221C2C7×#100412yesdirect product factor
222C2C24×D3×C711yescharacteristic
subgroupG/H
centerC22×C7C23×D3
commutator, cosocleC2×C3C25×C7
FrattiniC2C24×D3×C7
FittingC22×C3×C7×D4C2
socleC22×C3×C7C24
hypercenterC22×C7×D4D3
hypocenterC3C2×C7×H(5,𝔽2)
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C7 ◁ C22×C7×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C7×D4C2semidirect
2. C22×C7×#67C2semidirect
3. C7×#1004C2direct (N × H)
4. C2×C7×#237C2semidirect
5. C2×D3×C7×D4C2semidirect
6. C2×C7×#235C2semidirect
8. C24×C3×C7C22semidirect
9. C23×D3×C7C22semidirect
10. C2×C3×C7×D4C22semidirect
11. C2×C3×C7×D4C22semidirect
12. C22×C3×C4×C7C22semidirect
13. C2×C7×#67C22semidirect
14. C2×C7×#67C22semidirect
15. C2×C7×#67C22semidirect
16. C22×C7×Dic3C22semidirect
17. C7×#237C22semidirect
18. D3×C7×D4C22semidirect
19. C7×#235C22semidirect
20. C2×C7×D12C22semidirect
21. C2×C4×D3×C7C22semidirect
22. C2×C7×Dic6C22semidirect
28. C22×C7×D4D3semidirect
29. C2×#1004C7direct (N × H)
33. C23×C3×C7C23semidirect
36. C22×D3×C7C23semidirect
37. C3×C7×D4C23semidirect
39. C2×C3×C4×C7C23semidirect
40. C7×#67C23semidirect
41. C7×#67C23semidirect
43. C2×C7×Dic3C23semidirect
44. C7×D12C23semidirect
45. C4×D3×C7C23semidirect
46. C7×Dic6C23semidirect
48. C24×C7C2×D3semidirect
57. C2×C7×D4C2×D3semidirect
58. C2×C7×D4C2×D3semidirect
61. C22×C4×C7C2×D3semidirect
62. C22×C3×D4C2×C7semidirect
63. C22×#67C2×C7semidirect
64. #1004C2×C7direct (N × H)
65. C2×#237C2×C7semidirect
66. C2×D3×D4C2×C7semidirect
67. C2×#235C2×C7semidirect
86. C23×C7C22×D3semidirect
91. C7×D4C22×D3semidirect
96. C2×C4×C7C22×D3semidirect
97. C24×C3C22×C7semidirect
98. C23×D3C22×C7semidirect
99. C2×C3×D4C22×C7semidirect
100. C2×C3×D4C22×C7semidirect
101. C22×C3×C4C22×C7semidirect
102. C2×#67C22×C7semidirect
103. C2×#67C22×C7semidirect
104. C2×#67C22×C7semidirect
105. C22×Dic3C22×C7semidirect
106. #237C22×C7semidirect
107. D3×D4C22×C7semidirect
108. #235C22×C7semidirect
109. C2×D12C22×C7semidirect
110. C2×C4×D3C22×C7semidirect
111. C2×Dic6C22×C7semidirect
114. C2×C3×C7H(5,𝔽2)semidirect
122. C22×D4D3×C7semidirect
137. C23×C3C23×C7semidirect
140. C22×D3C23×C7semidirect
141. C3×D4C23×C7semidirect
143. C2×C3×C4C23×C7semidirect
144. #67C23×C7semidirect
145. #67C23×C7semidirect
147. C2×Dic3C23×C7semidirect
148. D12C23×C7semidirect
149. C4×D3C23×C7semidirect
150. Dic6C23×C7semidirect
151. C3×C7C2×H(5,𝔽2)semidirect
153. C24C2×D3×C7semidirect
162. C2×D4C2×D3×C7semidirect
163. C2×D4C2×D3×C7semidirect
166. C22×C4C2×D3×C7semidirect
170. C2×C7#1004direct (N × H)
189. C23C22×D3×C7semidirect
194. D4C22×D3×C7semidirect
199. C2×C4C22×D3×C7semidirect
200. C7C2×#1004direct (N × H)
203. C2×C3C7×H(5,𝔽2)semidirect
217. C3C2×C7×H(5,𝔽2)semidirect
221. C2C7×#1004direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C7×D42
C22×C7×#672
C7×#10042
C2×C7×#2372
C2×D3×C7×D42
C2×C7×#2352
C2×C7×H(5,𝔽2)3
C2×#10047