Finite Group Databasebeta

← C7×#5835 #13061551

Subgroups — C7×#5835 #13061551

Order 1344.

536 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 214 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
1C3×C7×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C7×#1009C211yesmaximal, characteristic, complemented
3C7×#950C213yesmaximal, complemented
4C7×#949C213yesmaximal, complemented
5C7×#948C211yesmaximal, characteristic, complemented
6C7×#932C213yesmaximal, complemented
7C2×C7×#213C213yesmaximal, complemented
8C7×#576—31nomaximal, Hall
9C3×C7×#41—21no—
10C3×C7×#41C2211yescharacteristic
11C3×C7×#41C2213yescomplemented
12C2×C3×C7×D4—23no—
13C2×C3×C7×D4C2213yescomplemented
14C7×#239—21no—
15D3×C7×D4—23no—
16C7×#235C2213yescomplemented
17C2×C7×D12C2213yescomplemented
18C7×#216C2211yescharacteristic, complemented
19C7×#215C2213yescomplemented
20C7×#214C2213yescomplemented
21C7×#213C2213yescomplemented
22C7×#213C2216yescomplemented
23C7×#208C2213yescomplemented
24C2×C7×C3⋊C8C2213yescomplemented
25C7×H(5,𝔽2)—31no—
26C7×H(5,𝔽2)D311yescharacteristic, complemented
27C7×AGL(1,ℤ/8)—33no—
28C7×AGL(1,ℤ/8)—33no—
29C7×#135—33no—
30C2×C7×D8—33no—
31C7×#131—31no—
32#5835C711yesmaximal, characteristic, direct product factor, Hall
33C23×C3×C7—23no—
34C22×D3×C7—23no—
35C3×C7×Q8D411yescharacteristic
36C3×C7×Q8C2311yescharacteristic
37C3×C7×D4—23no—
38C3×C7×D4—23no—
39C3×C7×D4D413yes—
40C3×C7×D4C2313yes—
41C2×C3×C4×C7—23no—
42C2×C3×C4×C7C2313yes—
43C7×#67—23no—
44C7×D12—23no—
45C7×D12C2313yes—
46C4×D3×C7—23no—
47C7×Dic6C2311yescharacteristic
48C7×C3⋊C8C2311yescharacteristic
49C7×C3⋊C8C2313yes—
50C7×#41—21no—
51C7×#41—33no—
52C7×#41—61no—
53C7×#41C2×D311yescharacteristic
54C7×#41C2×D313yescomplemented
55C2×C7×D4—23no—
56C2×C7×D4—33no—
57C2×C7×D4—63no—
58C2×C7×D4C2×D313yescomplemented
59C7×Q16—31no—
60C7×SD16—33no—
61C7×SD16—33no—
62C7×D8—33no—
63C7×D8—36no—
64C7×M16—33no—
65C2×C7×C8—33no—
66C3×H(5,𝔽2)C2×C711yescharacteristic, complemented
67#1009C2×C711yescharacteristic, complemented
68#950C2×C713yescomplemented
69#949C2×C713yescomplemented
70#948C2×C711yescharacteristic, complemented
71#932C2×C713yescomplemented
72C2×#213C2×C713yescomplemented
73C22×C3×C7—23no—
74C22×C3×C7—83no—
75C22×C3×C7C2×D413yes—
76C2×D3×C7—23no—
77C2×D3×C7—83no—
78C3×C4×C7—21no—
79C3×C4×C7C2×D413yes—
80C3×C4×C7C2411yescharacteristic
81C7×Dic3—21no—
82#576—31noSylow
83C23×C7—23no—
84C23×C7—63no—
85C7×Q8—31no—
86C7×Q8#6711yescharacteristic
87C7×Q8C22×D311yescharacteristic
88C7×D4—23no—
89C7×D4—23no—
90C7×D4—33no—
91C7×D4—63no—
92C7×D4—63no—
93C7×D4#6713yes—
94C7×D4C22×D313yes—
95C2×C4×C7—23no—
96C2×C4×C7—63no—
97C2×C4×C7C22×D313yes—
98C7×C8—31no—
99C7×C8—33no—
100C3×#41—21no—
101C3×#41C22×C711yescharacteristic
102C3×#41C22×C713yescomplemented
103C2×C3×D4—23no—
104C2×C3×D4C22×C713yescomplemented
105#239—21no—
106D3×D4—23no—
107#235C22×C713yescomplemented
108C2×D12C22×C713yescomplemented
109#216C22×C711yescharacteristic, complemented
110#215C22×C713yescomplemented
111#214C22×C713yescomplemented
112#213C22×C713yescomplemented
113#213C22×C716yescomplemented
114#208C22×C713yescomplemented
115C2×C3⋊C8C22×C713yescomplemented
116C2×C3×C7—23no—
117C2×C3×C7—43no—
118C2×C3×C7C22×D411yescharacteristic
119D3×C7—43no—
120H(5,𝔽2)—31no—
121H(5,𝔽2)D3×C711yescharacteristic, complemented
122AGL(1,ℤ/8)—33no—
123AGL(1,ℤ/8)—33no—
124#135—33no—
125C2×D8—33no—
126#131—31no—
127C22×C7—23no—
128C22×C7—63no—
129C22×C7—83no—
130C22×C7—243no—
131C22×C7C2×#6713yes—
132C4×C7—21no—
133C4×C7—61no—
134C4×C7C2×#6713yes—
135C4×C7C23×D311yescharacteristic
136C23×C3—23no—
137C22×D3—23no—
138C3×Q8C7×D411yescharacteristic
139C3×Q8C23×C711yescharacteristic
140C3×D4—23no—
141C3×D4—23no—
142C3×D4C7×D413yes—
143C3×D4C23×C713yes—
144C2×C3×C4—23no—
145C2×C3×C4C23×C713yes—
146#67—23no—
147D12—23no—
148D12C23×C713yes—
149C4×D3—23no—
150Dic6C23×C711yescharacteristic
151C3⋊C8C23×C711yescharacteristic
152C3⋊C8C23×C713yes—
153C3×C7#57611yescharacteristic, complemented, Hall
154#41—21no—
155#41—33no—
156#41—61no—
157#41C2×D3×C711yescharacteristic
158#41C2×D3×C713yescomplemented
159C2×D4—23no—
160C2×D4—33no—
161C2×D4—63no—
162C2×D4C2×D3×C713yescomplemented
163Q16—31no—
164SD16—33no—
165SD16—33no—
166D8—33no—
167D8—36no—
168M16—33no—
169C2×C8—33no—
170C2×C7—23no—
171C2×C7—43no—
172C2×C7—123no—
173C2×C7C22×#6711yescharacteristic
174C22×C3—23no—
175C22×C3—83no—
176C22×C3C2×C7×D413yes—
177C2×D3—23no—
178C2×D3—83no—
179C3×C4—21no—
180C3×C4C2×C7×D413yes—
181C3×C4C24×C711yescharacteristic
182Dic3—21no—
183C23—23no—
184C23—63no—
185Q8—31no—
186Q8C7×#6711yescharacteristic
187Q8C22×D3×C711yescharacteristic
188D4—23no—
189D4—23no—
190D4—33no—
191D4—63no—
192D4—63no—
193D4C7×#6713yes—
194D4C22×D3×C713yes—
195C2×C4—23no—
196C2×C4—63no—
197C2×C4C22×D3×C713yes—
198C8—31no—
199C8—33no—
200C7#583511yescharacteristic, direct product factor, Sylow
201C2×C3—23no—
202C2×C3—43no—
203C2×C3C22×C7×D411yescharacteristic
204D3—43no—
205C22—23no—
206C22—63no—
207C22—83no—
208C22—243no—
209C22C2×C7×#6713yes—
210C4—21no—
211C4—61no—
212C4C2×C7×#6713yes—
213C4C23×D3×C711yescharacteristic
214C3C7×#57611yescharacteristic, complemented, Sylow
215C2—23no—
216C2—43no—
217C2—123no—
218C2C22×C7×#6711yescharacteristic
subgroupG/H
centerC2×C7C22×#67
commutator, cosocleC3×C4C24×C7
FrattiniC4C23×D3×C7
FittingC3×C7×H(5,𝔽2)C2
socleC2×C3×C7C22×D4
hypercenterC7×H(5,𝔽2)D3
hypocenterC3C7×#576
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7 ◁ C7×#41 ◁ C7×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C3×C7×H(5,𝔽2)C2semidirect
2. C7×#1009C2semidirect
3. C7×#950C2semidirect
4. C7×#949C2semidirect
5. C7×#948C2semidirect
6. C7×#932C2semidirect
7. C2×C7×#213C2semidirect
11. C3×C7×#41C22semidirect
13. C2×C3×C7×D4C22semidirect
16. C7×#235C22semidirect
17. C2×C7×D12C22semidirect
18. C7×#216C22semidirect
19. C7×#215C22semidirect
20. C7×#214C22semidirect
21. C7×#213C22semidirect
22. C7×#213C22semidirect
23. C7×#208C22semidirect
24. C2×C7×C3⋊C8C22semidirect
26. C7×H(5,𝔽2)D3semidirect
32. #5835C7direct (N × H)
54. C7×#41C2×D3semidirect
58. C2×C7×D4C2×D3semidirect
66. C3×H(5,𝔽2)C2×C7semidirect
67. #1009C2×C7semidirect
68. #950C2×C7semidirect
69. #949C2×C7semidirect
70. #948C2×C7semidirect
71. #932C2×C7semidirect
72. C2×#213C2×C7semidirect
102. C3×#41C22×C7semidirect
104. C2×C3×D4C22×C7semidirect
107. #235C22×C7semidirect
108. C2×D12C22×C7semidirect
109. #216C22×C7semidirect
110. #215C22×C7semidirect
111. #214C22×C7semidirect
112. #213C22×C7semidirect
113. #213C22×C7semidirect
114. #208C22×C7semidirect
115. C2×C3⋊C8C22×C7semidirect
121. H(5,𝔽2)D3×C7semidirect
153. C3×C7#576semidirect
158. #41C2×D3×C7semidirect
162. C2×D4C2×D3×C7semidirect
200. C7#5835direct (N × H)
214. C3C7×#576semidirect

Maximal subgroups

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

subgroupindex
C3×C7×H(5,𝔽2)2
C7×#10092
C7×#9502
C7×#9492
C7×#9482
C7×#9322
C2×C7×#2132
C7×#5763
#58357