Finite Group Databasebeta

← C7×#5836 #13061552

Subgroups — C7×#5836 #13061552

Order 1344.

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

normal subgroups only

This group has 516 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×#1010C211yesmaximal, characteristic, complemented
3C7×#951C213yesmaximal, complemented
4C7×#950C213yesmaximal, complemented
5C7×#948C211yesmaximal, characteristic, complemented
6C2×C7×#214C213yesmaximal, complemented
7C7×#932C213yesmaximal, complemented
8C7×#577—31nomaximal, Hall
9C3×C7×#41—21no—
10C3×C7×#41C2213yes—
11C3×C7×#41C2211yescharacteristic
12C2×C3×C7×D4—23no—
13C2×C3×C7×D4C2213yes—
14D3×C7×Q8—21no—
15C7×#237—23no—
16C7×#235C2213yescomplemented
17C2×C7×Dic6C2213yescomplemented
18C7×#216C2213yescomplemented
19C7×#215C2211yescharacteristic, complemented
20C7×#214C2216yescomplemented
21C7×#214C2213yescomplemented
22C7×#213C2213yescomplemented
23C7×#208C2213yescomplemented
24C2×C7×C3⋊C8C2213yescomplemented
25C7×2−1+4—31no—
26C7×H(5,𝔽2)D311yescharacteristic, complemented
27C7×#137—33no—
28C7×AGL(1,ℤ/8)—33no—
29C7×#135—33no—
30C2×C7×SD16—33no—
31C7×#131—31no—
32#5836C711yesmaximal, characteristic, direct product factor, Hall
33C23×C3×C7—23no—
34C3×C7×Q8D411yescharacteristic
35C3×C7×Q8C2311yescharacteristic
36C3×C7×D4—23no—
37C3×C7×D4—23no—
38C3×C7×D4D413yes—
39C3×C7×D4C2313yes—
40C2×C3×C4×C7—23no—
41C2×C3×C4×C7C2313yes—
42C7×#67—23no—
43C2×C7×Dic3—23no—
44C7×D12C2311yescharacteristic
45C4×D3×C7—23no—
46C7×Dic6—23no—
47C7×Dic6C2313yes—
48C7×C3⋊C8C2311yescharacteristic
49C7×C3⋊C8C2313yes—
50C7×#41—21no—
51C7×#41—33no—
52C7×#41—63no—
53C7×#41C2×D313yes—
54C7×#41C2×D311yescharacteristic
55C2×C7×Q8—33no—
56C2×C7×Q8—61no—
57C2×C7×D4—23no—
58C2×C7×D4C2×D313yes—
59C7×Q16—33no—
60C7×SD16—31no—
61C7×SD16—33no—
62C7×SD16—36no—
63C7×D8—33no—
64C7×M16—33no—
65C2×C7×C8—33no—
66C3×H(5,𝔽2)C2×C711yescharacteristic, complemented
67#1010C2×C711yescharacteristic, complemented
68#951C2×C713yescomplemented
69#950C2×C713yescomplemented
70#948C2×C711yescharacteristic, complemented
71C2×#214C2×C713yescomplemented
72#932C2×C713yescomplemented
73C22×C3×C7—23no—
74C22×C3×C7—83no—
75C22×C3×C7C2×D413yes—
76C2×D3×C7—21no—
77C3×C4×C7—21no—
78C3×C4×C7C2×D413yes—
79C3×C4×C7C2411yescharacteristic
80C7×Dic3—23no—
81#577—31noSylow
82C23×C7—23no—
83C7×Q8—33no—
84C7×Q8—63no—
85C7×Q8#6711yescharacteristic
86C7×Q8C22×D311yescharacteristic
87C7×D4—23no—
88C7×D4—23no—
89C7×D4—31no—
90C7×D4—63no—
91C7×D4#6713yes—
92C7×D4C22×D313yes—
93C2×C4×C7—23no—
94C2×C4×C7—63no—
95C2×C4×C7—63no—
96C2×C4×C7C22×D313yes—
97C7×C8—31no—
98C7×C8—33no—
99C3×#41—21no—
100C3×#41C22×C713yes—
101C3×#41C22×C711yescharacteristic
102C2×C3×D4—23no—
103C2×C3×D4C22×C713yes—
104D3×Q8—21no—
105#237—23no—
106#235C22×C713yescomplemented
107C2×Dic6C22×C713yescomplemented
108#216C22×C713yescomplemented
109#215C22×C711yescharacteristic, complemented
110#214C22×C716yescomplemented
111#214C22×C713yescomplemented
112#213C22×C713yescomplemented
113#208C22×C713yescomplemented
114C2×C3⋊C8C22×C713yescomplemented
115C2×C3×C7—23no—
116C2×C3×C7—43no—
117C2×C3×C7C22×D411yescharacteristic
118D3×C7—41no—
1192−1+4—31no—
120H(5,𝔽2)D3×C711yescharacteristic, complemented
121#137—33no—
122AGL(1,ℤ/8)—33no—
123#135—33no—
124C2×SD16—33no—
125#131—31no—
126C22×C7—23no—
127C22×C7—61no—
128C22×C7—83no—
129C22×C7C2×#6713yes—
130C4×C7—21no—
131C4×C7—63no—
132C4×C7C2×#6713yes—
133C4×C7C23×D311yescharacteristic
134C23×C3—23no—
135C3×Q8C7×D411yescharacteristic
136C3×Q8C23×C711yescharacteristic
137C3×D4—23no—
138C3×D4—23no—
139C3×D4C7×D413yes—
140C3×D4C23×C713yes—
141C2×C3×C4—23no—
142C2×C3×C4C23×C713yes—
143#67—23no—
144C2×Dic3—23no—
145D12C23×C711yescharacteristic
146C4×D3—23no—
147Dic6—23no—
148Dic6C23×C713yes—
149C3⋊C8C23×C711yescharacteristic
150C3⋊C8C23×C713yes—
151C3×C7#57711yescharacteristic, complemented, Hall
152#41—21no—
153#41—33no—
154#41—63no—
155#41C2×D3×C713yes—
156#41C2×D3×C711yescharacteristic
157C2×Q8—33no—
158C2×Q8—61no—
159C2×D4—23no—
160C2×D4C2×D3×C713yes—
161Q16—33no—
162SD16—31no—
163SD16—33no—
164SD16—36no—
165D8—33no—
166M16—33no—
167C2×C8—33no—
168C2×C7—23no—
169C2×C7—43no—
170C2×C7—121no—
171C2×C7C22×#6711yescharacteristic
172C22×C3—23no—
173C22×C3—83no—
174C22×C3C2×C7×D413yes—
175C2×D3—21no—
176C3×C4—21no—
177C3×C4C2×C7×D413yes—
178C3×C4C24×C711yescharacteristic
179Dic3—23no—
180C23—23no—
181Q8—33no—
182Q8—63no—
183Q8C7×#6711yescharacteristic
184Q8C22×D3×C711yescharacteristic
185D4—23no—
186D4—23no—
187D4—31no—
188D4—63no—
189D4C7×#6713yes—
190D4C22×D3×C713yes—
191C2×C4—23no—
192C2×C4—63no—
193C2×C4—63no—
194C2×C4C22×D3×C713yes—
195C8—31no—
196C8—33no—
197C7#583611yescharacteristic, direct product factor, Sylow
198C2×C3—23no—
199C2×C3—43no—
200C2×C3C22×C7×D411yescharacteristic
201D3—41no—
202C22—23no—
203C22—61no—
204C22—83no—
205C22C2×C7×#6713yes—
206C4—21no—
207C4—63no—
208C4C2×C7×#6713yes—
209C4C23×D3×C711yescharacteristic
210C3C7×#57711yescharacteristic, complemented, Sylow
211C2—23no—
212C2—43no—
213C2—121no—
214C2C22×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×#577
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×#1010C2semidirect
3. C7×#951C2semidirect
4. C7×#950C2semidirect
5. C7×#948C2semidirect
6. C2×C7×#214C2semidirect
7. C7×#932C2semidirect
16. C7×#235C22semidirect
17. C2×C7×Dic6C22semidirect
18. C7×#216C22semidirect
19. C7×#215C22semidirect
20. C7×#214C22semidirect
21. C7×#214C22semidirect
22. C7×#213C22semidirect
23. C7×#208C22semidirect
24. C2×C7×C3⋊C8C22semidirect
26. C7×H(5,𝔽2)D3semidirect
32. #5836C7direct (N × H)
66. C3×H(5,𝔽2)C2×C7semidirect
67. #1010C2×C7semidirect
68. #951C2×C7semidirect
69. #950C2×C7semidirect
70. #948C2×C7semidirect
71. C2×#214C2×C7semidirect
72. #932C2×C7semidirect
106. #235C22×C7semidirect
107. C2×Dic6C22×C7semidirect
108. #216C22×C7semidirect
109. #215C22×C7semidirect
110. #214C22×C7semidirect
111. #214C22×C7semidirect
112. #213C22×C7semidirect
113. #208C22×C7semidirect
114. C2×C3⋊C8C22×C7semidirect
120. H(5,𝔽2)D3×C7semidirect
151. C3×C7#577semidirect
197. C7#5836direct (N × H)
210. C3C7×#577semidirect

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×#10102
C7×#9512
C7×#9502
C7×#9482
C2×C7×#2142
C7×#9322
C7×#5773
#58367