Finite Group Databasebeta

← C7×#5751 #13061467

Subgroups — C7×#5751 #13061467

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
1C7×#1010C211yesmaximal, characteristic, complemented
2C7×#1009C211yesmaximal, characteristic, complemented
3C3×C7×#131C211yesmaximal, characteristic, complemented
4C7×#909C213yesmaximal, complemented
5C7×#908C213yesmaximal, complemented
6C7×#904C213yesmaximal, complemented
7C2×C7×#204C213yesmaximal, complemented
8C7×#577—31nomaximal, Hall
9C3×C7×#41C2211yescharacteristic
10C7×#239—21no—
11D3×C7×Q8—21no—
12C7×#237—23no—
13D3×C7×D4—23no—
14C7×#235C2213yes—
15C7×#235C2213yescomplemented
16C2×C7×D12C2213yes—
17C2×C7×Dic6C2213yescomplemented
18C3×C7×M16C2213yescomplemented
19C2×C3×C7×C8C2213yescomplemented
20C7×Dic12C2213yescomplemented
21C7×D24C2213yescomplemented
22C7×#204C2213yescomplemented
23C7×#204C2216yescomplemented
24C7×#204C2211yescharacteristic, complemented
25C7×2−1+4—31no—
26C7×H(5,𝔽2)—31no—
27C7×#137—33no—
28C7×AGL(1,ℤ/8)—33no—
29C7×#135—33no—
30C2×C7×SD16—33no—
31C7×#131D311yescharacteristic, complemented
32#5751C711yesmaximal, characteristic, direct product factor, Hall
33C22×D3×C7—23no—
34C3×C7×Q8D411yescharacteristic
35C3×C7×D4D413yes—
36C2×C3×C4×C7C2313yes—
37C7×#67—23no—
38C7×#67—23no—
39C2×C7×Dic3—23no—
40C7×D12—23no—
41C7×D12C2313yes—
42C7×D12C2311yescharacteristic
43C4×D3×C7—23no—
44C4×D3×C7—23no—
45C7×Dic6—23no—
46C7×Dic6C2313yes—
47C7×Dic6C2311yescharacteristic
48C3×C7×C8C2311yescharacteristic
49C3×C7×C8C2313yes—
50C7×#41—33no—
51C7×#41—33no—
52C7×#41—61no—
53C7×#41—63no—
54C7×#41C2×D311yescharacteristic
55C2×C7×Q8—33no—
56C2×C7×Q8—61no—
57C2×C7×D4—33no—
58C2×C7×D4—63no—
59C7×Q16—33no—
60C7×SD16—33no—
61C7×SD16—36no—
62C7×SD16—31no—
63C7×D8—33no—
64C7×M16C2×D313yescomplemented
65C2×C7×C8C2×D313yescomplemented
66#1010C2×C711yescharacteristic, complemented
67#1009C2×C711yescharacteristic, complemented
68C3×#131C2×C711yescharacteristic, complemented
69#909C2×C713yescomplemented
70#908C2×C713yescomplemented
71#904C2×C713yescomplemented
72C2×#204C2×C713yescomplemented
73C22×C3×C7C2×D413yes—
74C2×D3×C7—21no—
75C2×D3×C7—23no—
76C2×D3×C7—83no—
77C3×C4×C7C2×D413yes—
78C3×C4×C7C2411yescharacteristic
79C7×Dic3—23no—
80C7×Dic3—21no—
81#577—31noSylow
82C23×C7—63no—
83C7×Q8—33no—
84C7×Q8—31no—
85C7×Q8—63no—
86C7×Q8D1211yescharacteristic
87C7×D4—31no—
88C7×D4—33no—
89C7×D4—63no—
90C7×D4—63no—
91C7×D4—63no—
92C7×D4D1213yes—
93C2×C4×C7—63no—
94C2×C4×C7—63no—
95C2×C4×C7—63no—
96C2×C4×C7C22×D313yes—
97C7×C8C22×D311yescharacteristic
98C7×C8C22×D313yes—
99C3×#41C22×C711yescharacteristic
100#239—21no—
101D3×Q8—21no—
102#237—23no—
103D3×D4—23no—
104#235C22×C713yes—
105#235C22×C713yescomplemented
106C2×D12C22×C713yes—
107C2×Dic6C22×C713yescomplemented
108C3×M16C22×C713yescomplemented
109C2×C3×C8C22×C713yescomplemented
110Dic12C22×C713yescomplemented
111D24C22×C713yescomplemented
112#204C22×C713yescomplemented
113#204C22×C716yescomplemented
114#204C22×C711yescharacteristic, complemented
115C2×C3×C7—23no—
116C2×C3×C7C22×D411yescharacteristic
117D3×C7—41no—
118D3×C7—43no—
1192−1+4—31no—
120H(5,𝔽2)—31no—
121#137—33no—
122AGL(1,ℤ/8)—33no—
123#135—33no—
124C2×SD16—33no—
125#131D3×C711yescharacteristic, complemented
126C22×C7—61no—
127C22×C7—63no—
128C22×C7—243no—
129C22×C7C2×D1213yes—
130C4×C7—61no—
131C4×C7—63no—
132C4×C7C2×D1213yes—
133C4×C7C23×D311yescharacteristic
134C22×D3—23no—
135C3×Q8C7×D411yescharacteristic
136C3×D4C7×D413yes—
137C2×C3×C4C23×C713yes—
138#67—23no—
139#67—23no—
140C2×Dic3—23no—
141D12—23no—
142D12C23×C713yes—
143D12C23×C711yescharacteristic
144C4×D3—23no—
145C4×D3—23no—
146Dic6—23no—
147Dic6C23×C713yes—
148Dic6C23×C711yescharacteristic
149C3×C8C23×C711yescharacteristic
150C3×C8C23×C713yes—
151C3×C7#57711yescharacteristic, complemented, Hall
152#41—33no—
153#41—33no—
154#41—61no—
155#41—63no—
156#41C2×D3×C711yescharacteristic
157C2×Q8—33no—
158C2×Q8—61no—
159C2×D4—33no—
160C2×D4—63no—
161Q16—33no—
162SD16—33no—
163SD16—36no—
164SD16—31no—
165D8—33no—
166M16C2×D3×C713yescomplemented
167C2×C8C2×D3×C713yescomplemented
168C2×C7—23no—
169C2×C7—123no—
170C2×C7—121no—
171C2×C7C22×D1211yescharacteristic
172C22×C3C2×C7×D413yes—
173C2×D3—21no—
174C2×D3—23no—
175C2×D3—83no—
176C3×C4C2×C7×D413yes—
177C3×C4C24×C711yescharacteristic
178Dic3—23no—
179Dic3—21no—
180C23—63no—
181Q8—33no—
182Q8—31no—
183Q8—63no—
184Q8C7×D1211yescharacteristic
185D4—31no—
186D4—33no—
187D4—63no—
188D4—63no—
189D4—63no—
190D4C7×D1213yes—
191C2×C4—63no—
192C2×C4—63no—
193C2×C4—63no—
194C2×C4C22×D3×C713yes—
195C8C22×D3×C711yescharacteristic
196C8C22×D3×C713yes—
197C7#575111yescharacteristic, direct product factor, Sylow
198C2×C3—23no—
199C2×C3C22×C7×D411yescharacteristic
200D3—43no—
201D3—41no—
202C22—61no—
203C22—63no—
204C22—243no—
205C22C2×C7×D1213yes—
206C4—61no—
207C4—63no—
208C4C2×C7×D1213yes—
209C4C23×D3×C711yescharacteristic
210C3C7×#57711yescharacteristic, complemented, Sylow
211C2—23no—
212C2—123no—
213C2—121no—
214C2C22×C7×D1211yescharacteristic
subgroupG/H
centerC2×C7C22×D12
commutator, cosocleC3×C4C24×C7
FrattiniC4C23×D3×C7
FittingC3×C7×#131C2
socleC2×C3×C7C22×D4
hypercenterC7×#131D3
hypocenterC3C7×#577
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7 ◁ C7×#41 ◁ C7×#131

Complements

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

NH ≅ G/Ntype
1. C7×#1010C2semidirect
2. C7×#1009C2semidirect
3. C3×C7×#131C2semidirect
4. C7×#909C2semidirect
5. C7×#908C2semidirect
6. C7×#904C2semidirect
7. C2×C7×#204C2semidirect
15. C7×#235C22semidirect
17. C2×C7×Dic6C22semidirect
18. C3×C7×M16C22semidirect
19. C2×C3×C7×C8C22semidirect
20. C7×Dic12C22semidirect
21. C7×D24C22semidirect
22. C7×#204C22semidirect
23. C7×#204C22semidirect
24. C7×#204C22semidirect
31. C7×#131D3semidirect
32. #5751C7direct (N × H)
64. C7×M16C2×D3semidirect
65. C2×C7×C8C2×D3semidirect
66. #1010C2×C7semidirect
67. #1009C2×C7semidirect
68. C3×#131C2×C7semidirect
69. #909C2×C7semidirect
70. #908C2×C7semidirect
71. #904C2×C7semidirect
72. C2×#204C2×C7semidirect
105. #235C22×C7semidirect
107. C2×Dic6C22×C7semidirect
108. C3×M16C22×C7semidirect
109. C2×C3×C8C22×C7semidirect
110. Dic12C22×C7semidirect
111. D24C22×C7semidirect
112. #204C22×C7semidirect
113. #204C22×C7semidirect
114. #204C22×C7semidirect
125. #131D3×C7semidirect
151. C3×C7#577semidirect
166. M16C2×D3×C7semidirect
167. C2×C8C2×D3×C7semidirect
197. C7#5751direct (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
C7×#10102
C7×#10092
C3×C7×#1312
C7×#9092
C7×#9082
C7×#9042
C2×C7×#2042
C7×#5773
#57517