Finite Group Databasebeta

← C71×#3026 #22585588

Subgroups — C71×#3026 #22585588

Order 9088.

800 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 212 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×H(5,𝔽2)×C71C211yesmaximal, characteristic, complemented
2#534×C71C213yesmaximal, complemented
3#458×C71C214yesmaximal, complemented
4B(4,𝔽2)×C71C214yesmaximal, complemented
5C2×#99×C71C213yesmaximal, complemented
6H(5,𝔽2)×C71—26no—
7H(5,𝔽2)×C71C2214yes—
8C2×#41×C71—23no—
9C22×D4×C71—23no—
10C22×D4×C71C2213yes—
11C22×D4×C71C2211yescharacteristic, complemented
12#123×C71—23no—
13#123×C71C2216yescomplemented
14#121×C71—26no—
15#120×C71—23no—
16#120×C71C2216yescomplemented
17C2×C22⋊C4×C71C2213yescomplemented
18#99×C71C22112yescomplemented
19C24×C71—21no—
20C24×C71D413yescomplemented
21C24×C71C2311yescharacteristic
22C24×C71C2311yescharacteristic, complemented
23#41×C71—212no—
24#41×C71—212no—
25C2×Q8×C71—21no—
26C2×D4×C71—23no—
27C2×D4×C71—212no—
28C2×D4×C71—26no—
29C2×D4×C71—212no—
30C2×D4×C71—26no—
31C2×D4×C71—23no—
32C2×D4×C71—23no—
33C2×D4×C71—26no—
34C2×D4×C71D416yescomplemented
35C2×D4×C71C2316yes—
36C22×C4×C71—23no—
37C22×C4×C71—23no—
38C22×C4×C71D413yescomplemented
39C22×C4×C71C2311yescharacteristic, complemented
40C4⋊C4×C71—26no—
41C22⋊C4×C71—23no—
42C22⋊C4×C71—26no—
43C22⋊C4×C71—26no—
44C22⋊C4×C71C2316yescomplemented
45C23×C71—24no—
46C23×C71—23no—
47C23×C71—23no—
48C23×C71—23no—
49C23×C71—81no—
50C23×C71—81no—
51C23×C71—83no—
52C23×C71—82no—
53C23×C71C2×D416yes—
54C23×C71C2×D413yes—
55C23×C71C2×D413yes—
56C23×C71C2411yescharacteristic
57Q8×C71—24no—
58D4×C71—212no—
59D4×C71—212no—
60D4×C71—212no—
61D4×C71—41no—
62D4×C71—43no—
63D4×C71—86no—
64C2×C4×C71—212no—
65C2×C4×C71—23no—
66C2×C4×C71—23no—
67C2×C4×C71—26no—
68C2×C4×C71—23no—
69C2×C4×C71—83no—
70C2×C4×C71C2×D416yes—
71C22×C71—26no—
72C22×C71—43no—
73C22×C71—43no—
74C22×C71—41no—
75C22×C71—46no—
76C22×C71—46no—
77C22×C71—46no—
78C22×C71—41no—
79C22×C71—83no—
80C22×C71—84no—
81C22×C71#12011yescharacteristic
82C22×C71#12013yes—
83C22×C71C22×D413yes—
84C4×C71—26no—
85C4×C71—46no—
86C4×C71—41no—
87C2×C71—23no—
88C2×C71—21no—
89C2×C71—23no—
90C2×C71—46no—
91C2×C71—81no—
92C2×C71C2×#12011yescharacteristic
93#3026C7111yesmaximal, characteristic, direct product factor, Sylow
94C71#302611yescharacteristic, direct product factor, Sylow
95C2×H(5,𝔽2)C2×C7111yescharacteristic, complemented
96#534C2×C7113yescomplemented
97#458C2×C7114yescomplemented
98B(4,𝔽2)C2×C7114yescomplemented
99C2×#99C2×C7113yescomplemented
100H(5,𝔽2)—26no—
101H(5,𝔽2)C22×C7114yes—
102C2×#41—23no—
103C22×D4—23no—
104C22×D4C22×C7113yes—
105C22×D4C22×C7111yescharacteristic, complemented
106#123—23no—
107#123C22×C7116yescomplemented
108#121—26no—
109#120—23no—
110#120C22×C7116yescomplemented
111C2×C22⋊C4C22×C7113yescomplemented
112#99C22×C71112yescomplemented
113C24—21no—
114C24D4×C7113yescomplemented
115C24C23×C7111yescharacteristic
116C24C23×C7111yescharacteristic, complemented
117#41—212no—
118#41—212no—
119C2×Q8—21no—
120C2×D4—23no—
121C2×D4—212no—
122C2×D4—26no—
123C2×D4—212no—
124C2×D4—26no—
125C2×D4—23no—
126C2×D4—23no—
127C2×D4—26no—
128C2×D4D4×C7116yescomplemented
129C2×D4C23×C7116yes—
130C22×C4—23no—
131C22×C4—23no—
132C22×C4D4×C7113yescomplemented
133C22×C4C23×C7111yescharacteristic, complemented
134C4⋊C4—26no—
135C22⋊C4—23no—
136C22⋊C4—26no—
137C22⋊C4—26no—
138C22⋊C4C23×C7116yescomplemented
139C23—24no—
140C23—23no—
141C23—23no—
142C23—23no—
143C23—81no—
144C23—81no—
145C23—83no—
146C23—82no—
147C23C2×D4×C7116yes—
148C23C2×D4×C7113yes—
149C23C2×D4×C7113yes—
150C23C24×C7111yescharacteristic
151Q8—24no—
152D4—212no—
153D4—212no—
154D4—212no—
155D4—41no—
156D4—43no—
157D4—86no—
158C2×C4—212no—
159C2×C4—23no—
160C2×C4—23no—
161C2×C4—26no—
162C2×C4—23no—
163C2×C4—83no—
164C2×C4C2×D4×C7116yes—
165C22—26no—
166C22—43no—
167C22—43no—
168C22—41no—
169C22—46no—
170C22—46no—
171C22—46no—
172C22—41no—
173C22—83no—
174C22—84no—
175C22#120×C7111yescharacteristic
176C22#120×C7113yes—
177C22C22×D4×C7113yes—
178C4—26no—
179C4—46no—
180C4—41no—
181C2—23no—
182C2—21no—
183C2—23no—
184C2—46no—
185C2—81no—
186C2C2×#120×C7111yescharacteristic
subgroupG/H
center, socleC2×C71C2×#120
commutator, Frattini, cosocleC23C24×C71
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C71 ◁ C24×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×H(5,𝔽2)×C71C2semidirect
2. #534×C71C2semidirect
3. #458×C71C2semidirect
4. B(4,𝔽2)×C71C2semidirect
5. C2×#99×C71C2semidirect
11. C22×D4×C71C22semidirect
13. #123×C71C22semidirect
16. #120×C71C22semidirect
17. C2×C22⋊C4×C71C22semidirect
18. #99×C71C22semidirect
20. C24×C71D4semidirect
22. C24×C71C23semidirect
34. C2×D4×C71D4semidirect
38. C22×C4×C71D4semidirect
39. C22×C4×C71C23semidirect
44. C22⋊C4×C71C23semidirect
93. #3026C71direct (N × H)
94. C71#3026direct (N × H)
95. C2×H(5,𝔽2)C2×C71semidirect
96. #534C2×C71semidirect
97. #458C2×C71semidirect
98. B(4,𝔽2)C2×C71semidirect
99. C2×#99C2×C71semidirect
105. C22×D4C22×C71semidirect
107. #123C22×C71semidirect
110. #120C22×C71semidirect
111. C2×C22⋊C4C22×C71semidirect
112. #99C22×C71semidirect
114. C24D4×C71semidirect
116. C24C23×C71semidirect
128. C2×D4D4×C71semidirect
132. C22×C4D4×C71semidirect
133. C22×C4C23×C71semidirect
138. C22⋊C4C23×C71semidirect

Maximal subgroups

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

subgroupindex
C2×H(5,𝔽2)×C712
#534×C712
#458×C712
B(4,𝔽2)×C712
C2×#99×C712
#302671