Finite Group Databasebeta

← C2×B(4,𝔽2)×C71 #22585585

Subgroups — C2×B(4,𝔽2)×C71 #22585585

Order 9088.

1018 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 216 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
2C2×#120×C71C213yesmaximal, complemented
3B(4,𝔽2)×C71C218yesmaximal, direct product factor
4C2×#99×C71C213yesmaximal, complemented
5C25×C71C2211yescharacteristic, complemented
6H(5,𝔽2)×C71—26no—
7H(5,𝔽2)×C71C2214yescomplemented
8C2×#41×C71—23no—
9C22×D4×C71—23no—
10C22×D4×C71—23no—
11C22×D4×C71C2213yes—
12#120×C71—26no—
13#120×C71C22112yescomplemented
14C2×C22⋊C4×C71—23no—
15C2×C22⋊C4×C71C2213yescomplemented
16#99×C71C22112yescomplemented
17C24×C71—21no—
18C24×C71—23no—
19C24×C71—23no—
20C24×C71—44no—
21C24×C71D413yescomplemented
22C24×C71C2311yescharacteristic
23C24×C71C2312yescomplemented
24#41×C71—212no—
25#41×C71—212no—
26C2×Q8×C71—21no—
27C2×D4×C71—23no—
28C2×D4×C71—212no—
29C2×D4×C71—26no—
30C2×D4×C71—26no—
31C2×D4×C71—212no—
32C2×D4×C71—212no—
33C2×D4×C71—412no—
34C2×D4×C71D416yescomplemented
35C2×D4×C71C2316yes—
36C22×C4×C71—23no—
37C22×C4×C71—23no—
38C22×C4×C71D413yescomplemented
39C22⋊C4×C71—23no—
40C22⋊C4×C71—212no—
41C22⋊C4×C71C2316yescomplemented
42C23×C71—212no—
43C23×C71—24no—
44C23×C71—23no—
45C23×C71—26no—
46C23×C71—23no—
47C23×C71—42no—
48C23×C71—44no—
49C23×C71—46no—
50C23×C71—412no—
51C23×C71—46no—
52C23×C71—43no—
53C23×C71—41no—
54C23×C71—82no—
55C23×C71C2×D413yes—
56C23×C71C2×D413yes—
57C23×C71C2×D416yescomplemented
58C23×C71C2411yescharacteristic
59Q8×C71—24no—
60D4×C71—212no—
61D4×C71—212no—
62D4×C71—212no—
63D4×C71—424no—
64C2×C4×C71—212no—
65C2×C4×C71—23no—
66C2×C4×C71—23no—
67C2×C4×C71—26no—
68C2×C4×C71—46no—
69C2×C4×C71C2×D416yescomplemented
70C22×C71—26no—
71C22×C71—212no—
72C22×C71—26no—
73C22×C71—43no—
74C22×C71—41no—
75C22×C71—46no—
76C22×C71—46no—
77C22×C71—412no—
78C22×C71—42no—
79C22×C71—44no—
80C22×C71—46no—
81C22×C71—83no—
82C22×C71—84no—
83C22×C71#12013yes—
84C22×C71#12011yescharacteristic
85C22×C71C22×D413yes—
86C4×C71—26no—
87C4×C71—46no—
88C2×C71—23no—
89C2×C71—23no—
90C2×C71—44no—
91C2×C71—46no—
92C2×C71B(4,𝔽2)12yesdirect product factor
93C2×C71C2×#12011yescharacteristic
94C2×B(4,𝔽2)C7111yesmaximal, characteristic, direct product factor, Sylow
95C71C2×B(4,𝔽2)11yescharacteristic, direct product factor, Sylow
96C2×H(5,𝔽2)C2×C7111yescharacteristic, complemented
97C2×#120C2×C7113yescomplemented
98B(4,𝔽2)C2×C7118yesdirect product factor
99C2×#99C2×C7113yescomplemented
100C25C22×C7111yescharacteristic, complemented
101H(5,𝔽2)—26no—
102H(5,𝔽2)C22×C7114yescomplemented
103C2×#41—23no—
104C22×D4—23no—
105C22×D4—23no—
106C22×D4C22×C7113yes—
107#120—26no—
108#120C22×C71112yescomplemented
109C2×C22⋊C4—23no—
110C2×C22⋊C4C22×C7113yescomplemented
111#99C22×C71112yescomplemented
112C24—21no—
113C24—23no—
114C24—23no—
115C24—44no—
116C24D4×C7113yescomplemented
117C24C23×C7111yescharacteristic
118C24C23×C7112yescomplemented
119#41—212no—
120#41—212no—
121C2×Q8—21no—
122C2×D4—23no—
123C2×D4—212no—
124C2×D4—26no—
125C2×D4—26no—
126C2×D4—212no—
127C2×D4—212no—
128C2×D4—412no—
129C2×D4D4×C7116yescomplemented
130C2×D4C23×C7116yes—
131C22×C4—23no—
132C22×C4—23no—
133C22×C4D4×C7113yescomplemented
134C22⋊C4—23no—
135C22⋊C4—212no—
136C22⋊C4C23×C7116yescomplemented
137C23—212no—
138C23—24no—
139C23—23no—
140C23—26no—
141C23—23no—
142C23—42no—
143C23—44no—
144C23—46no—
145C23—412no—
146C23—46no—
147C23—43no—
148C23—41no—
149C23—82no—
150C23C2×D4×C7113yes—
151C23C2×D4×C7113yes—
152C23C2×D4×C7116yescomplemented
153C23C24×C7111yescharacteristic
154Q8—24no—
155D4—212no—
156D4—212no—
157D4—212no—
158D4—424no—
159C2×C4—212no—
160C2×C4—23no—
161C2×C4—23no—
162C2×C4—26no—
163C2×C4—46no—
164C2×C4C2×D4×C7116yescomplemented
165C22—26no—
166C22—212no—
167C22—26no—
168C22—43no—
169C22—41no—
170C22—46no—
171C22—46no—
172C22—412no—
173C22—44no—
174C22—46no—
175C22—42no—
176C22—83no—
177C22—84no—
178C22#120×C7113yes—
179C22#120×C7111yescharacteristic
180C22C22×D4×C7113yes—
181C4—26no—
182C4—46no—
183C2—23no—
184C2—23no—
185C2—44no—
186C2—46no—
187C2B(4,𝔽2)×C7112yesdirect product factor
188C2C2×#120×C7111yescharacteristic
subgroupG/H
center, socleC22×C71#120
commutator, Frattini, cosocleC23C24×C71
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×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. C2×#120×C71C2semidirect
3. B(4,𝔽2)×C71C2direct (N × H)
4. C2×#99×C71C2semidirect
5. C25×C71C22semidirect
7. H(5,𝔽2)×C71C22semidirect
13. #120×C71C22semidirect
15. C2×C22⋊C4×C71C22semidirect
16. #99×C71C22semidirect
21. C24×C71D4semidirect
23. C24×C71C23semidirect
34. C2×D4×C71D4semidirect
38. C22×C4×C71D4semidirect
41. C22⋊C4×C71C23semidirect
57. C23×C71C2×D4semidirect
69. C2×C4×C71C2×D4semidirect
92. C2×C71B(4,𝔽2)direct (N × H)
94. C2×B(4,𝔽2)C71direct (N × H)
95. C71C2×B(4,𝔽2)direct (N × H)
96. C2×H(5,𝔽2)C2×C71semidirect
97. C2×#120C2×C71semidirect
98. B(4,𝔽2)C2×C71direct (N × H)
99. C2×#99C2×C71semidirect
100. C25C22×C71semidirect
102. H(5,𝔽2)C22×C71semidirect
108. #120C22×C71semidirect
110. C2×C22⋊C4C22×C71semidirect
111. #99C22×C71semidirect
116. C24D4×C71semidirect
118. C24C23×C71semidirect
129. C2×D4D4×C71semidirect
133. C22×C4D4×C71semidirect
136. C22⋊C4C23×C71semidirect
152. C23C2×D4×C71semidirect
164. C2×C4C2×D4×C71semidirect
187. C2B(4,𝔽2)×C71direct (N × H)

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
C2×#120×C712
B(4,𝔽2)×C712
C2×#99×C712
C2×B(4,𝔽2)71