Finite Group Databasebeta

← C2×#453×C71 #22585576

Subgroups — C2×#453×C71 #22585576

Order 9088.

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

normal subgroups only

This group has 844 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×AGL(1,ℤ/8)×C71C212yesmaximal, complemented
3C2×#127×C71C211yesmaximal, characteristic, complemented
4#453×C71C218yesmaximal, direct product factor
5C2×#104×C71C212yesmaximal, complemented
6C2×#100×C71C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C71—24no—
8H(5,𝔽2)×C71—22no—
9H(5,𝔽2)×C71C2214yescomplemented
10C2×#41×C71—22no—
11C2×#41×C71C2212yes—
12C22×D4×C71—22no—
13C22×D4×C71—22no—
14C22×D4×C71—22no—
15C22×D4×C71C2211yescharacteristic
16C22×D4×C71C2211yescharacteristic, complemented
17AGL(1,ℤ/8)×C71—24no—
18AGL(1,ℤ/8)×C71C2218yescomplemented
19C2×SD16×C71—22no—
20C2×D8×C71—22no—
21C2×M16×C71C2212yescomplemented
22#127×C71—22no—
23#127×C71C2214yescomplemented
24C2×C42×C71C2211yescharacteristic, complemented
25#104×C71C2218yescomplemented
26#100×C71C2214yescomplemented
27C24×C71—22no—
28C24×C71—21no—
29C24×C71D412yes—
30#41×C71—28no—
31#41×C71—28no—
32#41×C71—22no—
33#41×C71—22no—
34#41×C71D414yescomplemented
35#41×C71C2314yes—
36C2×Q8×C71D412yescomplemented
37C2×D4×C71—24no—
38C2×D4×C71—28no—
39C2×D4×C71—28no—
40C2×D4×C71—22no—
41C2×D4×C71—28no—
42C2×D4×C71—28no—
43C2×D4×C71—28no—
44C2×D4×C71—22no—
45C2×D4×C71—22no—
46C2×D4×C71—22no—
47C2×D4×C71—22no—
48C2×D4×C71—21no—
49C2×D4×C71—48no—
50C2×D4×C71D412yes—
51C2×D4×C71D412yescomplemented
52C2×D4×C71C2312yes—
53C2×D4×C71C2312yescomplemented
54C22×C4×C71—22no—
55C22×C4×C71—21no—
56C22×C4×C71—22no—
57C22×C4×C71C2311yescharacteristic
58SD16×C71—28no—
59D8×C71—28no—
60M16×C71—22no—
61M16×C71C2314yescomplemented
62C2×C8×C71—22no—
63C42×C71—21no—
64C42×C71C2312yescomplemented
65C23×C71—28no—
66C23×C71—22no—
67C23×C71—22no—
68C23×C71—22no—
69C23×C71—22no—
70C23×C71—44no—
71C23×C71—44no—
72C23×C71—41no—
73C23×C71—41no—
74C23×C71—84no—
75C23×C71C2×D411yescharacteristic
76C23×C71C2×D414yes—
77Q8×C71—22no—
78Q8×C71C2×D414yescomplemented
79D4×C71—216no—
80D4×C71—24no—
81D4×C71—24no—
82D4×C71—24no—
83D4×C71—22no—
84D4×C71—24no—
85D4×C71—24no—
86D4×C71—416no—
87D4×C71—42no—
88D4×C71C2×D414yescomplemented
89C2×C4×C71—28no—
90C2×C4×C71—24no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—24no—
94C2×C4×C71—24no—
95C2×C4×C71C2×D411yescharacteristic
96C2×C4×C71C2×D412yes—
97C2×C4×C71C2×D412yes—
98C2×C4×C71C2411yescharacteristic
99C8×C71—24no—
100C22×C71—22no—
101C22×C71—24no—
102C22×C71—24no—
103C22×C71—44no—
104C22×C71—44no—
105C22×C71—44no—
106C22×C71—44no—
107C22×C71—42no—
108C22×C71—82no—
109C22×C71—82no—
110C22×C71—88no—
111C22×C71—82no—
112C22×C71#12011yescharacteristic
113C22×C71#12011yescharacteristic
114C22×C71C22×D411yescharacteristic
115C4×C71—24no—
116C4×C71—24no—
117C4×C71#12012yes—
118C4×C71C22×D412yes—
119C2×C71—21no—
120C2×C71—21no—
121C2×C71—44no—
122C2×C71—44no—
123C2×C71—82no—
124C2×C71#45312yesdirect product factor
125C2×C71C2×#12011yescharacteristic
126C2×#453C7111yesmaximal, characteristic, direct product factor, Sylow
127C71C2×#45311yescharacteristic, direct product factor, Sylow
128C2×H(5,𝔽2)C2×C7111yescharacteristic, complemented
129C2×AGL(1,ℤ/8)C2×C7112yescomplemented
130C2×#127C2×C7111yescharacteristic, complemented
131#453C2×C7118yesdirect product factor
132C2×#104C2×C7112yescomplemented
133C2×#100C2×C7111yescharacteristic, complemented
134H(5,𝔽2)—24no—
135H(5,𝔽2)—22no—
136H(5,𝔽2)C22×C7114yescomplemented
137C2×#41—22no—
138C2×#41C22×C7112yes—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4C22×C7111yescharacteristic
143C22×D4C22×C7111yescharacteristic, complemented
144AGL(1,ℤ/8)—24no—
145AGL(1,ℤ/8)C22×C7118yescomplemented
146C2×SD16—22no—
147C2×D8—22no—
148C2×M16C22×C7112yescomplemented
149#127—22no—
150#127C22×C7114yescomplemented
151C2×C42C22×C7111yescharacteristic, complemented
152#104C22×C7118yescomplemented
153#100C22×C7114yescomplemented
154C24—22no—
155C24—21no—
156C24D4×C7112yes—
157#41—28no—
158#41—28no—
159#41—22no—
160#41—22no—
161#41D4×C7114yescomplemented
162#41C23×C7114yes—
163C2×Q8D4×C7112yescomplemented
164C2×D4—24no—
165C2×D4—28no—
166C2×D4—28no—
167C2×D4—22no—
168C2×D4—28no—
169C2×D4—28no—
170C2×D4—28no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—21no—
176C2×D4—48no—
177C2×D4D4×C7112yes—
178C2×D4D4×C7112yescomplemented
179C2×D4C23×C7112yes—
180C2×D4C23×C7112yescomplemented
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4C23×C7111yescharacteristic
185SD16—28no—
186D8—28no—
187M16—22no—
188M16C23×C7114yescomplemented
189C2×C8—22no—
190C42—21no—
191C42C23×C7112yescomplemented
192C23—28no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—44no—
198C23—44no—
199C23—41no—
200C23—41no—
201C23—84no—
202C23C2×D4×C7111yescharacteristic
203C23C2×D4×C7114yes—
204Q8—22no—
205Q8C2×D4×C7114yescomplemented
206D4—216no—
207D4—24no—
208D4—24no—
209D4—24no—
210D4—22no—
211D4—24no—
212D4—24no—
213D4—416no—
214D4—42no—
215D4C2×D4×C7114yescomplemented
216C2×C4—28no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4C2×D4×C7111yescharacteristic
223C2×C4C2×D4×C7112yes—
224C2×C4C2×D4×C7112yes—
225C2×C4C24×C7111yescharacteristic
226C8—24no—
227C22—22no—
228C22—24no—
229C22—24no—
230C22—44no—
231C22—44no—
232C22—44no—
233C22—44no—
234C22—42no—
235C22—82no—
236C22—82no—
237C22—88no—
238C22—82no—
239C22#120×C7111yescharacteristic
240C22#120×C7111yescharacteristic
241C22C22×D4×C7111yescharacteristic
242C4—24no—
243C4—24no—
244C4#120×C7112yes—
245C4C22×D4×C7112yes—
246C2—21no—
247C2—21no—
248C2—44no—
249C2—44no—
250C2—82no—
251C2#453×C7112yesdirect product factor
252C2C2×#120×C7111yescharacteristic
subgroupG/H
center, socleC22×C71#120
commutator, Frattini, cosocleC2×C4C24×C71
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C22×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×H(5,𝔽2)×C71C2semidirect
2. C2×AGL(1,ℤ/8)×C71C2semidirect
3. C2×#127×C71C2semidirect
4. #453×C71C2direct (N × H)
5. C2×#104×C71C2semidirect
6. C2×#100×C71C2semidirect
9. H(5,𝔽2)×C71C22semidirect
16. C22×D4×C71C22semidirect
18. AGL(1,ℤ/8)×C71C22semidirect
21. C2×M16×C71C22semidirect
23. #127×C71C22semidirect
24. C2×C42×C71C22semidirect
25. #104×C71C22semidirect
26. #100×C71C22semidirect
34. #41×C71D4semidirect
36. C2×Q8×C71D4semidirect
51. C2×D4×C71D4semidirect
53. C2×D4×C71C23semidirect
61. M16×C71C23semidirect
64. C42×C71C23semidirect
78. Q8×C71C2×D4semidirect
88. D4×C71C2×D4semidirect
124. C2×C71#453direct (N × H)
126. C2×#453C71direct (N × H)
127. C71C2×#453direct (N × H)
128. C2×H(5,𝔽2)C2×C71semidirect
129. C2×AGL(1,ℤ/8)C2×C71semidirect
130. C2×#127C2×C71semidirect
131. #453C2×C71direct (N × H)
132. C2×#104C2×C71semidirect
133. C2×#100C2×C71semidirect
136. H(5,𝔽2)C22×C71semidirect
143. C22×D4C22×C71semidirect
145. AGL(1,ℤ/8)C22×C71semidirect
148. C2×M16C22×C71semidirect
150. #127C22×C71semidirect
151. C2×C42C22×C71semidirect
152. #104C22×C71semidirect
153. #100C22×C71semidirect
161. #41D4×C71semidirect
163. C2×Q8D4×C71semidirect
178. C2×D4D4×C71semidirect
180. C2×D4C23×C71semidirect
188. M16C23×C71semidirect
191. C42C23×C71semidirect
205. Q8C2×D4×C71semidirect
215. D4C2×D4×C71semidirect
251. C2#453×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×AGL(1,ℤ/8)×C712
C2×#127×C712
#453×C712
C2×#104×C712
C2×#100×C712
C2×#45371