Finite Group Databasebeta

← C71×#3020 #22585582

Subgroups — C71×#3020 #22585582

Order 9088.

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

normal subgroups only

This group has 698 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
1C2×2−1+4×C71C211yesmaximal, characteristic, complemented
2#575×C71C212yesmaximal, complemented
3#536×C71C211yesmaximal, characteristic, complemented
4#456×C71C214yesmaximal, complemented
5#454×C71C214yesmaximal, complemented
6#421×C71C212yesmaximal, complemented
7C2×#101×C71C211yesmaximal, characteristic, complemented
82−1+4×C71—24no—
92−1+4×C71—22no—
102−1+4×C71C2214yes—
11C2×#41×C71—22no—
12C2×#41×C71—22no—
13C2×#41×C71C2212yes—
14C2×#41×C71C2211yescharacteristic
15C22×Q8×C71—22no—
16C22×Q8×C71C2211yescharacteristic
17#137×C71—22no—
18#137×C71C2214yescomplemented
19AGL(1,ℤ/8)×C71—22no—
20AGL(1,ℤ/8)×C71C2214yescomplemented
21#135×C71—24no—
22C2×M16×C71C2212yes—
23C4⋊Q8×C71C2212yescomplemented
24#124×C71C2212yescomplemented
25#123×C71—22no—
26C22⋊Q8×C71—22no—
27#117×C71C2211yescharacteristic
28#104×C71C2218yescomplemented
29#101×C71C2214yescomplemented
30#41×C71—28no—
31#41×C71—28no—
32#41×C71—28no—
33#41×C71—28no—
34#41×C71—22no—
35#41×C71—22no—
36#41×C71—22no—
37#41×C71—22no—
38#41×C71D414yes—
39#41×C71C2314yes—
40C2×Q8×C71—22no—
41C2×Q8×C71—28no—
42C2×Q8×C71—28no—
43C2×Q8×C71—22no—
44C2×Q8×C71—22no—
45C2×Q8×C71D412yes—
46C2×Q8×C71D412yes—
47C2×Q8×C71C2311yescharacteristic
48C2×Q8×C71C2312yes—
49C2×D4×C71—22no—
50C2×D4×C71—22no—
51C2×D4×C71—21no—
52C2×D4×C71D412yes—
53C2×D4×C71C2311yescharacteristic
54C22×C4×C71—24no—
55C22×C4×C71—22no—
56C22×C4×C71—21no—
57C22×C4×C71D412yes—
58C22×C4×C71C2311yescharacteristic
59Q16×C71—24no—
60SD16×C71—24no—
61SD16×C71—24no—
62D8×C71—24no—
63M16×C71—22no—
64M16×C71C2314yes—
65C2×C8×C71—22no—
66C4⋊C4×C71—21no—
67C4⋊C4×C71—24no—
68C22⋊C4×C71—21no—
69C22⋊C4×C71—24no—
70C42×C71C2312yes—
71C23×C71—22no—
72C23×C71—21no—
73C23×C71C2×D411yescharacteristic
74Q8×C71—28no—
75Q8×C71—22no—
76Q8×C71—24no—
77Q8×C71—24no—
78Q8×C71—22no—
79Q8×C71C2×D414yes—
80D4×C71—28no—
81D4×C71—24no—
82D4×C71—24no—
83D4×C71—22no—
84D4×C71—22no—
85D4×C71—41no—
86D4×C71—41no—
87D4×C71C2×D414yes—
88C2×C4×C71—216no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—24no—
93C2×C4×C71—24no—
94C2×C4×C71—22no—
95C2×C4×C71—21no—
96C2×C4×C71—42no—
97C2×C4×C71C2×D414yes—
98C2×C4×C71C2×D411yescharacteristic
99C2×C4×C71C2×D412yes—
100C2×C4×C71C2×D412yes—
101C2×C4×C71C2411yescharacteristic
102C8×C71—24no—
103C22×C71—24no—
104C22×C71—41no—
105C22×C71—44no—
106C22×C71—42no—
107C22×C71—41no—
108C22×C71#12011yescharacteristic
109C22×C71#12011yescharacteristic
110C22×C71C22×D411yescharacteristic
111C4×C71—24no—
112C4×C71—24no—
113C4×C71—42no—
114C4×C71—41no—
115C4×C71#12012yes—
116C4×C71C22×D412yes—
117C2×C71—21no—
118C2×C71—21no—
119C2×C71—21no—
120C2×C71—44no—
121C2×C71—81no—
122C2×C71C2×#12011yescharacteristic
123#3020C7111yesmaximal, characteristic, direct product factor, Sylow
124C71#302011yescharacteristic, direct product factor, Sylow
125C2×2−1+4C2×C7111yescharacteristic, complemented
126#575C2×C7112yescomplemented
127#536C2×C7111yescharacteristic, complemented
128#456C2×C7114yescomplemented
129#454C2×C7114yescomplemented
130#421C2×C7112yescomplemented
131C2×#101C2×C7111yescharacteristic, complemented
1322−1+4—24no—
1332−1+4—22no—
1342−1+4C22×C7114yes—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41C22×C7112yes—
138C2×#41C22×C7111yescharacteristic
139C22×Q8—22no—
140C22×Q8C22×C7111yescharacteristic
141#137—22no—
142#137C22×C7114yescomplemented
143AGL(1,ℤ/8)—22no—
144AGL(1,ℤ/8)C22×C7114yescomplemented
145#135—24no—
146C2×M16C22×C7112yes—
147C4⋊Q8C22×C7112yescomplemented
148#124C22×C7112yescomplemented
149#123—22no—
150C22⋊Q8—22no—
151#117C22×C7111yescharacteristic
152#104C22×C7118yescomplemented
153#101C22×C7114yescomplemented
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—22no—
161#41—22no—
162#41D4×C7114yes—
163#41C23×C7114yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8D4×C7112yes—
170C2×Q8D4×C7112yes—
171C2×Q8C23×C7111yescharacteristic
172C2×Q8C23×C7112yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—21no—
176C2×D4D4×C7112yes—
177C2×D4C23×C7111yescharacteristic
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4D4×C7112yes—
182C22×C4C23×C7111yescharacteristic
183Q16—24no—
184SD16—24no—
185SD16—24no—
186D8—24no—
187M16—22no—
188M16C23×C7114yes—
189C2×C8—22no—
190C4⋊C4—21no—
191C4⋊C4—24no—
192C22⋊C4—21no—
193C22⋊C4—24no—
194C42C23×C7112yes—
195C23—22no—
196C23—21no—
197C23C2×D4×C7111yescharacteristic
198Q8—28no—
199Q8—22no—
200Q8—24no—
201Q8—24no—
202Q8—22no—
203Q8C2×D4×C7114yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—22no—
209D4—41no—
210D4—41no—
211D4C2×D4×C7114yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—42no—
221C2×C4C2×D4×C7114yes—
222C2×C4C2×D4×C7111yescharacteristic
223C2×C4C2×D4×C7112yes—
224C2×C4C2×D4×C7112yes—
225C2×C4C24×C7111yescharacteristic
226C8—24no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—42no—
231C22—41no—
232C22#120×C7111yescharacteristic
233C22#120×C7111yescharacteristic
234C22C22×D4×C7111yescharacteristic
235C4—24no—
236C4—24no—
237C4—42no—
238C4—41no—
239C4#120×C7112yes—
240C4C22×D4×C7112yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—81no—
246C2C2×#120×C7111yescharacteristic
subgroupG/H
center, socleC2×C71C2×#120
commutator, Frattini, cosocleC2×C4C24×C71
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C71 ◁ C22×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×2−1+4×C71C2semidirect
2. #575×C71C2semidirect
3. #536×C71C2semidirect
4. #456×C71C2semidirect
5. #454×C71C2semidirect
6. #421×C71C2semidirect
7. C2×#101×C71C2semidirect
18. #137×C71C22semidirect
20. AGL(1,ℤ/8)×C71C22semidirect
23. C4⋊Q8×C71C22semidirect
24. #124×C71C22semidirect
28. #104×C71C22semidirect
29. #101×C71C22semidirect
123. #3020C71direct (N × H)
124. C71#3020direct (N × H)
125. C2×2−1+4C2×C71semidirect
126. #575C2×C71semidirect
127. #536C2×C71semidirect
128. #456C2×C71semidirect
129. #454C2×C71semidirect
130. #421C2×C71semidirect
131. C2×#101C2×C71semidirect
142. #137C22×C71semidirect
144. AGL(1,ℤ/8)C22×C71semidirect
147. C4⋊Q8C22×C71semidirect
148. #124C22×C71semidirect
152. #104C22×C71semidirect
153. #101C22×C71semidirect

Maximal subgroups

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

subgroupindex
C2×2−1+4×C712
#575×C712
#536×C712
#456×C712
#454×C712
#421×C712
C2×#101×C712
#302071