Finite Group Databasebeta

← C71×#11670 #31832542

Subgroups — C71×#11670 #31832542

Order 18176.

348 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C71×#2956C211yesmaximal, characteristic
2C71×#2165C211yesmaximal, characteristic, complemented
3C71×#1453C211yesmaximal, characteristic, complemented
4C71×#1336C212yesmaximal, complemented
5C71×#1335C212yesmaximal, complemented
6#575×C71C411yescharacteristic
7#568×C71—21no—
8#444×C71—21no—
9#444×C71C412yes—
10#443×C71—21no—
11#443×C71C2212yes—
12#431×C71—21no—
13#430×C71C411yescharacteristic
14C2×#104×C71—21no—
15C4×M16×C71C2211yescharacteristic
16#364×C71C2212yescomplemented
17#349×C71—21no—
18#347×C71—21no—
19#322×C71—21no—
20#322×C71C2211yescharacteristic, complemented
21C82×C71C2211yescharacteristic, complemented
22C2×#41×C71—21no—
23#137×C71—22no—
24AGL(1,ℤ/8)×C71—22no—
25#135×C71—21no—
26#135×C71C2×C412yes—
27#131×C71—22no—
28#131×C71—22no—
29#131×C71—42no—
30C2×M16×C71—21no—
31C2×M16×C71—42no—
32C2×M16×C71C2×C411yescharacteristic
33C2×M16×C71D411yescharacteristic
34C2×C42×C71D411yescharacteristic
35M32×C71—22no—
36#108×C71—21no—
37#108×C71C2×C412yes—
38C4⋊C8×C71—22no—
39#104×C71—22no—
40#104×C71—22no—
41C22⋊C8×C71—22no—
42#97×C71C2×C411yescharacteristic
43#97×C71D411yescharacteristic
44C4×C8×C71—21no—
45C4×C8×C71—21no—
46C4×C8×C71D411yescharacteristic
47C4×C8×C71C2311yescharacteristic
48#41×C71—22no—
49#41×C71—22no—
50#41×C71—41no—
51#41×C71—41no—
52C2×Q8×C71—21no—
53C2×D4×C71—21no—
54C2×D4×C71—41no—
55C22×C4×C71—21no—
56C22×C4×C71—41no—
57C22×C4×C71C22⋊C411yescharacteristic
58Q16×C71—21no—
59Q16×C71M1612yes—
60SD16×C71—22no—
61SD16×C71—22no—
62D8×C71—21no—
63D8×C71M1612yes—
64M16×C71—22no—
65M16×C71—22no—
66M16×C71—22no—
67M16×C71—42no—
68M16×C71—41no—
69M16×C71—42no—
70C2×C8×C71—22no—
71C2×C8×C71—22no—
72C2×C8×C71—42no—
73C2×C8×C71—41no—
74C2×C8×C71—42no—
75C2×C8×C71C22⋊C411yescharacteristic
76C2×C8×C71C22⋊C411yescharacteristic
77C2×C8×C71C22×C411yescharacteristic
78C2×C8×C71C2×D411yescharacteristic
79C42×C71—21no—
80C42×C71C22⋊C411yescharacteristic
81C42×C71C2×D411yescharacteristic
82C16×C71—42no—
83C23×C71—41no—
84C23×C71#10411yescharacteristic
85Q8×C71—22no—
86Q8×C71—41no—
87D4×C71—22no—
88D4×C71—41no—
89D4×C71—42no—
90D4×C71—42no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—41no—
98C2×C4×C71—42no—
99C2×C4×C71#10411yescharacteristic
100C2×C4×C71C2×C22⋊C411yescharacteristic
101C8×C71—21no—
102C8×C71—21no—
103C8×C71—42no—
104C8×C71—42no—
105C8×C71—42no—
106C8×C71C2×M1612yes—
107C8×C71AGL(1,ℤ/8)12yes—
108C22×C71—21no—
109C22×C71—41no—
110C22×C71—42no—
111C22×C71—161no—
112C22×C71C2×#10411yescharacteristic
113C4×C71—22no—
114C4×C71—21no—
115C4×C71—41no—
116C4×C71—42no—
117C4×C71#40711yescharacteristic
118C4×C71#41811yescharacteristic
119#11670C7111yesmaximal, characteristic, direct product factor, Sylow
120C2×C71—21no—
121C2×C71—41no—
122C2×C71—82no—
123C2×C71#149011yescharacteristic
124#2956C2×C7111yescharacteristic
125#2165C2×C7111yescharacteristic, complemented
126#1453C2×C7111yescharacteristic, complemented
127#1336C2×C7112yescomplemented
128#1335C2×C7112yescomplemented
129C71#1167011yescharacteristic, direct product factor, Sylow
130#575C4×C7111yescharacteristic
131#568—21no—
132#444—21no—
133#444C4×C7112yes—
134#443—21no—
135#443C22×C7112yes—
136#431—21no—
137#430C4×C7111yescharacteristic
138C2×#104—21no—
139C4×M16C22×C7111yescharacteristic
140#364C22×C7112yescomplemented
141#349—21no—
142#347—21no—
143#322—21no—
144#322C22×C7111yescharacteristic, complemented
145C82C22×C7111yescharacteristic, complemented
146C2×#41—21no—
147#137—22no—
148AGL(1,ℤ/8)—22no—
149#135—21no—
150#135C2×C4×C7112yes—
151#131—22no—
152#131—22no—
153#131—42no—
154C2×M16—21no—
155C2×M16—42no—
156C2×M16C2×C4×C7111yescharacteristic
157C2×M16D4×C7111yescharacteristic
158C2×C42D4×C7111yescharacteristic
159M32—22no—
160#108—21no—
161#108C2×C4×C7112yes—
162C4⋊C8—22no—
163#104—22no—
164#104—22no—
165C22⋊C8—22no—
166#97C2×C4×C7111yescharacteristic
167#97D4×C7111yescharacteristic
168C4×C8—21no—
169C4×C8—21no—
170C4×C8D4×C7111yescharacteristic
171C4×C8C23×C7111yescharacteristic
172#41—22no—
173#41—22no—
174#41—41no—
175#41—41no—
176C2×Q8—21no—
177C2×D4—21no—
178C2×D4—41no—
179C22×C4—21no—
180C22×C4—41no—
181C22×C4C22⋊C4×C7111yescharacteristic
182Q16—21no—
183Q16M16×C7112yes—
184SD16—22no—
185SD16—22no—
186D8—21no—
187D8M16×C7112yes—
188M16—22no—
189M16—22no—
190M16—22no—
191M16—42no—
192M16—41no—
193M16—42no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—42no—
197C2×C8—41no—
198C2×C8—42no—
199C2×C8C22⋊C4×C7111yescharacteristic
200C2×C8C22⋊C4×C7111yescharacteristic
201C2×C8C22×C4×C7111yescharacteristic
202C2×C8C2×D4×C7111yescharacteristic
203C42—21no—
204C42C22⋊C4×C7111yescharacteristic
205C42C2×D4×C7111yescharacteristic
206C16—42no—
207C23—41no—
208C23#104×C7111yescharacteristic
209Q8—22no—
210Q8—41no—
211D4—22no—
212D4—41no—
213D4—42no—
214D4—42no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4#104×C7111yescharacteristic
224C2×C4C2×C22⋊C4×C7111yescharacteristic
225C8—21no—
226C8—21no—
227C8—42no—
228C8—42no—
229C8—42no—
230C8C2×M16×C7112yes—
231C8AGL(1,ℤ/8)×C7112yes—
232C22—21no—
233C22—41no—
234C22—42no—
235C22—161no—
236C22C2×#104×C7111yescharacteristic
237C4—22no—
238C4—21no—
239C4—41no—
240C4—42no—
241C4#407×C7111yescharacteristic
242C4#418×C7111yescharacteristic
243C2—21no—
244C2—41no—
245C2—82no—
246C2C71×#149011yescharacteristic
subgroupG/H
centerC4×C71#418
commutatorC2×C8C22×C4×C71
Frattini, cosocleC4×C8C23×C71
socleC2×C71#1490
prime radical#575×C71C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C71 ◁ C2×C8×C71 ◁ C4×M16×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#2165C2semidirect
3. C71×#1453C2semidirect
4. C71×#1336C2semidirect
5. C71×#1335C2semidirect
16. #364×C71C22semidirect
20. #322×C71C22semidirect
21. C82×C71C22semidirect
119. #11670C71direct (N × H)
125. #2165C2×C71semidirect
126. #1453C2×C71semidirect
127. #1336C2×C71semidirect
128. #1335C2×C71semidirect
129. C71#11670direct (N × H)
140. #364C22×C71semidirect
144. #322C22×C71semidirect
145. C82C22×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#29562
C71×#21652
C71×#14532
C71×#13362
C71×#13352
#1167071