Finite Group Databasebeta

← C71×#10280 #31831152

Subgroups — C71×#10280 #31831152

Order 18176.

950 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 130 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×#3205C212yesmaximal, complemented
2C71×#2867C211yesmaximal, characteristic, complemented
3C71×#1962C211yesmaximal, characteristic, complemented
4C71×#1545C212yesmaximal, complemented
5C71×#1276C211yesmaximal, characteristic, complemented
6#532×C71C2211yescharacteristic
7C2×#127×C71—24no—
8C2×#127×C71—22no—
9C2×#127×C71C2212yes—
10#493×C71—22no—
11#486×C71—22no—
12#459×C71—24no—
13#447×C71—24no—
14#431×C71C2212yes—
15#423×C71C2211yescharacteristic
16C2×#104×C71—22no—
17C43×C71C2211yescharacteristic
18#331×C71—24no—
19C2×#41×C71—21no—
20C22×D4×C71—28no—
21C22×D4×C71—22no—
22C22×D4×C71—22no—
23C22×D4×C71D414yes—
24C2×D8×C71—28no—
25C2×M16×C71—21no—
26C4⋊Q8×C71D411yescharacteristic, complemented
27#127×C71—22no—
28#127×C71—22no—
29#127×C71—22no—
30#127×C71—22no—
31#127×C71—416no—
32#127×C71—44no—
33#127×C71—44no—
34#127×C71D411yescharacteristic, complemented
35#124×C71—21no—
36#121×C71—22no—
37C4×D4×C71—22no—
38C2×C42×C71—22no—
39C2×C42×C71D412yes—
40C2×C42×C71C2311yescharacteristic
41C4⋊C8×C71—22no—
42C4⋊C8×C71D414yescomplemented
43#104×C71—44no—
44#102×C71—28no—
45C22⋊C8×C71—22no—
46C4×C8×C71—22no—
47C24×C71—24no—
48#41×C71—42no—
49#41×C71—42no—
50C2×Q8×C71—21no—
51C2×D4×C71—24no—
52C2×D4×C71—22no—
53C2×D4×C71—21no—
54C2×D4×C71—24no—
55C2×D4×C71—48no—
56C2×D4×C71—432no—
57C2×D4×C71—416no—
58C2×D4×C71—48no—
59C2×D4×C71—41no—
60C2×D4×C71—42no—
61C2×D4×C71—42no—
62C2×D4×C71—42no—
63C2×D4×C71—42no—
64C2×D4×C71—44no—
65C2×D4×C71—44no—
66C22×C4×C71—22no—
67C22×C4×C71—41no—
68C22×C4×C71C2×D412yes—
69C22×C4×C71C2×D411yescharacteristic
70D8×C71—88no—
71M16×C71—42no—
72C2×C8×C71—24no—
73C2×C8×C71—41no—
74C4⋊C4×C71—22no—
75C22⋊C4×C71—42no—
76C42×C71—28no—
77C42×C71—22no—
78C42×C71D814yescomplemented
79C42×C71C2×D411yescharacteristic
80C42×C71C2×D412yes—
81C42×C71#4111yescharacteristic
82C23×C71—48no—
83C23×C71—48no—
84C23×C71—41no—
85C23×C71—44no—
86C23×C71—48no—
87C23×C71#12011yescharacteristic
88Q8×C71—42no—
89D4×C71—44no—
90D4×C71—48no—
91D4×C71—42no—
92D4×C71—84no—
93D4×C71—832no—
94D4×C71—84no—
95D4×C71—82no—
96D4×C71—84no—
97D4×C71—84no—
98D4×C71—82no—
99D4×C71—82no—
100C2×C4×C71—28no—
101C2×C4×C71—24no—
102C2×C4×C71—24no—
103C2×C4×C71—41no—
104C2×C4×C71—81no—
105C2×C4×C71—81no—
106C2×C4×C71#12012yes—
107C2×C4×C71#12112yes—
108C2×C4×C71#12111yescharacteristic
109C2×C4×C71#12711yescharacteristic
110C2×C4×C71C2×D812yes—
111C2×C4×C71AGL(1,ℤ/8)12yes—
112C8×C71—44no—
113C8×C71—81no—
114C22×C71—22no—
115C22×C71—816no—
116C22×C71—84no—
117C22×C71—82no—
118C22×C71—84no—
119C22×C71—81no—
120C22×C71—84no—
121C22×C71#39211yescharacteristic
122C22×C71#44712yes—
123C4×C71—28no—
124C4×C71—22no—
125C4×C71—81no—
126C4×C71#45314yes—
127C4×C71#45912yes—
128C4×C71#49311yescharacteristic
129C4×C71#49611yescharacteristic
130#10280C7111yesmaximal, characteristic, direct product factor, Sylow
131C2×C71—22no—
132C2×C71—161no—
133C2×C71—164no—
134C2×C71#200212yes—
135C2×C71#201111yescharacteristic
136#3205C2×C7112yescomplemented
137#2867C2×C7111yescharacteristic, complemented
138#1962C2×C7111yescharacteristic, complemented
139#1545C2×C7112yescomplemented
140#1276C2×C7111yescharacteristic, complemented
141C71#1028011yescharacteristic, direct product factor, Sylow
142#532C22×C7111yescharacteristic
143C2×#127—24no—
144C2×#127—22no—
145C2×#127C22×C7112yes—
146#493—22no—
147#486—22no—
148#459—24no—
149#447—24no—
150#431C22×C7112yes—
151#423C22×C7111yescharacteristic
152C2×#104—22no—
153C43C22×C7111yescharacteristic
154#331—24no—
155C2×#41—21no—
156C22×D4—28no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4D4×C7114yes—
160C2×D8—28no—
161C2×M16—21no—
162C4⋊Q8D4×C7111yescharacteristic, complemented
163#127—22no—
164#127—22no—
165#127—22no—
166#127—22no—
167#127—416no—
168#127—44no—
169#127—44no—
170#127D4×C7111yescharacteristic, complemented
171#124—21no—
172#121—22no—
173C4×D4—22no—
174C2×C42—22no—
175C2×C42D4×C7112yes—
176C2×C42C23×C7111yescharacteristic
177C4⋊C8—22no—
178C4⋊C8D4×C7114yescomplemented
179#104—44no—
180#102—28no—
181C22⋊C8—22no—
182C4×C8—22no—
183C24—24no—
184#41—42no—
185#41—42no—
186C2×Q8—21no—
187C2×D4—24no—
188C2×D4—22no—
189C2×D4—21no—
190C2×D4—24no—
191C2×D4—48no—
192C2×D4—432no—
193C2×D4—416no—
194C2×D4—48no—
195C2×D4—41no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—44no—
201C2×D4—44no—
202C22×C4—22no—
203C22×C4—41no—
204C22×C4C2×D4×C7112yes—
205C22×C4C2×D4×C7111yescharacteristic
206D8—88no—
207M16—42no—
208C2×C8—24no—
209C2×C8—41no—
210C4⋊C4—22no—
211C22⋊C4—42no—
212C42—28no—
213C42—22no—
214C42D8×C7114yescomplemented
215C42C2×D4×C7111yescharacteristic
216C42C2×D4×C7112yes—
217C42#41×C7111yescharacteristic
218C23—48no—
219C23—48no—
220C23—48no—
221C23—41no—
222C23—44no—
223C23#120×C7111yescharacteristic
224Q8—42no—
225D4—44no—
226D4—48no—
227D4—42no—
228D4—84no—
229D4—832no—
230D4—84no—
231D4—82no—
232D4—84no—
233D4—84no—
234D4—82no—
235D4—82no—
236C2×C4—28no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—41no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4#120×C7112yes—
243C2×C4#121×C7112yes—
244C2×C4#121×C7111yescharacteristic
245C2×C4#127×C7111yescharacteristic
246C2×C4C2×D8×C7112yes—
247C2×C4AGL(1,ℤ/8)×C7112yes—
248C8—44no—
249C8—81no—
250C22—22no—
251C22—816no—
252C22—84no—
253C22—82no—
254C22—84no—
255C22—81no—
256C22—84no—
257C22#392×C7111yescharacteristic
258C22#447×C7112yes—
259C4—28no—
260C4—22no—
261C4—81no—
262C4#453×C7114yes—
263C4#459×C7112yes—
264C4#493×C7111yescharacteristic
265C4#496×C7111yescharacteristic
266C2—22no—
267C2—164no—
268C2—161no—
269C2C71×#200212yes—
270C2C71×#201111yescharacteristic
subgroupG/H
center, socleC22×C71#392
commutator, Frattini, cosocleC2×C42C23×C71
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C71×#3205C2semidirect
2. C71×#2867C2semidirect
3. C71×#1962C2semidirect
4. C71×#1545C2semidirect
5. C71×#1276C2semidirect
26. C4⋊Q8×C71D4semidirect
34. #127×C71D4semidirect
42. C4⋊C8×C71D4semidirect
78. C42×C71D8semidirect
130. #10280C71direct (N × H)
136. #3205C2×C71semidirect
137. #2867C2×C71semidirect
138. #1962C2×C71semidirect
139. #1545C2×C71semidirect
140. #1276C2×C71semidirect
141. C71#10280direct (N × H)
162. C4⋊Q8D4×C71semidirect
170. #127D4×C71semidirect
178. C4⋊C8D4×C71semidirect
214. C42D8×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#32052
C71×#28672
C71×#19622
C71×#15452
C71×#12762
#1028071