Finite Group Databasebeta

← C71×#12969 #31833841

Subgroups — C71×#12969 #31833841

Order 18176.

416 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2980C211yesmaximal, characteristic
2C71×#2126C211yesmaximal, characteristic
3C71×#1884C211yesmaximal, characteristic
4C71×#1868C212yesmaximal, complemented
5C71×#1391C212yesmaximal, complemented
6#568×C71C2211yescharacteristic
7#532×C71C2211yescharacteristic
8#435×C71—22no—
9#435×C71—22no—
10#432×C71—21no—
11#424×C71—22no—
12#421×C71—21no—
13#410×C71C2211yescharacteristic
14#410×C71C2212yes—
15#408×C71—21no—
16#408×C71C2212yes—
17#354×C71C412yescomplemented
18#353×C71C412yescomplemented
19#344×C71—22no—
20#339×C71—21no—
21C2×#41×C71D411yescharacteristic
22C2×#41×C71C2311yescharacteristic
23#131×C71—42no—
24C2×M16×C71—22no—
25C2×M16×C71—21no—
26C2×M16×C71—21no—
27C2×M16×C71D412yes—
28C22×C8×C71—21no—
29C22×C8×C71D412yes—
30C4⋊Q8×C71Q811yescharacteristic
31#127×C71C2×C411yescharacteristic
32#124×C71C2×C411yescharacteristic
33#124×C71Q811yescharacteristic
34#121×C71—22no—
35C4×D4×C71—22no—
36#117×C71—22no—
37#117×C71—21no—
38C2×C42×C71D411yescharacteristic
39C4⋊C8×C71—21no—
40C4⋊C8×C71—21no—
41C4⋊C8×C71—44no—
42#104×C71—42no—
43#99×C71—22no—
44#99×C71—22no—
45#99×C71C2×C414yes—
46C22⋊C8×C71—22no—
47C22⋊C8×C71—22no—
48C22⋊C8×C71—22no—
49#97×C71—22no—
50#41×C71—41no—
51#41×C71—41no—
52#41×C71—41no—
53#41×C71—41no—
54C2×Q8×C71C4⋊C411yescharacteristic
55C2×Q8×C71C2×Q811yescharacteristic
56C2×D4×C71—22no—
57C2×D4×C71—21no—
58C2×D4×C71C4⋊C411yescharacteristic
59C2×D4×C71C22×C411yescharacteristic
60C2×D4×C71#4112yes—
61C22×C4×C71—21no—
62C22×C4×C71—21no—
63C22×C4×C71C2×D411yescharacteristic
64C22×C4×C71C2×D412yes—
65M16×C71—22no—
66M16×C71—42no—
67M16×C71—41no—
68M16×C71—42no—
69M16×C71—81no—
70C2×C8×C71—24no—
71C2×C8×C71—22no—
72C2×C8×C71—22no—
73C2×C8×C71—41no—
74C2×C8×C71—42no—
75C2×C8×C71—81no—
76C4⋊C4×C71—22no—
77C4⋊C4×C71—42no—
78C4⋊C4×C71—41no—
79C22⋊C4×C71—22no—
80C22⋊C4×C71—22no—
81C22⋊C4×C71—24no—
82C22⋊C4×C71—41no—
83C42×C71—21no—
84C42×C71—42no—
85C42×C71—41no—
86C42×C71C4⋊C411yescharacteristic
87C42×C71C4⋊C411yescharacteristic
88C23×C71—21no—
89C23×C71C4×D412yes—
90C23×C71#12311yescharacteristic
91Q8×C71—41no—
92Q8×C71—41no—
93D4×C71—41no—
94D4×C71—42no—
95D4×C71—41no—
96D4×C71—81no—
97D4×C71—82no—
98C2×C4×C71—22no—
99C2×C4×C71—21no—
100C2×C4×C71—22no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71—44no—
108C2×C4×C71—81no—
109C2×C4×C71—81no—
110C2×C4×C71C2×C4⋊C411yescharacteristic
111C2×C4×C71#12011yescharacteristic
112C2×C4×C71C22⋊Q812yes—
113C8×C71—42no—
114C8×C71—42no—
115C8×C71—42no—
116C22×C71—22no—
117C22×C71—21no—
118C22×C71—41no—
119C22×C71—42no—
120C22×C71—42no—
121C22×C71—41no—
122C22×C71#38511yescharacteristic
123C4×C71—21no—
124C4×C71—22no—
125C4×C71—41no—
126C4×C71—41no—
127C4×C71—41no—
128C4×C71—84no—
129C4×C71—161no—
130C4×C71B(4,𝔽2)11yescharacteristic
131C4×C71#45811yescharacteristic
132#12969C7111yesmaximal, characteristic, direct product factor, Sylow
133C2×C71—21no—
134C2×C71—42no—
135C2×C71—41no—
136C2×C71—81no—
137C2×C71#186711yescharacteristic
138#2980C2×C7111yescharacteristic
139#2126C2×C7111yescharacteristic
140#1884C2×C7111yescharacteristic
141#1868C2×C7112yescomplemented
142#1391C2×C7112yescomplemented
143C71#1296911yescharacteristic, direct product factor, Sylow
144#568C22×C7111yescharacteristic
145#532C22×C7111yescharacteristic
146#435—22no—
147#435—22no—
148#432—21no—
149#424—22no—
150#421—21no—
151#410C22×C7111yescharacteristic
152#410C22×C7112yes—
153#408—21no—
154#408C22×C7112yes—
155#354C4×C7112yescomplemented
156#353C4×C7112yescomplemented
157#344—22no—
158#339—21no—
159C2×#41D4×C7111yescharacteristic
160C2×#41C23×C7111yescharacteristic
161#131—42no—
162C2×M16—22no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16D4×C7112yes—
166C22×C8—21no—
167C22×C8D4×C7112yes—
168C4⋊Q8Q8×C7111yescharacteristic
169#127C2×C4×C7111yescharacteristic
170#124C2×C4×C7111yescharacteristic
171#124Q8×C7111yescharacteristic
172#121—22no—
173C4×D4—22no—
174#117—22no—
175#117—21no—
176C2×C42D4×C7111yescharacteristic
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—44no—
180#104—42no—
181#99—22no—
182#99—22no—
183#99C2×C4×C7114yes—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187#97—22no—
188#41—41no—
189#41—41no—
190#41—41no—
191#41—41no—
192C2×Q8C4⋊C4×C7111yescharacteristic
193C2×Q8C2×Q8×C7111yescharacteristic
194C2×D4—22no—
195C2×D4—21no—
196C2×D4C4⋊C4×C7111yescharacteristic
197C2×D4C22×C4×C7111yescharacteristic
198C2×D4#41×C7112yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4C2×D4×C7111yescharacteristic
202C22×C4C2×D4×C7112yes—
203M16—22no—
204M16—42no—
205M16—41no—
206M16—42no—
207M16—81no—
208C2×C8—24no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—41no—
212C2×C8—42no—
213C2×C8—81no—
214C4⋊C4—22no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—24no—
220C22⋊C4—41no—
221C42—21no—
222C42—42no—
223C42—41no—
224C42C4⋊C4×C7111yescharacteristic
225C42C4⋊C4×C7111yescharacteristic
226C23—21no—
227C23C4×D4×C7112yes—
228C23#123×C7111yescharacteristic
229Q8—41no—
230Q8—41no—
231D4—41no—
232D4—42no—
233D4—41no—
234D4—81no—
235D4—82no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—44no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4C2×C4⋊C4×C7111yescharacteristic
249C2×C4#120×C7111yescharacteristic
250C2×C4C22⋊Q8×C7112yes—
251C8—42no—
252C8—42no—
253C8—42no—
254C22—22no—
255C22—21no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22#385×C7111yescharacteristic
261C4—21no—
262C4—22no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—84no—
267C4—161no—
268C4B(4,𝔽2)×C7111yescharacteristic
269C4#458×C7111yescharacteristic
270C2—21no—
271C2—42no—
272C2—41no—
273C2—81no—
274C2C71×#186711yescharacteristic
subgroupG/H
centerC4×C71B(4,𝔽2)
commutatorC2×D4C22×C4×C71
Frattini, cosocleC2×#41C23×C71
socleC2×C71#1867
prime radical#532×C71C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C71 ◁ C2×C4×C71 ◁ C2×#41×C71 ◁ G

Complements

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

NH ≅ G/Ntype
4. C71×#1868C2semidirect
5. C71×#1391C2semidirect
17. #354×C71C4semidirect
18. #353×C71C4semidirect
132. #12969C71direct (N × H)
141. #1868C2×C71semidirect
142. #1391C2×C71semidirect
143. C71#12969direct (N × H)
155. #354C4×C71semidirect
156. #353C4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#29802
C71×#21262
C71×#18842
C71×#18682
C71×#13912
#1296971