Finite Group Databasebeta

← C71×#11556 #31832428

Subgroups — C71×#11556 #31832428

Order 18176.

366 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2499C211yesmaximal, characteristic
2C71×#1464C211yesmaximal, characteristic
3C71×#1458C211yesmaximal, characteristic
4C71×#1324C212yesmaximal, complemented
5C71×#1319C212yesmaximal, complemented
6C2×C22⋊Q8×C71C411yescharacteristic
7C2×#117×C71C2211yescharacteristic
8#432×C71—21no—
9#431×C71—21no—
10#397×C71—21no—
11#397×C71—21no—
12#391×C71C411yescharacteristic
13#391×C71C2211yescharacteristic
14#387×C71—21no—
15#387×C71—21no—
16#384×C71C411yescharacteristic
17#382×C71C2211yescharacteristic
18C4×C22⋊C4×C71C411yescharacteristic
19#324×C71C2212yes—
20#324×C71C2212yes—
21#323×C71—21no—
22#323×C71—21no—
23C22×Q8×C71C2×C411yescharacteristic
24C23×C4×C71C2×C411yescharacteristic
25C22⋊Q8×C71—41no—
26C22⋊Q8×C71—41no—
27#117×C71—41no—
28#117×C71—41no—
29C2×C4⋊C4×C71—21no—
30C2×C4⋊C4×C71—21no—
31C2×C4⋊C4×C71—21no—
32C2×C4⋊C4×C71C2×C411yescharacteristic
33C2×C4⋊C4×C71C2×C411yescharacteristic
34C2×C4⋊C4×C71C2311yescharacteristic
35C2×C22⋊C4×C71—21no—
36C2×C22⋊C4×C71—21no—
37C2×C22⋊C4×C71D411yescharacteristic
38C2×C22⋊C4×C71D411yescharacteristic
39C2×C42×C71C2×C411yescharacteristic
40C2×C42×C71C2×C411yescharacteristic
41C2×C42×C71D411yescharacteristic
42C2×C42×C71D411yescharacteristic
43C4⋊C8×C71—41no—
44C4⋊C8×C71—41no—
45C22⋊C8×C71—22no—
46C22⋊C8×C71—22no—
47#97×C71—41no—
48C4×C8×C71—41no—
49#95×C71—21no—
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56C24×C71C22⋊C411yescharacteristic
57C2×Q8×C71—22no—
58C2×Q8×C71—41no—
59C2×Q8×C71—41no—
60C22×C4×C71—21no—
61C22×C4×C71—21no—
62C22×C4×C71—21no—
63C22×C4×C71—22no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—41no—
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C22×C411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C2×C8×C71—42no—
75C2×C8×C71—42no—
76C4⋊C4×C71—41no—
77C4⋊C4×C71—41no—
78C4⋊C4×C71—41no—
79C4⋊C4×C71—42no—
80C4⋊C4×C71—42no—
81C4⋊C4×C71—41no—
82C4⋊C4×C71—41no—
83C22⋊C4×C71—41no—
84C22⋊C4×C71—41no—
85C22⋊C4×C71—42no—
86C22⋊C4×C71—42no—
87C42×C71—21no—
88C42×C71—21no—
89C42×C71—21no—
90C42×C71—21no—
91C42×C71—22no—
92C42×C71—41no—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—21no—
96C23×C71—41no—
97C23×C71—41no—
98C23×C71C2×C22⋊C411yescharacteristic
99Q8×C71—82no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—22no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—42no—
110C2×C4×C71—42no—
111C2×C4×C71—41no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71#9911yescharacteristic
127C2×C4×C71#9911yescharacteristic
128C2×C4×C71#13112yes—
129C8×C71—82no—
130C8×C71—82no—
131C22×C71—21no—
132C22×C71—21no—
133C22×C71—41no—
134C22×C71—41no—
135C22×C71—41no—
136C22×C71—81no—
137C22×C71—81no—
138C22×C71#40811yescharacteristic
139C22×C71C2×#9911yescharacteristic
140C22×C71#41311yescharacteristic
141C4×C71—21no—
142C4×C71—21no—
143C4×C71—22no—
144C4×C71—42no—
145C4×C71—41no—
146C4×C71—41no—
147C4×C71—41no—
148C4×C71—41no—
149C4×C71—81no—
150C4×C71—81no—
151C4×C71—81no—
152#11556C7111yesmaximal, characteristic, direct product factor, Sylow
153C2×C71—21no—
154C2×C71—21no—
155C2×C71—81no—
156C2×C71#146811yescharacteristic
157C2×C71#212911yescharacteristic
158C2×C71#213211yescharacteristic
159#2499C2×C7111yescharacteristic
160#1464C2×C7111yescharacteristic
161#1458C2×C7111yescharacteristic
162#1324C2×C7112yescomplemented
163#1319C2×C7112yescomplemented
164C71#1155611yescharacteristic, direct product factor, Sylow
165C2×C22⋊Q8C4×C7111yescharacteristic
166C2×#117C22×C7111yescharacteristic
167#432—21no—
168#431—21no—
169#397—21no—
170#397—21no—
171#391C4×C7111yescharacteristic
172#391C22×C7111yescharacteristic
173#387—21no—
174#387—21no—
175#384C4×C7111yescharacteristic
176#382C22×C7111yescharacteristic
177C4×C22⋊C4C4×C7111yescharacteristic
178#324C22×C7112yes—
179#324C22×C7112yes—
180#323—21no—
181#323—21no—
182C22×Q8C2×C4×C7111yescharacteristic
183C23×C4C2×C4×C7111yescharacteristic
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186#117—41no—
187#117—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C2×C4×C7111yescharacteristic
192C2×C4⋊C4C2×C4×C7111yescharacteristic
193C2×C4⋊C4C23×C7111yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4D4×C7111yescharacteristic
197C2×C22⋊C4D4×C7111yescharacteristic
198C2×C42C2×C4×C7111yescharacteristic
199C2×C42C2×C4×C7111yescharacteristic
200C2×C42D4×C7111yescharacteristic
201C2×C42D4×C7111yescharacteristic
202C4⋊C8—41no—
203C4⋊C8—41no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206#97—41no—
207C4×C8—41no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215C24C22⋊C4×C7111yescharacteristic
216C2×Q8—22no—
217C2×Q8—41no—
218C2×Q8—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4C22⋊C4×C7111yescharacteristic
228C22×C4C22⋊C4×C7111yescharacteristic
229C22×C4C22⋊C4×C7111yescharacteristic
230C22×C4C22×C4×C7111yescharacteristic
231C22×C4C2×D4×C7111yescharacteristic
232C22×C4C2×D4×C7111yescharacteristic
233C2×C8—42no—
234C2×C8—42no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—22no—
251C42—41no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23C2×C22⋊C4×C7111yescharacteristic
258Q8—82no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4#99×C7111yescharacteristic
286C2×C4#99×C7111yescharacteristic
287C2×C4#131×C7112yes—
288C8—82no—
289C8—82no—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#408×C7111yescharacteristic
298C22C2×#99×C7111yescharacteristic
299C22#413×C7111yescharacteristic
300C4—21no—
301C4—21no—
302C4—22no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—81no—
309C4—81no—
310C4—81no—
311C2—21no—
312C2—21no—
313C2—81no—
314C2C71×#146811yescharacteristic
315C2C71×#212911yescharacteristic
316C2C71×#213211yescharacteristic
subgroupG/H
center, socleC22×C71C2×#99
commutatorC22×C4C22×C4×C71
Frattini, cosocleC2×C4⋊C4C23×C71
prime radicalC24×C71C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C22×C4×C71 ◁ C2×#117×C71 ◁ G

Complements

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

NH ≅ G/Ntype
4. C71×#1324C2semidirect
5. C71×#1319C2semidirect
152. #11556C71direct (N × H)
162. #1324C2×C71semidirect
163. #1319C2×C71semidirect
164. C71#11556direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#24992
C71×#14642
C71×#14582
C71×#13242
C71×#13192
#1155671