Finite Group Databasebeta

← C71×#11717 #31832589

Subgroups — C71×#11717 #31832589

Order 18176.

448 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2628C211yesmaximal, characteristic
2C71×#1740C211yesmaximal, characteristic
3C71×#1458C211yesmaximal, characteristic
4C71×#1351C214yesmaximal
5C2×C4⋊Q8×C71C411yescharacteristic, complemented
6C2×#117×C71—21no—
7C2×#117×C71C2211yescharacteristic
8#431×C71—21no—
9#400×C71—21no—
10#399×C71—21no—
11#395×C71C412yescomplemented
12#393×C71C2212yes—
13#389×C71—21no—
14#382×C71C411yescharacteristic, complemented
15#380×C71—21no—
16#342×C71—21no—
17#342×C71C2212yes—
18#324×C71—21no—
19#323×C71C2212yes—
20C22×Q8×C71C2×C412yes—
21C23×C4×C71D411yescharacteristic
22C4⋊Q8×C71—41no—
23C4⋊Q8×C71—41no—
24#117×C71—24no—
25#117×C71—41no—
26#117×C71—41no—
27#117×C71—42no—
28C2×C4⋊C4×C71—21no—
29C2×C4⋊C4×C71—21no—
30C2×C4⋊C4×C71—21no—
31C2×C4⋊C4×C71—22no—
32C2×C4⋊C4×C71—41no—
33C2×C4⋊C4×C71C2×C411yescharacteristic
34C2×C4⋊C4×C71D411yescharacteristic
35C2×C22⋊C4×C71—22no—
36C2×C22⋊C4×C71—22no—
37C2×C22⋊C4×C71D411yescharacteristic
38C2×C22⋊C4×C71C2311yescharacteristic
39C2×C42×C71—41no—
40C2×C42×C71C2×C411yescharacteristic
41C2×C42×C71D411yescharacteristic
42C4⋊C8×C71—41no—
43C22⋊C8×C71—22no—
44C4×C8×C71—41no—
45#95×C71—22no—
46#95×C71—21no—
47#95×C71—22no—
48#95×C71C2×C412yes—
49C24×C71C2×D411yescharacteristic
50C2×Q8×C71—42no—
51C2×Q8×C71—82no—
52C22×C4×C71—21no—
53C22×C4×C71—21no—
54C22×C4×C71—22no—
55C22×C4×C71—22no—
56C22×C4×C71—22no—
57C22×C4×C71—42no—
58C22×C4×C71C22⋊C411yescharacteristic
59C22×C4×C71C22⋊C411yescharacteristic
60C22×C4×C71C22⋊C411yescharacteristic
61C22×C4×C71C22⋊C411yescharacteristic
62C22×C4×C71SD1612yes—
63C22×C4×C71Q1612yes—
64C22×C4×C71C22×C411yescharacteristic
65C22×C4×C71C2×D411yescharacteristic
66C2×C8×C71—42no—
67C4⋊C4×C71—41no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—44no—
71C4⋊C4×C71—41no—
72C4⋊C4×C71—41no—
73C4⋊C4×C71—42no—
74C4⋊C4×C71—82no—
75C22⋊C4×C71—24no—
76C22⋊C4×C71—41no—
77C22⋊C4×C71—41no—
78C22⋊C4×C71—42no—
79C22⋊C4×C71—82no—
80C42×C71—21no—
81C42×C71—21no—
82C42×C71—21no—
83C42×C71—21no—
84C42×C71—44no—
85C23×C71—21no—
86C23×C71—21no—
87C23×C71—41no—
88C23×C71—41no—
89C23×C71C2×C22⋊C411yescharacteristic
90C23×C71C2×SD1611yescharacteristic
91C23×C71C2×Q1611yescharacteristic
92Q8×C71—84no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—41no—
104C2×C4×C71—41no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71—44no—
108C2×C4×C71—42no—
109C2×C4×C71—42no—
110C2×C4×C71—42no—
111C2×C4×C71—42no—
112C2×C4×C71—42no—
113C2×C4×C71—42no—
114C2×C4×C71—82no—
115C2×C4×C71—82no—
116C2×C4×C71#9911yescharacteristic
117C2×C4×C71#9911yescharacteristic
118C2×C4×C71#10312yes—
119C2×C4×C71#10312yes—
120C2×C4×C71#10412yes—
121C8×C71—82no—
122C22×C71—21no—
123C22×C71—21no—
124C22×C71—21no—
125C22×C71—21no—
126C22×C71—41no—
127C22×C71—41no—
128C22×C71—41no—
129C22×C71—41no—
130C22×C71—81no—
131C22×C71C2×#9911yescharacteristic
132C22×C71C2×#10311yescharacteristic
133C22×C71C2×#10411yescharacteristic
134C4×C71—22no—
135C4×C71—41no—
136C4×C71—41no—
137C4×C71—41no—
138C4×C71—41no—
139C4×C71—82no—
140C4×C71—84no—
141C4×C71—82no—
142C4×C71#32812yes—
143C4×C71#32812yes—
144#11717C7111yesmaximal, characteristic, direct product factor, Sylow
145C2×C71—21no—
146C2×C71—21no—
147C2×C71—41no—
148C2×C71—41no—
149C2×C71C2×#32811yescharacteristic
150C2×C71#212211yescharacteristic
151C2×C71#212911yescharacteristic
152#2628C2×C7111yescharacteristic
153#1740C2×C7111yescharacteristic
154#1458C2×C7111yescharacteristic
155#1351C2×C7114yes—
156C71#1171711yescharacteristic, direct product factor, Sylow
157C2×C4⋊Q8C4×C7111yescharacteristic, complemented
158C2×#117—21no—
159C2×#117C22×C7111yescharacteristic
160#431—21no—
161#400—21no—
162#399—21no—
163#395C4×C7112yescomplemented
164#393C22×C7112yes—
165#389—21no—
166#382C4×C7111yescharacteristic, complemented
167#380—21no—
168#342—21no—
169#342C22×C7112yes—
170#324—21no—
171#323C22×C7112yes—
172C22×Q8C2×C4×C7112yes—
173C23×C4D4×C7111yescharacteristic
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176#117—24no—
177#117—41no—
178#117—41no—
179#117—42no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4C2×C4×C7111yescharacteristic
186C2×C4⋊C4D4×C7111yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4D4×C7111yescharacteristic
190C2×C22⋊C4C23×C7111yescharacteristic
191C2×C42—41no—
192C2×C42C2×C4×C7111yescharacteristic
193C2×C42D4×C7111yescharacteristic
194C4⋊C8—41no—
195C22⋊C8—22no—
196C4×C8—41no—
197#95—22no—
198#95—21no—
199#95—22no—
200#95C2×C4×C7112yes—
201C24C2×D4×C7111yescharacteristic
202C2×Q8—42no—
203C2×Q8—82no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4C22⋊C4×C7111yescharacteristic
211C22×C4C22⋊C4×C7111yescharacteristic
212C22×C4C22⋊C4×C7111yescharacteristic
213C22×C4C22⋊C4×C7111yescharacteristic
214C22×C4SD16×C7112yes—
215C22×C4Q16×C7112yes—
216C22×C4C22×C4×C7111yescharacteristic
217C22×C4C2×D4×C7111yescharacteristic
218C2×C8—42no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—82no—
227C22⋊C4—24no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C22⋊C4—82no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—44no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23C2×C22⋊C4×C7111yescharacteristic
242C23C2×SD16×C7111yescharacteristic
243C23C2×Q16×C7111yescharacteristic
244Q8—84no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4#99×C7111yescharacteristic
269C2×C4#99×C7111yescharacteristic
270C2×C4#103×C7112yes—
271C2×C4#103×C7112yes—
272C2×C4#104×C7112yes—
273C8—82no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22C2×#99×C7111yescharacteristic
284C22C2×#103×C7111yescharacteristic
285C22C2×#104×C7111yescharacteristic
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—82no—
292C4—84no—
293C4—82no—
294C4#328×C7112yes—
295C4#328×C7112yes—
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2C2×#328×C7111yescharacteristic
301C2C71×#212211yescharacteristic
302C2C71×#212911yescharacteristic
subgroupG/H
center, socleC22×C71C2×#99
commutatorC22×C4C22×C4×C71
Frattini, cosocleC2×C22⋊C4C23×C71
prime radicalC24×C71C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ 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
5. C2×C4⋊Q8×C71C4semidirect
11. #395×C71C4semidirect
14. #382×C71C4semidirect
144. #11717C71direct (N × H)
156. C71#11717direct (N × H)
157. C2×C4⋊Q8C4×C71semidirect
163. #395C4×C71semidirect
166. #382C4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#26282
C71×#17402
C71×#14582
C71×#13512
#1171771