Finite Group Databasebeta

← C71×#7696 #31828568

Subgroups — C71×#7696 #31828568

Order 18176.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#2861C211yesmaximal, characteristic
2C71×#1831C211yesmaximal, characteristic, complemented
3C71×#1767C211yesmaximal, characteristic, complemented
4C71×#1272C212yesmaximal, complemented
5C71×#1270C212yesmaximal
6C2×#125×C71C411yescharacteristic
7C2×#124×C71C411yescharacteristic
8C2×C4⋊C8×C71—21no—
9C2×C4⋊C8×C71—21no—
10C2×#97×C71—21no—
11#391×C71C411yescharacteristic
12#391×C71C2211yescharacteristic
13#390×C71C411yescharacteristic
14#390×C71C2211yescharacteristic
15#388×C71—21no—
16#388×C71—21no—
17#388×C71—21no—
18#388×C71—21no—
19C43×C71C2211yescharacteristic
20#336×C71—21no—
21#336×C71C2212yes—
22#336×C71C2212yes—
23C22×Q8×C71C2×C411yescharacteristic
24C22×D4×C71C2×C411yescharacteristic
25C22×C8×C71—22no—
26C22×C8×C71—22no—
27#125×C71—42no—
28#124×C71—42no—
29C2×C4⋊C4×C71—21no—
30C2×C4⋊C4×C71—21no—
31C2×C4⋊C4×C71C2×C411yescharacteristic
32C2×C4⋊C4×C71C2×C411yescharacteristic
33C2×C22⋊C4×C71—21no—
34C2×C22⋊C4×C71—21no—
35C2×C22⋊C4×C71—21no—
36C2×C22⋊C4×C71—21no—
37C2×C22⋊C4×C71—21no—
38C2×C22⋊C4×C71—21no—
39C2×C42×C71C2×C411yescharacteristic
40C2×C42×C71C2×C411yescharacteristic
41C2×C42×C71D411yescharacteristic
42C2×C42×C71D411yescharacteristic
43C2×C42×C71D411yescharacteristic
44C2×C42×C71D411yescharacteristic
45C2×C42×C71C2311yescharacteristic
46C4⋊C8×C71—24no—
47C4⋊C8×C71—42no—
48#97×C71—42no—
49#95×C71—21no—
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53C24×C71—21no—
54C2×Q8×C71—22no—
55C2×Q8×C71—41no—
56C2×Q8×C71—41no—
57C2×D4×C71—22no—
58C2×D4×C71—41no—
59C2×D4×C71—41no—
60C22×C4×C71—21no—
61C22×C4×C71—21no—
62C22×C4×C71—21no—
63C22×C4×C71C22⋊C411yescharacteristic
64C22×C4×C71C22⋊C411yescharacteristic
65C22×C4×C71C22⋊C411yescharacteristic
66C22×C4×C71C22⋊C411yescharacteristic
67C22×C4×C71C22×C411yescharacteristic
68C22×C4×C71C2×D411yescharacteristic
69C22×C4×C71C2×D411yescharacteristic
70C2×C8×C71—24no—
71C2×C8×C71—42no—
72C2×C8×C71—42no—
73C2×C8×C71—42no—
74C2×C8×C71—42no—
75C2×C8×C71—42no—
76C4⋊C4×C71—41no—
77C4⋊C4×C71—41no—
78C4⋊C4×C71—42no—
79C4⋊C4×C71—42no—
80C22⋊C4×C71—42no—
81C22⋊C4×C71—42no—
82C22⋊C4×C71—42no—
83C22⋊C4×C71—42no—
84C22⋊C4×C71—41no—
85C22⋊C4×C71—41no—
86C22⋊C4×C71—41no—
87C22⋊C4×C71—41no—
88C42×C71—22no—
89C42×C71—22no—
90C42×C71—22no—
91C42×C71—21no—
92C42×C71—21no—
93C42×C71—21no—
94C42×C71—21no—
95C42×C71—41no—
96C42×C71—41no—
97C23×C71—41no—
98C23×C71—41no—
99C23×C71—41no—
100C23×C71—41no—
101C23×C71—41no—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71C2×C22⋊C411yescharacteristic
105Q8×C71—82no—
106D4×C71—82no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71#10012yes—
133C2×C4×C71#10412yes—
134C2×C4×C71#13112yes—
135C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C71#13711yescharacteristic
138C2×C4×C71#13711yescharacteristic
139C8×C71—44no—
140C8×C71—82no—
141C22×C71—81no—
142C22×C71—81no—
143C22×C71—81no—
144C22×C71—81no—
145C22×C71—81no—
146C22×C71—81no—
147C22×C71—81no—
148C22×C71#40811yescharacteristic
149C22×C71#41011yescharacteristic
150C22×C71C2×#10011yescharacteristic
151C22×C71#41711yescharacteristic
152C22×C71#41711yescharacteristic
153C22×C71C2×#10411yescharacteristic
154C22×C71#42111yescharacteristic
155C4×C71—22no—
156C4×C71—22no—
157C4×C71—21no—
158C4×C71—21no—
159C4×C71—21no—
160C4×C71—21no—
161C4×C71—22no—
162C4×C71—41no—
163C4×C71—41no—
164C4×C71—81no—
165C4×C71—81no—
166C4×C71—81no—
167#7696C7111yesmaximal, characteristic, direct product factor, Sylow
168C2×C71—161no—
169C2×C71#146911yescharacteristic
170C2×C71#148311yescharacteristic
171C2×C71#149511yescharacteristic
172C2×C71#150911yescharacteristic
173C2×C71#151211yescharacteristic
174C2×C71#153011yescharacteristic
175C2×C71#156011yescharacteristic
176#2861C2×C7111yescharacteristic
177#1831C2×C7111yescharacteristic, complemented
178#1767C2×C7111yescharacteristic, complemented
179#1272C2×C7112yescomplemented
180#1270C2×C7112yes—
181C71#769611yescharacteristic, direct product factor, Sylow
182C2×#125C4×C7111yescharacteristic
183C2×#124C4×C7111yescharacteristic
184C2×C4⋊C8—21no—
185C2×C4⋊C8—21no—
186C2×#97—21no—
187#391C4×C7111yescharacteristic
188#391C22×C7111yescharacteristic
189#390C4×C7111yescharacteristic
190#390C22×C7111yescharacteristic
191#388—21no—
192#388—21no—
193#388—21no—
194#388—21no—
195C43C22×C7111yescharacteristic
196#336—21no—
197#336C22×C7112yes—
198#336C22×C7112yes—
199C22×Q8C2×C4×C7111yescharacteristic
200C22×D4C2×C4×C7111yescharacteristic
201C22×C8—22no—
202C22×C8—22no—
203#125—42no—
204#124—42no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C2×C4×C7111yescharacteristic
208C2×C4⋊C4C2×C4×C7111yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C42C2×C4×C7111yescharacteristic
216C2×C42C2×C4×C7111yescharacteristic
217C2×C42D4×C7111yescharacteristic
218C2×C42D4×C7111yescharacteristic
219C2×C42D4×C7111yescharacteristic
220C2×C42D4×C7111yescharacteristic
221C2×C42C23×C7111yescharacteristic
222C4⋊C8—24no—
223C4⋊C8—42no—
224#97—42no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229C24—21no—
230C2×Q8—22no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×D4—22no—
234C2×D4—41no—
235C2×D4—41no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4C22⋊C4×C7111yescharacteristic
240C22×C4C22⋊C4×C7111yescharacteristic
241C22×C4C22⋊C4×C7111yescharacteristic
242C22×C4C22⋊C4×C7111yescharacteristic
243C22×C4C22×C4×C7111yescharacteristic
244C22×C4C2×D4×C7111yescharacteristic
245C22×C4C2×D4×C7111yescharacteristic
246C2×C8—24no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—41no—
272C42—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×C22⋊C4×C7111yescharacteristic
281Q8—82no—
282D4—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4#100×C7112yes—
309C2×C4#104×C7112yes—
310C2×C4#131×C7112yes—
311C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
312C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
313C2×C4#137×C7111yescharacteristic
314C2×C4#137×C7111yescharacteristic
315C8—44no—
316C8—82no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#408×C7111yescharacteristic
325C22#410×C7111yescharacteristic
326C22C2×#100×C7111yescharacteristic
327C22#417×C7111yescharacteristic
328C22#417×C7111yescharacteristic
329C22C2×#104×C7111yescharacteristic
330C22#421×C7111yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—22no—
336C4—22no—
337C4—22no—
338C4—41no—
339C4—41no—
340C4—81no—
341C4—81no—
342C4—81no—
343C2—161no—
344C2C71×#146911yescharacteristic
345C2C71×#148311yescharacteristic
346C2C71×#149511yescharacteristic
347C2C71×#150911yescharacteristic
348C2C71×#151211yescharacteristic
349C2C71×#153011yescharacteristic
350C2C71×#156011yescharacteristic
subgroupG/H
center, socleC23×C71C2×C22⋊C4
commutatorC22×C4C22×C4×C71
Frattini, cosocleC2×C42C23×C71
prime radicalC22×D4×C71C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C71 ◁ C43×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#1831C2semidirect
3. C71×#1767C2semidirect
4. C71×#1272C2semidirect
167. #7696C71direct (N × H)
177. #1831C2×C71semidirect
178. #1767C2×C71semidirect
179. #1272C2×C71semidirect
181. C71#7696direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#28612
C71×#18312
C71×#17672
C71×#12722
C71×#12702
#769671