Finite Group Databasebeta

← C71×#9294 #31830166

Subgroups — C71×#9294 #31830166

Order 18176.

490 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3208C211yesmaximal, characteristic
2C71×#2961C211yesmaximal, characteristic, complemented
3C71×#1961C211yesmaximal, characteristic
4C71×#1545C211yesmaximal, characteristic
5C71×#1518C211yesmaximal, characteristic, complemented
6C71×#1286C211yesmaximal, characteristic, complemented
7C71×#1275C211yesmaximal, characteristic, complemented
8#533×C71C2211yescharacteristic
9C2×#127×C71C2211yescharacteristic
10#492×C71C412yescomplemented
11#486×C71C412yescomplemented
12#460×C71—22no—
13#450×C71—22no—
14#446×C71—22no—
15#446×C71—22no—
16#431×C71C2211yescharacteristic
17#431×C71C2211yescharacteristic
18#424×C71—21no—
19#423×C71C2211yescharacteristic
20C2×#102×C71—21no—
21C2×#102×C71—21no—
22#405×C71—21no—
23C2×#97×C71C2211yescharacteristic
24#382×C71C2211yescharacteristic
25#331×C71—22no—
26#328×C71—22no—
27C22×D4×C71—21no—
28C22×D4×C71—21no—
29C22×D4×C71—21no—
30C2×SD16×C71—24no—
31C2×M16×C71—21no—
32C22×C8×C71D411yescharacteristic
33C22×C8×C71D411yescharacteristic
34C4⋊Q8×C71C2×C411yescharacteristic
35C4⋊Q8×C71C2×C411yescharacteristic
36#127×C71—21no—
37#127×C71—21no—
38#127×C71—21no—
39#127×C71C2×C412yes—
40#125×C71—21no—
41C22⋊Q8×C71—21no—
42C22⋊Q8×C71D412yes—
43C4×Q8×C71—21no—
44C4×Q8×C71D412yes—
45#117×C71—21no—
46C2×C4⋊C4×C71—21no—
47C2×C42×C71C2311yescharacteristic
48C4⋊C8×C71—24no—
49C4⋊C8×C71—21no—
50C4⋊C8×C71—21no—
51C4⋊C8×C71—21no—
52C4⋊C8×C71D412yes—
53#102×C71—24no—
54#102×C71—42no—
55#102×C71—42no—
56C22⋊C8×C71—21no—
57C22⋊C8×C71—21no—
58#97×C71—22no—
59#97×C71—21no—
60#97×C71—21no—
61C4×C8×C71—22no—
62C4×C8×C71—21no—
63C4×C8×C71C2×C412yes—
64#95×C71—21no—
65#95×C71—21no—
66C24×C71—41no—
67C2×Q8×C71—21no—
68C2×Q8×C71C22⋊C412yes—
69C2×D4×C71—22no—
70C2×D4×C71—22no—
71C2×D4×C71—22no—
72C2×D4×C71—41no—
73C2×D4×C71—41no—
74C2×D4×C71—41no—
75C2×D4×C71—41no—
76C2×D4×C71—41no—
77C2×D4×C71—41no—
78C2×D4×C71—41no—
79C2×D4×C71—82no—
80C2×D4×C71—82no—
81C22×C4×C71—41no—
82C22×C4×C71C2×D411yescharacteristic
83C22×C4×C71C2×D411yescharacteristic
84C22×C4×C71#4111yescharacteristic
85SD16×C71—84no—
86M16×C71—41no—
87M16×C71—41no—
88C2×C8×C71—21no—
89C2×C8×C71—21no—
90C2×C8×C71—22no—
91C2×C8×C71—22no—
92C2×C8×C71—21no—
93C2×C8×C71—21no—
94C2×C8×C71—21no—
95C2×C8×C71—41no—
96C2×C8×C71SD1612yescomplemented
97C4⋊C4×C71—21no—
98C4⋊C4×C71—21no—
99C4⋊C4×C71—21no—
100C4⋊C4×C71—22no—
101C4⋊C4×C71—22no—
102C4⋊C4×C71—41no—
103C4⋊C4×C71—41no—
104C4⋊C4×C71C22⋊C412yes—
105C22⋊C4×C71—22no—
106C42×C71—22no—
107C42×C71C22×C411yescharacteristic
108C42×C71C2×D411yescharacteristic
109C42×C71C2×D411yescharacteristic
110C42×C71#4111yescharacteristic
111C23×C71—42no—
112C23×C71—82no—
113C23×C71—82no—
114C23×C71—81no—
115C23×C71—81no—
116C23×C71#12111yescharacteristic
117Q8×C71—42no—
118Q8×C71—81no—
119D4×C71—44no—
120D4×C71—42no—
121D4×C71—42no—
122D4×C71—82no—
123D4×C71—82no—
124D4×C71—82no—
125D4×C71—81no—
126D4×C71—81no—
127C2×C4×C71—22no—
128C2×C4×C71—22no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—81no—
132C2×C4×C71—81no—
133C2×C4×C71—81no—
134C2×C4×C71C2×C22⋊C411yescharacteristic
135C2×C4×C71C4×D411yescharacteristic
136C2×C4×C71C4×D411yescharacteristic
137C2×C4×C71#12011yescharacteristic
138C2×C4×C71#12111yescharacteristic
139C2×C4×C71#12311yescharacteristic
140C2×C4×C71C2×SD1611yescharacteristic
141C2×C4×C71#13511yescharacteristic
142C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C71#13711yescharacteristic
144C8×C71—22no—
145C8×C71—42no—
146C8×C71—41no—
147C8×C71—42no—
148C8×C71—81no—
149C22×C71—21no—
150C22×C71—21no—
151C22×C71—82no—
152C22×C71—82no—
153C22×C71—82no—
154C22×C71—162no—
155C22×C71—162no—
156C22×C71#38611yescharacteristic
157C22×C71#46011yescharacteristic
158C22×C71#46411yescharacteristic
159C4×C71—21no—
160C4×C71—21no—
161C4×C71—21no—
162C4×C71—21no—
163C4×C71—42no—
164C4×C71—42no—
165C4×C71—161no—
166C4×C71C4×SD1611yescharacteristic
167C4×C71#44011yescharacteristic
168C4×C71#44811yescharacteristic
169C4×C71#44911yescharacteristic
170#9294C7111yesmaximal, characteristic, direct product factor, Sylow
171C2×C71—21no—
172C2×C71—21no—
173C2×C71—162no—
174C2×C71#188011yescharacteristic
175C2×C71#190311yescharacteristic
176C2×C71#191011yescharacteristic
177#3208C2×C7111yescharacteristic
178#2961C2×C7111yescharacteristic, complemented
179#1961C2×C7111yescharacteristic
180#1545C2×C7111yescharacteristic
181#1518C2×C7111yescharacteristic, complemented
182#1286C2×C7111yescharacteristic, complemented
183#1275C2×C7111yescharacteristic, complemented
184C71#929411yescharacteristic, direct product factor, Sylow
185#533C22×C7111yescharacteristic
186C2×#127C22×C7111yescharacteristic
187#492C4×C7112yescomplemented
188#486C4×C7112yescomplemented
189#460—22no—
190#450—22no—
191#446—22no—
192#446—22no—
193#431C22×C7111yescharacteristic
194#431C22×C7111yescharacteristic
195#424—21no—
196#423C22×C7111yescharacteristic
197C2×#102—21no—
198C2×#102—21no—
199#405—21no—
200C2×#97C22×C7111yescharacteristic
201#382C22×C7111yescharacteristic
202#331—22no—
203#328—22no—
204C22×D4—21no—
205C22×D4—21no—
206C22×D4—21no—
207C2×SD16—24no—
208C2×M16—21no—
209C22×C8D4×C7111yescharacteristic
210C22×C8D4×C7111yescharacteristic
211C4⋊Q8C2×C4×C7111yescharacteristic
212C4⋊Q8C2×C4×C7111yescharacteristic
213#127—21no—
214#127—21no—
215#127—21no—
216#127C2×C4×C7112yes—
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8D4×C7112yes—
220C4×Q8—21no—
221C4×Q8D4×C7112yes—
222#117—21no—
223C2×C4⋊C4—21no—
224C2×C42C23×C7111yescharacteristic
225C4⋊C8—24no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8D4×C7112yes—
230#102—24no—
231#102—42no—
232#102—42no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235#97—22no—
236#97—21no—
237#97—21no—
238C4×C8—22no—
239C4×C8—21no—
240C4×C8C2×C4×C7112yes—
241#95—21no—
242#95—21no—
243C24—41no—
244C2×Q8—21no—
245C2×Q8C22⋊C4×C7112yes—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—82no—
257C2×D4—82no—
258C22×C4—41no—
259C22×C4C2×D4×C7111yescharacteristic
260C22×C4C2×D4×C7111yescharacteristic
261C22×C4#41×C7111yescharacteristic
262SD16—84no—
263M16—41no—
264M16—41no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—41no—
273C2×C8SD16×C7112yescomplemented
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4C22⋊C4×C7112yes—
282C22⋊C4—22no—
283C42—22no—
284C42C22×C4×C7111yescharacteristic
285C42C2×D4×C7111yescharacteristic
286C42C2×D4×C7111yescharacteristic
287C42#41×C7111yescharacteristic
288C23—42no—
289C23—82no—
290C23—82no—
291C23—81no—
292C23—81no—
293C23#121×C7111yescharacteristic
294Q8—42no—
295Q8—81no—
296D4—44no—
297D4—42no—
298D4—42no—
299D4—82no—
300D4—82no—
301D4—82no—
302D4—81no—
303D4—81no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4C2×C22⋊C4×C7111yescharacteristic
312C2×C4C4×D4×C7111yescharacteristic
313C2×C4C4×D4×C7111yescharacteristic
314C2×C4#120×C7111yescharacteristic
315C2×C4#121×C7111yescharacteristic
316C2×C4#123×C7111yescharacteristic
317C2×C4C2×SD16×C7111yescharacteristic
318C2×C4#135×C7111yescharacteristic
319C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
320C2×C4#137×C7111yescharacteristic
321C8—22no—
322C8—42no—
323C8—41no—
324C8—42no—
325C8—81no—
326C22—21no—
327C22—21no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—162no—
332C22—162no—
333C22#386×C7111yescharacteristic
334C22#460×C7111yescharacteristic
335C22#464×C7111yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—161no—
343C4C4×SD16×C7111yescharacteristic
344C4#440×C7111yescharacteristic
345C4#448×C7111yescharacteristic
346C4#449×C7111yescharacteristic
347C2—21no—
348C2—21no—
349C2—162no—
350C2C71×#188011yescharacteristic
351C2C71×#190311yescharacteristic
352C2C71×#191011yescharacteristic
subgroupG/H
center, socleC22×C71#386
commutatorC42C22×C4×C71
Frattini, cosocleC2×C42C23×C71
prime radicalC2×#127×C71C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#2961C2semidirect
5. C71×#1518C2semidirect
6. C71×#1286C2semidirect
7. C71×#1275C2semidirect
10. #492×C71C4semidirect
11. #486×C71C4semidirect
96. C2×C8×C71SD16semidirect
170. #9294C71direct (N × H)
178. #2961C2×C71semidirect
181. #1518C2×C71semidirect
182. #1286C2×C71semidirect
183. #1275C2×C71semidirect
184. C71#9294direct (N × H)
187. #492C4×C71semidirect
188. #486C4×C71semidirect
273. C2×C8SD16×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#32082
C71×#29612
C71×#19612
C71×#15452
C71×#15182
C71×#12862
C71×#12752
#929471