Finite Group Databasebeta

← C71×#19530 #31840402

Subgroups — C71×#19530 #31840402

Order 18176.

582 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3226C211yesmaximal, characteristic
2C71×#3216C211yesmaximal, characteristic
3C71×#3209C211yesmaximal, characteristic, complemented
4C71×#2985C211yesmaximal, characteristic, complemented
5C71×#1702C212yesmaximal, complemented
6C71×#1694C212yesmaximal, complemented
7C71×#1682C212yesmaximal, complemented
8C71×#1674C212yesmaximal, complemented
9C71×#1572C211yesmaximal, characteristic, complemented
10C71×#1556C211yesmaximal, characteristic, complemented
11C71×#1553C211yesmaximal, characteristic
12#533×C71C2211yescharacteristic
13#532×C71C2211yescharacteristic
14C2×C4⋊Q8×C71C2211yescharacteristic
15#500×C71C2211yescharacteristic
16#500×C71C2211yescharacteristic
17#497×C71—21no—
18#494×C71—21no—
19#489×C71—21no—
20#487×C71—21no—
21#486×C71C2211yescharacteristic
22#486×C71C2211yescharacteristic
23#480×C71—22no—
24#474×C71—22no—
25#462×C71C2212yes—
26#462×C71C2212yes—
27#462×C71C2212yes—
28#460×C71C2212yes—
29#452×C71—22no—
30#451×C71—22no—
31#446×C71—22no—
32#446×C71C2212yes—
33C8×Q8×C71C2212yes—
34#433×C71—22no—
35#432×C71C2211yescharacteristic
36#432×C71C2211yescharacteristic
37#431×C71C2211yescharacteristic
38#431×C71C2211yescharacteristic
39#335×C71C2212yes—
40#334×C71C2212yes—
41#332×C71C2212yes—
42#332×C71C2212yes—
43#332×C71C2212yes—
44#331×C71C2212yes—
45C2×#41×C71—21no—
46C22×Q8×C71—21no—
47C2×Q16×C71—22no—
48C2×Q16×C71—22no—
49C2×Q16×C71—22no—
50C2×SD16×C71—22no—
51C4⋊Q8×C71—21no—
52C4⋊Q8×C71—21no—
53C4⋊Q8×C71—21no—
54C4⋊Q8×C71D411yescharacteristic
55C4⋊Q8×C71D411yescharacteristic
56C4⋊Q8×C71C2311yescharacteristic
57C4⋊Q8×C71C2311yescharacteristic
58C4⋊Q8×C71C2311yescharacteristic
59#127×C71C2311yescharacteristic
60#125×C71—21no—
61#124×C71—21no—
62C22⋊Q8×C71—21no—
63C22⋊Q8×C71D412yes—
64#121×C71—21no—
65#121×C71—21no—
66C4×Q8×C71—21no—
67C4×Q8×C71C2312yes—
68C4×D4×C71—21no—
69C4×D4×C71—21no—
70#117×C71—21no—
71C2×C4⋊C4×C71—21no—
72C2×C4⋊C4×C71—21no—
73C2×C42×C71C2311yescharacteristic
74#107×C71—22no—
75#107×C71—22no—
76C4⋊C8×C71—22no—
77C4⋊C8×C71—22no—
78C4⋊C8×C71D412yes—
79C4⋊C8×C71C2312yes—
80C4⋊C8×C71C2312yes—
81C4⋊C8×C71C2312yes—
82#103×C71—22no—
83#103×C71—22no—
84#103×C71—22no—
85#102×C71—22no—
86#102×C71—22no—
87#102×C71—22no—
88C22⋊C8×C71—21no—
89C22⋊C8×C71—21no—
90C22⋊C8×C71—21no—
91C22⋊C8×C71D412yes—
92#97×C71—22no—
93#97×C71—21no—
94#97×C71—21no—
95C4×C8×C71—22no—
96C4×C8×C71—21no—
97C4×C8×C71C2311yescharacteristic
98C4×C8×C71C2311yescharacteristic
99#41×C71—82no—
100C2×Q8×C71—21no—
101C2×Q8×C71—21no—
102C2×Q8×C71—21no—
103C2×Q8×C71—21no—
104C2×Q8×C71—41no—
105C2×Q8×C71—82no—
106C2×Q8×C71C2×D412yes—
107C2×D4×C71—21no—
108C2×D4×C71—21no—
109C2×D4×C71—21no—
110C2×D4×C71—41no—
111C22×C4×C71—41no—
112C22×C4×C71—41no—
113C22×C4×C71C2×D411yescharacteristic
114C22×C4×C71C2×D411yescharacteristic
115C22×C4×C71#4111yescharacteristic
116Q16×C71—82no—
117Q16×C71—82no—
118Q16×C71—82no—
119SD16×C71—82no—
120C2×C8×C71—22no—
121C2×C8×C71—22no—
122C2×C8×C71—22no—
123C2×C8×C71C2×D412yes—
124C2×C8×C71C2×D412yes—
125C4⋊C4×C71—21no—
126C4⋊C4×C71—21no—
127C4⋊C4×C71—21no—
128C4⋊C4×C71—22no—
129C4⋊C4×C71—22no—
130C4⋊C4×C71—21no—
131C4⋊C4×C71—21no—
132C4⋊C4×C71—21no—
133C4⋊C4×C71—21no—
134C4⋊C4×C71—21no—
135C4⋊C4×C71—21no—
136C4⋊C4×C71—41no—
137C4⋊C4×C71—41no—
138C4⋊C4×C71C2×D412yes—
139C22⋊C4×C71—22no—
140C22⋊C4×C71—41no—
141C22⋊C4×C71—41no—
142C42×C71—22no—
143C42×C71C2×D411yescharacteristic
144C42×C71C2×D411yescharacteristic
145C42×C71#4111yescharacteristic
146C42×C71C2411yescharacteristic
147C23×C71—41no—
148C23×C71#12111yescharacteristic
149Q8×C71—22no—
150Q8×C71—22no—
151Q8×C71—42no—
152Q8×C71—42no—
153Q8×C71—42no—
154Q8×C71—81no—
155Q8×C71—82no—
156D4×C71—42no—
157D4×C71—42no—
158D4×C71—81no—
159D4×C71—82no—
160C2×C4×C71—22no—
161C2×C4×C71—22no—
162C2×C4×C71—22no—
163C2×C4×C71—22no—
164C2×C4×C71—41no—
165C2×C4×C71—41no—
166C2×C4×C71—41no—
167C2×C4×C71—41no—
168C2×C4×C71—82no—
169C2×C4×C71—82no—
170C2×C4×C71#12111yescharacteristic
171C2×C4×C71#12111yescharacteristic
172C2×C4×C71#12111yescharacteristic
173C2×C4×C71C22×D411yescharacteristic
174C2×C4×C71C22×D411yescharacteristic
175C2×C4×C71C2×#4111yescharacteristic
176C8×C71—22no—
177C8×C71—22no—
178C8×C71—42no—
179C8×C71—42no—
180C8×C71—42no—
181C22×C71—21no—
182C22×C71—21no—
183C22×C71—21no—
184C22×C71—81no—
185C22×C71—81no—
186C22×C71—81no—
187C22×C71C2×#12111yescharacteristic
188C4×C71—21no—
189C4×C71—21no—
190C4×C71—21no—
191C4×C71—21no—
192C4×C71—42no—
193C4×C71—42no—
194C4×C71—81no—
195C4×C71—81no—
196C4×C71—81no—
197C4×C71#57612yes—
198C4×C71#57812yes—
199#19530C7111yesmaximal, characteristic, direct product factor, Sylow
200C2×C71—41no—
201C2×C71—161no—
202C2×C71#304411yescharacteristic
203C2×C71#305811yescharacteristic
204C2×C71#306911yescharacteristic
205#3226C2×C7111yescharacteristic
206#3216C2×C7111yescharacteristic
207#3209C2×C7111yescharacteristic, complemented
208#2985C2×C7111yescharacteristic, complemented
209#1702C2×C7112yescomplemented
210#1694C2×C7112yescomplemented
211#1682C2×C7112yescomplemented
212#1674C2×C7112yescomplemented
213#1572C2×C7111yescharacteristic, complemented
214#1556C2×C7111yescharacteristic, complemented
215#1553C2×C7111yescharacteristic
216C71#1953011yescharacteristic, direct product factor, Sylow
217#533C22×C7111yescharacteristic
218#532C22×C7111yescharacteristic
219C2×C4⋊Q8C22×C7111yescharacteristic
220#500C22×C7111yescharacteristic
221#500C22×C7111yescharacteristic
222#497—21no—
223#494—21no—
224#489—21no—
225#487—21no—
226#486C22×C7111yescharacteristic
227#486C22×C7111yescharacteristic
228#480—22no—
229#474—22no—
230#462C22×C7112yes—
231#462C22×C7112yes—
232#462C22×C7112yes—
233#460C22×C7112yes—
234#452—22no—
235#451—22no—
236#446—22no—
237#446C22×C7112yes—
238C8×Q8C22×C7112yes—
239#433—22no—
240#432C22×C7111yescharacteristic
241#432C22×C7111yescharacteristic
242#431C22×C7111yescharacteristic
243#431C22×C7111yescharacteristic
244#335C22×C7112yes—
245#334C22×C7112yes—
246#332C22×C7112yes—
247#332C22×C7112yes—
248#332C22×C7112yes—
249#331C22×C7112yes—
250C2×#41—21no—
251C22×Q8—21no—
252C2×Q16—22no—
253C2×Q16—22no—
254C2×Q16—22no—
255C2×SD16—22no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8D4×C7111yescharacteristic
260C4⋊Q8D4×C7111yescharacteristic
261C4⋊Q8C23×C7111yescharacteristic
262C4⋊Q8C23×C7111yescharacteristic
263C4⋊Q8C23×C7111yescharacteristic
264#127C23×C7111yescharacteristic
265#125—21no—
266#124—21no—
267C22⋊Q8—21no—
268C22⋊Q8D4×C7112yes—
269#121—21no—
270#121—21no—
271C4×Q8—21no—
272C4×Q8C23×C7112yes—
273C4×D4—21no—
274C4×D4—21no—
275#117—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C42C23×C7111yescharacteristic
279#107—22no—
280#107—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8D4×C7112yes—
284C4⋊C8C23×C7112yes—
285C4⋊C8C23×C7112yes—
286C4⋊C8C23×C7112yes—
287#103—22no—
288#103—22no—
289#103—22no—
290#102—22no—
291#102—22no—
292#102—22no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8D4×C7112yes—
297#97—22no—
298#97—21no—
299#97—21no—
300C4×C8—22no—
301C4×C8—21no—
302C4×C8C23×C7111yescharacteristic
303C4×C8C23×C7111yescharacteristic
304#41—82no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—41no—
310C2×Q8—82no—
311C2×Q8C2×D4×C7112yes—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4C2×D4×C7111yescharacteristic
319C22×C4C2×D4×C7111yescharacteristic
320C22×C4#41×C7111yescharacteristic
321Q16—82no—
322Q16—82no—
323Q16—82no—
324SD16—82no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8C2×D4×C7112yes—
329C2×C8C2×D4×C7112yes—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4C2×D4×C7112yes—
344C22⋊C4—22no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C42—22no—
348C42C2×D4×C7111yescharacteristic
349C42C2×D4×C7111yescharacteristic
350C42#41×C7111yescharacteristic
351C42C24×C7111yescharacteristic
352C23—41no—
353C23#121×C7111yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—42no—
357Q8—42no—
358Q8—42no—
359Q8—81no—
360Q8—82no—
361D4—42no—
362D4—42no—
363D4—81no—
364D4—82no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4#121×C7111yescharacteristic
376C2×C4#121×C7111yescharacteristic
377C2×C4#121×C7111yescharacteristic
378C2×C4C22×D4×C7111yescharacteristic
379C2×C4C22×D4×C7111yescharacteristic
380C2×C4C2×#41×C7111yescharacteristic
381C8—22no—
382C8—22no—
383C8—42no—
384C8—42no—
385C8—42no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22C2×#121×C7111yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4—42no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4#576×C7112yes—
403C4#578×C7112yes—
404C2—41no—
405C2—161no—
406C2C71×#304411yescharacteristic
407C2C71×#305811yescharacteristic
408C2C71×#306911yescharacteristic
subgroupG/H
center, socleC22×C71C2×#121
commutator, Frattini, cosocleC42C24×C71
prime radical#532×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
3. C71×#3209C2semidirect
4. C71×#2985C2semidirect
5. C71×#1702C2semidirect
6. C71×#1694C2semidirect
7. C71×#1682C2semidirect
8. C71×#1674C2semidirect
9. C71×#1572C2semidirect
10. C71×#1556C2semidirect
199. #19530C71direct (N × H)
207. #3209C2×C71semidirect
208. #2985C2×C71semidirect
209. #1702C2×C71semidirect
210. #1694C2×C71semidirect
211. #1682C2×C71semidirect
212. #1674C2×C71semidirect
213. #1572C2×C71semidirect
214. #1556C2×C71semidirect
216. C71#19530direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#32262
C71×#32162
C71×#32092
C71×#29852
C71×#17022
C71×#16942
C71×#16822
C71×#16742
C71×#15722
C71×#15562
C71×#15532
#1953071