Finite Group Databasebeta

← C71×#19546 #31840418

Subgroups — C71×#19546 #31840418

Order 18176.

598 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3226C211yesmaximal, characteristic, complemented
2C71×#3218C211yesmaximal, characteristic, complemented
3C71×#3212C211yesmaximal, characteristic
4C71×#2993C211yesmaximal, characteristic, complemented
5C71×#1702C211yesmaximal, characteristic, complemented
6C71×#1701C211yesmaximal, characteristic, complemented
7C71×#1694C211yesmaximal, characteristic, complemented
8C71×#1693C211yesmaximal, characteristic, complemented
9C71×#1682C211yesmaximal, characteristic, complemented
10C71×#1681C211yesmaximal, characteristic, complemented
11C71×#1674C211yesmaximal, characteristic, complemented
12C71×#1673C211yesmaximal, characteristic, complemented
13C71×#1572C211yesmaximal, characteristic, complemented
14C71×#1557C211yesmaximal, characteristic, complemented
15C71×#1549C211yesmaximal, characteristic, complemented
16#532×C71C2211yescharacteristic
17C2×C4⋊Q8×C71C2211yescharacteristic
18#529×C71C2211yescharacteristic
19#500×C71C2211yescharacteristic
20#500×C71C2211yescharacteristic
21#497×C71—21no—
22#495×C71—21no—
23#491×C71—21no—
24#488×C71—21no—
25#486×C71C2211yescharacteristic
26#486×C71C2211yescharacteristic
27#480×C71—21no—
28#480×C71—21no—
29#474×C71—21no—
30#474×C71—21no—
31#462×C71C2211yescharacteristic
32#462×C71C2211yescharacteristic
33#462×C71C2211yescharacteristic
34#461×C71C2211yescharacteristic
35#461×C71C2211yescharacteristic
36#461×C71C2211yescharacteristic
37#460×C71C2211yescharacteristic
38#459×C71C2211yescharacteristic
39#452×C71—21no—
40#451×C71—21no—
41#449×C71—21no—
42#448×C71—21no—
43#446×C71C2211yescharacteristic
44C8×Q8×C71C2211yescharacteristic
45#436×C71—21no—
46#436×C71C2211yescharacteristic
47#435×C71—21no—
48C8×D4×C71C2211yescharacteristic
49#433×C71—21no—
50#432×C71C2211yescharacteristic
51#431×C71C2211yescharacteristic
52#431×C71C2211yescharacteristic
53#424×C71—21no—
54#423×C71C2211yescharacteristic
55#335×C71C2211yescharacteristic
56#335×C71C2211yescharacteristic
57#334×C71C2211yescharacteristic
58#334×C71C2211yescharacteristic
59#332×C71C2211yescharacteristic
60#332×C71C2211yescharacteristic
61#332×C71C2211yescharacteristic
62#332×C71C2211yescharacteristic
63#332×C71C2211yescharacteristic
64#332×C71C2211yescharacteristic
65#331×C71C2211yescharacteristic
66#331×C71C2211yescharacteristic
67C2×#41×C71—21no—
68C22×Q8×C71—21no—
69C2×Q16×C71—21no—
70C2×Q16×C71—21no—
71C2×Q16×C71—21no—
72C2×SD16×C71—21no—
73C2×SD16×C71—21no—
74C2×SD16×C71—21no—
75C2×SD16×C71—21no—
76C2×D8×C71—21no—
77C2×M16×C71—21no—
78C2×M16×C71—21no—
79C22×C8×C71—21no—
80C4⋊Q8×C71—21no—
81C4⋊Q8×C71—21no—
82C4⋊Q8×C71—21no—
83C4⋊Q8×C71C2311yescharacteristic
84C4⋊Q8×C71C2311yescharacteristic
85C4⋊Q8×C71C2311yescharacteristic
86#127×C71C2311yescharacteristic
87#126×C71—21no—
88#125×C71D411yescharacteristic
89#124×C71—21no—
90#124×C71D411yescharacteristic
91#123×C71—21no—
92C22⋊Q8×C71D411yescharacteristic
93#121×C71—21no—
94#121×C71—21no—
95#121×C71D411yescharacteristic
96C4×Q8×C71C2311yescharacteristic
97C4×D4×C71—21no—
98C4×D4×C71—21no—
99C4×D4×C71—21no—
100C4×D4×C71C2311yescharacteristic
101#117×C71—21no—
102C2×C4⋊C4×C71—21no—
103C2×C4⋊C4×C71—21no—
104C2×C42×C71C2311yescharacteristic
105#107×C71—21no—
106#107×C71—21no—
107#107×C71—21no—
108#107×C71—21no—
109C4⋊C8×C71—21no—
110C4⋊C8×C71—21no—
111C4⋊C8×C71—21no—
112C4⋊C8×C71D411yescharacteristic
113C4⋊C8×C71D411yescharacteristic
114C4⋊C8×C71C2311yescharacteristic
115C4⋊C8×C71C2311yescharacteristic
116C4⋊C8×C71C2311yescharacteristic
117C4⋊C8×C71C2311yescharacteristic
118C4⋊C8×C71C2311yescharacteristic
119C4⋊C8×C71C2311yescharacteristic
120#103×C71—21no—
121#103×C71—21no—
122#103×C71—21no—
123#103×C71—21no—
124#103×C71—21no—
125#103×C71—21no—
126#102×C71—21no—
127#102×C71—21no—
128#102×C71—21no—
129#102×C71—21no—
130#102×C71—21no—
131#102×C71—21no—
132C22⋊C8×C71—21no—
133C22⋊C8×C71—21no—
134C22⋊C8×C71—21no—
135C22⋊C8×C71—21no—
136C22⋊C8×C71D411yescharacteristic
137C22⋊C8×C71D411yescharacteristic
138#97×C71—21no—
139#97×C71—21no—
140C4×C8×C71—21no—
141C4×C8×C71—21no—
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144#41×C71—81no—
145#41×C71—81no—
146C2×Q8×C71—21no—
147C2×Q8×C71—21no—
148C2×Q8×C71—21no—
149C2×Q8×C71—41no—
150C2×Q8×C71—81no—
151C2×Q8×C71—81no—
152C2×Q8×C71C2×D411yescharacteristic
153C2×D4×C71—21no—
154C2×D4×C71—21no—
155C2×D4×C71—21no—
156C2×D4×C71—21no—
157C2×D4×C71—41no—
158C2×D4×C71C2×D411yescharacteristic
159C22×C4×C71—21no—
160C22×C4×C71—41no—
161C22×C4×C71—41no—
162C22×C4×C71C2×D411yescharacteristic
163C22×C4×C71C2×D411yescharacteristic
164C22×C4×C71#4111yescharacteristic
165Q16×C71—81no—
166Q16×C71—81no—
167Q16×C71—81no—
168SD16×C71—81no—
169SD16×C71—81no—
170SD16×C71—81no—
171SD16×C71—81no—
172D8×C71—81no—
173M16×C71—81no—
174M16×C71—81no—
175C2×C8×C71—21no—
176C2×C8×C71—21no—
177C2×C8×C71—21no—
178C2×C8×C71—21no—
179C2×C8×C71—21no—
180C2×C8×C71—21no—
181C2×C8×C71—81no—
182C2×C8×C71C2×D411yescharacteristic
183C2×C8×C71C2×D411yescharacteristic
184C2×C8×C71C2×D411yescharacteristic
185C2×C8×C71C2×D411yescharacteristic
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—21no—
193C4⋊C4×C71—21no—
194C4⋊C4×C71—21no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71—41no—
197C4⋊C4×C71—41no—
198C4⋊C4×C71C2×D411yescharacteristic
199C4⋊C4×C71C2×D411yescharacteristic
200C22⋊C4×C71—21no—
201C22⋊C4×C71—21no—
202C22⋊C4×C71—21no—
203C22⋊C4×C71—21no—
204C22⋊C4×C71—21no—
205C22⋊C4×C71—41no—
206C22⋊C4×C71—41no—
207C42×C71—21no—
208C42×C71C2×D411yescharacteristic
209C42×C71C2×D411yescharacteristic
210C42×C71#4111yescharacteristic
211C42×C71C2411yescharacteristic
212C23×C71—21no—
213C23×C71—41no—
214C23×C71#12111yescharacteristic
215Q8×C71—21no—
216Q8×C71—21no—
217Q8×C71—41no—
218Q8×C71—41no—
219Q8×C71—41no—
220Q8×C71—41no—
221Q8×C71—41no—
222Q8×C71—41no—
223Q8×C71—82no—
224D4×C71—21no—
225D4×C71—21no—
226D4×C71—41no—
227D4×C71—41no—
228D4×C71—41no—
229D4×C71—41no—
230D4×C71—81no—
231D4×C71—81no—
232D4×C71—82no—
233C2×C4×C71—21no—
234C2×C4×C71—21no—
235C2×C4×C71—21no—
236C2×C4×C71—21no—
237C2×C4×C71—21no—
238C2×C4×C71—21no—
239C2×C4×C71—21no—
240C2×C4×C71—41no—
241C2×C4×C71—41no—
242C2×C4×C71—41no—
243C2×C4×C71—41no—
244C2×C4×C71—41no—
245C2×C4×C71—41no—
246C2×C4×C71—82no—
247C2×C4×C71—82no—
248C2×C4×C71#12111yescharacteristic
249C2×C4×C71#12111yescharacteristic
250C2×C4×C71#12111yescharacteristic
251C2×C4×C71C22×D411yescharacteristic
252C2×C4×C71C22×D411yescharacteristic
253C2×C4×C71C2×#4111yescharacteristic
254C8×C71—21no—
255C8×C71—21no—
256C8×C71—21no—
257C8×C71—21no—
258C8×C71—41no—
259C8×C71—41no—
260C8×C71—41no—
261C8×C71—41no—
262C8×C71—41no—
263C8×C71—41no—
264C22×C71—21no—
265C22×C71—21no—
266C22×C71—21no—
267C22×C71—41no—
268C22×C71—41no—
269C22×C71—41no—
270C22×C71—81no—
271C22×C71—81no—
272C22×C71—81no—
273C22×C71C2×#12111yescharacteristic
274C4×C71—21no—
275C4×C71—21no—
276C4×C71—21no—
277C4×C71—21no—
278C4×C71—41no—
279C4×C71—41no—
280C4×C71—41no—
281C4×C71—81no—
282C4×C71—81no—
283C4×C71—81no—
284C4×C71#57611yescharacteristic
285C4×C71#57611yescharacteristic
286C4×C71#57711yescharacteristic
287C4×C71#57811yescharacteristic
288#19546C7111yesmaximal, characteristic, direct product factor, Sylow
289C2×C71—41no—
290C2×C71—81no—
291C2×C71—161no—
292C2×C71#304611yescharacteristic
293C2×C71#305811yescharacteristic
294C2×C71#306911yescharacteristic
295#3226C2×C7111yescharacteristic, complemented
296#3218C2×C7111yescharacteristic, complemented
297#3212C2×C7111yescharacteristic
298#2993C2×C7111yescharacteristic, complemented
299#1702C2×C7111yescharacteristic, complemented
300#1701C2×C7111yescharacteristic, complemented
301#1694C2×C7111yescharacteristic, complemented
302#1693C2×C7111yescharacteristic, complemented
303#1682C2×C7111yescharacteristic, complemented
304#1681C2×C7111yescharacteristic, complemented
305#1674C2×C7111yescharacteristic, complemented
306#1673C2×C7111yescharacteristic, complemented
307#1572C2×C7111yescharacteristic, complemented
308#1557C2×C7111yescharacteristic, complemented
309#1549C2×C7111yescharacteristic, complemented
310C71#1954611yescharacteristic, direct product factor, Sylow
311#532C22×C7111yescharacteristic
312C2×C4⋊Q8C22×C7111yescharacteristic
313#529C22×C7111yescharacteristic
314#500C22×C7111yescharacteristic
315#500C22×C7111yescharacteristic
316#497—21no—
317#495—21no—
318#491—21no—
319#488—21no—
320#486C22×C7111yescharacteristic
321#486C22×C7111yescharacteristic
322#480—21no—
323#480—21no—
324#474—21no—
325#474—21no—
326#462C22×C7111yescharacteristic
327#462C22×C7111yescharacteristic
328#462C22×C7111yescharacteristic
329#461C22×C7111yescharacteristic
330#461C22×C7111yescharacteristic
331#461C22×C7111yescharacteristic
332#460C22×C7111yescharacteristic
333#459C22×C7111yescharacteristic
334#452—21no—
335#451—21no—
336#449—21no—
337#448—21no—
338#446C22×C7111yescharacteristic
339C8×Q8C22×C7111yescharacteristic
340#436—21no—
341#436C22×C7111yescharacteristic
342#435—21no—
343C8×D4C22×C7111yescharacteristic
344#433—21no—
345#432C22×C7111yescharacteristic
346#431C22×C7111yescharacteristic
347#431C22×C7111yescharacteristic
348#424—21no—
349#423C22×C7111yescharacteristic
350#335C22×C7111yescharacteristic
351#335C22×C7111yescharacteristic
352#334C22×C7111yescharacteristic
353#334C22×C7111yescharacteristic
354#332C22×C7111yescharacteristic
355#332C22×C7111yescharacteristic
356#332C22×C7111yescharacteristic
357#332C22×C7111yescharacteristic
358#332C22×C7111yescharacteristic
359#332C22×C7111yescharacteristic
360#331C22×C7111yescharacteristic
361#331C22×C7111yescharacteristic
362C2×#41—21no—
363C22×Q8—21no—
364C2×Q16—21no—
365C2×Q16—21no—
366C2×Q16—21no—
367C2×SD16—21no—
368C2×SD16—21no—
369C2×SD16—21no—
370C2×SD16—21no—
371C2×D8—21no—
372C2×M16—21no—
373C2×M16—21no—
374C22×C8—21no—
375C4⋊Q8—21no—
376C4⋊Q8—21no—
377C4⋊Q8—21no—
378C4⋊Q8C23×C7111yescharacteristic
379C4⋊Q8C23×C7111yescharacteristic
380C4⋊Q8C23×C7111yescharacteristic
381#127C23×C7111yescharacteristic
382#126—21no—
383#125D4×C7111yescharacteristic
384#124—21no—
385#124D4×C7111yescharacteristic
386#123—21no—
387C22⋊Q8D4×C7111yescharacteristic
388#121—21no—
389#121—21no—
390#121D4×C7111yescharacteristic
391C4×Q8C23×C7111yescharacteristic
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—21no—
395C4×D4C23×C7111yescharacteristic
396#117—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C42C23×C7111yescharacteristic
400#107—21no—
401#107—21no—
402#107—21no—
403#107—21no—
404C4⋊C8—21no—
405C4⋊C8—21no—
406C4⋊C8—21no—
407C4⋊C8D4×C7111yescharacteristic
408C4⋊C8D4×C7111yescharacteristic
409C4⋊C8C23×C7111yescharacteristic
410C4⋊C8C23×C7111yescharacteristic
411C4⋊C8C23×C7111yescharacteristic
412C4⋊C8C23×C7111yescharacteristic
413C4⋊C8C23×C7111yescharacteristic
414C4⋊C8C23×C7111yescharacteristic
415#103—21no—
416#103—21no—
417#103—21no—
418#103—21no—
419#103—21no—
420#103—21no—
421#102—21no—
422#102—21no—
423#102—21no—
424#102—21no—
425#102—21no—
426#102—21no—
427C22⋊C8—21no—
428C22⋊C8—21no—
429C22⋊C8—21no—
430C22⋊C8—21no—
431C22⋊C8D4×C7111yescharacteristic
432C22⋊C8D4×C7111yescharacteristic
433#97—21no—
434#97—21no—
435C4×C8—21no—
436C4×C8—21no—
437C4×C8C23×C7111yescharacteristic
438C4×C8C23×C7111yescharacteristic
439#41—81no—
440#41—81no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—41no—
445C2×Q8—81no—
446C2×Q8—81no—
447C2×Q8C2×D4×C7111yescharacteristic
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—41no—
453C2×D4C2×D4×C7111yescharacteristic
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4C2×D4×C7111yescharacteristic
458C22×C4C2×D4×C7111yescharacteristic
459C22×C4#41×C7111yescharacteristic
460Q16—81no—
461Q16—81no—
462Q16—81no—
463SD16—81no—
464SD16—81no—
465SD16—81no—
466SD16—81no—
467D8—81no—
468M16—81no—
469M16—81no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—81no—
477C2×C8C2×D4×C7111yescharacteristic
478C2×C8C2×D4×C7111yescharacteristic
479C2×C8C2×D4×C7111yescharacteristic
480C2×C8C2×D4×C7111yescharacteristic
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4C2×D4×C7111yescharacteristic
494C4⋊C4C2×D4×C7111yescharacteristic
495C22⋊C4—21no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C42—21no—
503C42C2×D4×C7111yescharacteristic
504C42C2×D4×C7111yescharacteristic
505C42#41×C7111yescharacteristic
506C42C24×C7111yescharacteristic
507C23—21no—
508C23—41no—
509C23#121×C7111yescharacteristic
510Q8—21no—
511Q8—21no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—82no—
519D4—21no—
520D4—21no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—41no—
525D4—81no—
526D4—81no—
527D4—82no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4#121×C7111yescharacteristic
544C2×C4#121×C7111yescharacteristic
545C2×C4#121×C7111yescharacteristic
546C2×C4C22×D4×C7111yescharacteristic
547C2×C4C22×D4×C7111yescharacteristic
548C2×C4C2×#41×C7111yescharacteristic
549C8—21no—
550C8—21no—
551C8—21no—
552C8—21no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8—41no—
557C8—41no—
558C8—41no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22C2×#121×C7111yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4#576×C7111yescharacteristic
580C4#576×C7111yescharacteristic
581C4#577×C7111yescharacteristic
582C4#578×C7111yescharacteristic
583C2—41no—
584C2—81no—
585C2—161no—
586C2C71×#304611yescharacteristic
587C2C71×#305811yescharacteristic
588C2C71×#306911yescharacteristic
subgroupG/H
center, socleC22×C71C2×#121
commutator, Frattini, cosocleC42C24×C71
prime radicalC71×#3212C2
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
1. C71×#3226C2semidirect
2. C71×#3218C2semidirect
4. C71×#2993C2semidirect
5. C71×#1702C2semidirect
6. C71×#1701C2semidirect
7. C71×#1694C2semidirect
8. C71×#1693C2semidirect
9. C71×#1682C2semidirect
10. C71×#1681C2semidirect
11. C71×#1674C2semidirect
12. C71×#1673C2semidirect
13. C71×#1572C2semidirect
14. C71×#1557C2semidirect
15. C71×#1549C2semidirect
288. #19546C71direct (N × H)
295. #3226C2×C71semidirect
296. #3218C2×C71semidirect
298. #2993C2×C71semidirect
299. #1702C2×C71semidirect
300. #1701C2×C71semidirect
301. #1694C2×C71semidirect
302. #1693C2×C71semidirect
303. #1682C2×C71semidirect
304. #1681C2×C71semidirect
305. #1674C2×C71semidirect
306. #1673C2×C71semidirect
307. #1572C2×C71semidirect
308. #1557C2×C71semidirect
309. #1549C2×C71semidirect
310. C71#19546direct (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×#32182
C71×#32122
C71×#29932
C71×#17022
C71×#17012
C71×#16942
C71×#16932
C71×#16822
C71×#16812
C71×#16742
C71×#16732
C71×#15722
C71×#15572
C71×#15492
#1954671