Finite Group Databasebeta

← C71×#11093 #31831965

Subgroups — C71×#11093 #31831965

Order 18176.

910 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#2896C211yesmaximal, characteristic
2C2×#417×C71C211yesmaximal, characteristic
3C2×#410×C71C211yesmaximal, characteristic
4C71×#2317C211yesmaximal, characteristic, complemented
5C2×#344×C71C211yesmaximal, characteristic, complemented
6C71×#1736C211yesmaximal, characteristic, complemented
7C71×#1733C211yesmaximal, characteristic, complemented
8C22×#41×C71C2211yescharacteristic
9C22×M16×C71C2211yescharacteristic
10C22×M16×C71C2211yescharacteristic
11C2×#117×C71C2211yescharacteristic
12C2×#117×C71C2211yescharacteristic
13C2×#117×C71C2211yescharacteristic
14C22×C4⋊C4×C71C2211yescharacteristic
15#419×C71—21no—
16#418×C71—21no—
17#417×C71—22no—
18#417×C71—22no—
19#417×C71—21no—
20#417×C71—21no—
21#416×C71—22no—
22C2×#103×C71—21no—
23C2×#102×C71—21no—
24#410×C71—22no—
25#410×C71C414yescomplemented
26C2×#99×C71—22no—
27#389×C71—22no—
28#385×C71—22no—
29#384×C71—21no—
30#382×C71—21no—
31#344×C71—22no—
32#340×C71—22no—
33#339×C71—22no—
34C2×#41×C71—21no—
35C2×#41×C71—21no—
36C2×#41×C71—21no—
37C2×#41×C71—21no—
38C2×#41×C71—21no—
39C2×#41×C71—21no—
40C2×#41×C71—21no—
41C2×#41×C71—21no—
42C2×#41×C71—21no—
43C2×#41×C71—21no—
44C2×#41×C71C2×C412yes—
45C2×#41×C71Q812yes—
46C22×Q8×C71D411yescharacteristic
47C22×D4×C71—21no—
48C22×D4×C71D411yescharacteristic
49C23×C4×C71—21no—
50C23×C4×C71—21no—
51C23×C4×C71C2311yescharacteristic
52C2×M16×C71—21no—
53C2×M16×C71—21no—
54C2×M16×C71—21no—
55C2×M16×C71—21no—
56C2×M16×C71—21no—
57C2×M16×C71—21no—
58C2×M16×C71—21no—
59C2×M16×C71—21no—
60C2×M16×C71—21no—
61C2×M16×C71—21no—
62C2×M16×C71—21no—
63C2×M16×C71D412yes—
64C22×C8×C71—21no—
65C22×C8×C71D412yes—
66#117×C71—21no—
67#117×C71—21no—
68#117×C71—21no—
69#117×C71—21no—
70#117×C71—21no—
71#117×C71—21no—
72#117×C71—21no—
73#117×C71—21no—
74#117×C71—21no—
75#117×C71C2×C412yes—
76#117×C71C2×C412yes—
77#117×C71D412yes—
78C2×C4⋊C4×C71—21no—
79C2×C4⋊C4×C71—21no—
80C2×C4⋊C4×C71—21no—
81C2×C4⋊C4×C71—21no—
82C2×C4⋊C4×C71—21no—
83C2×C4⋊C4×C71—21no—
84C2×C4⋊C4×C71—21no—
85C2×C4⋊C4×C71—24no—
86C2×C4⋊C4×C71D412yes—
87C2×C4⋊C4×C71D411yescharacteristic
88C2×C4⋊C4×C71D411yescharacteristic
89C2×C22⋊C4×C71—21no—
90C2×C22⋊C4×C71—21no—
91C2×C22⋊C4×C71D412yes—
92C2×C42×C71—21no—
93C2×C42×C71—21no—
94C2×C42×C71—21no—
95#103×C71—24no—
96#103×C71—22no—
97#103×C71—22no—
98#102×C71—24no—
99#102×C71—22no—
100#102×C71—22no—
101#99×C71—28no—
102#95×C71—24no—
103C24×C71—21no—
104C24×C71C2×D411yescharacteristic
105#41×C71—22no—
106#41×C71—22no—
107#41×C71—22no—
108#41×C71—22no—
109#41×C71—22no—
110#41×C71—22no—
111#41×C71—22no—
112#41×C71—22no—
113#41×C71—41no—
114#41×C71—41no—
115#41×C71—41no—
116#41×C71—41no—
117#41×C71—41no—
118#41×C71—41no—
119#41×C71—41no—
120#41×C71—41no—
121C2×Q8×C71—21no—
122C2×Q8×C71—21no—
123C2×Q8×C71—21no—
124C2×Q8×C71—21no—
125C2×Q8×C71—21no—
126C2×Q8×C71C4⋊C412yescomplemented
127C2×D4×C71—22no—
128C2×D4×C71—22no—
129C2×D4×C71—21no—
130C2×D4×C71—21no—
131C2×D4×C71—21no—
132C2×D4×C71—21no—
133C2×D4×C71—21no—
134C2×D4×C71—41no—
135C2×D4×C71—41no—
136C2×D4×C71—41no—
137C2×D4×C71—41no—
138C2×D4×C71C4⋊C412yescomplemented
139C22×C4×C71—22no—
140C22×C4×C71—21no—
141C22×C4×C71—22no—
142C22×C4×C71—22no—
143C22×C4×C71—21no—
144C22×C4×C71—22no—
145C22×C4×C71—21no—
146C22×C4×C71—41no—
147C22×C4×C71—41no—
148C22×C4×C71—41no—
149C22×C4×C71—41no—
150C22×C4×C71—41no—
151C22×C4×C71—41no—
152C22×C4×C71—41no—
153C22×C4×C71—82no—
154C22×C4×C71C22×C411yescharacteristic
155C22×C4×C71C2×D411yescharacteristic
156C22×C4×C71C2×D411yescharacteristic
157C22×C4×C71C2×D411yescharacteristic
158C22×C4×C71C2×D411yescharacteristic
159C22×C4×C71C2×D411yescharacteristic
160C22×C4×C71C2×D411yescharacteristic
161C22×C4×C71C2×Q811yescharacteristic
162C22×C4×C71#4111yescharacteristic
163C22×C4×C71#4111yescharacteristic
164C22×C4×C71#4111yescharacteristic
165C22×C4×C71#4111yescharacteristic
166C22×C4×C71#4111yescharacteristic
167C22×C4×C71#4111yescharacteristic
168M16×C71—22no—
169M16×C71—22no—
170M16×C71—41no—
171M16×C71—41no—
172M16×C71—41no—
173M16×C71—41no—
174M16×C71—41no—
175M16×C71—41no—
176C2×C8×C71—22no—
177C2×C8×C71—22no—
178C2×C8×C71—22no—
179C2×C8×C71—22no—
180C2×C8×C71—41no—
181C2×C8×C71—41no—
182C4⋊C4×C71—22no—
183C4⋊C4×C71—22no—
184C4⋊C4×C71—22no—
185C4⋊C4×C71—22no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—22no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71—41no—
192C4⋊C4×C71—41no—
193C4⋊C4×C71—41no—
194C4⋊C4×C71—84no—
195C22⋊C4×C71—22no—
196C22⋊C4×C71—22no—
197C22⋊C4×C71—41no—
198C22⋊C4×C71—41no—
199C22⋊C4×C71—41no—
200C22⋊C4×C71C22⋊C414yes—
201C42×C71—22no—
202C42×C71—22no—
203C42×C71—22no—
204C42×C71—41no—
205C42×C71—41no—
206C42×C71—41no—
207C23×C71—22no—
208C23×C71—41no—
209C23×C71—41no—
210C23×C71—41no—
211C23×C71—41no—
212C23×C71—81no—
213C23×C71—81no—
214C23×C71C2×C22⋊C411yescharacteristic
215C23×C71#12011yescharacteristic
216C23×C71#12111yescharacteristic
217C23×C71C22⋊Q811yescharacteristic
218C23×C71#12311yescharacteristic
219C23×C71#12411yescharacteristic
220C23×C71#12711yescharacteristic
221Q8×C71—22no—
222Q8×C71—22no—
223Q8×C71—41no—
224Q8×C71—41no—
225D4×C71—22no—
226D4×C71—22no—
227D4×C71—42no—
228D4×C71—42no—
229D4×C71—42no—
230D4×C71—42no—
231D4×C71—41no—
232D4×C71—41no—
233C2×C4×C71—22no—
234C2×C4×C71—24no—
235C2×C4×C71—42no—
236C2×C4×C71—42no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71—42no—
240C2×C4×C71—42no—
241C2×C4×C71—42no—
242C2×C4×C71—42no—
243C2×C4×C71—42no—
244C2×C4×C71—44no—
245C2×C4×C71—42no—
246C2×C4×C71—42no—
247C2×C4×C71—41no—
248C2×C4×C71—41no—
249C2×C4×C71—42no—
250C2×C4×C71—81no—
251C2×C4×C71—81no—
252C2×C4×C71—82no—
253C2×C4×C71—82no—
254C2×C4×C71C2×C4⋊C411yescharacteristic
255C2×C4×C71#11711yescharacteristic
256C2×C4×C71C4×D411yescharacteristic
257C2×C4×C71C4×D411yescharacteristic
258C2×C4×C71C4×D411yescharacteristic
259C2×C4×C71C4×D411yescharacteristic
260C2×C4×C71#12011yescharacteristic
261C2×C4×C71#12011yescharacteristic
262C2×C4×C71#12111yescharacteristic
263C2×C4×C71#12111yescharacteristic
264C2×C4×C71#12111yescharacteristic
265C2×C4×C71#12111yescharacteristic
266C2×C4×C71#12111yescharacteristic
267C2×C4×C71#12111yescharacteristic
268C2×C4×C71#12111yescharacteristic
269C2×C4×C71C22⋊Q811yescharacteristic
270C2×C4×C71C22⋊Q811yescharacteristic
271C2×C4×C71C22⋊Q811yescharacteristic
272C2×C4×C71C22⋊Q811yescharacteristic
273C2×C4×C71C22⋊Q811yescharacteristic
274C2×C4×C71#12311yescharacteristic
275C2×C4×C71#12311yescharacteristic
276C2×C4×C71#12311yescharacteristic
277C2×C4×C71#12311yescharacteristic
278C2×C4×C71#12411yescharacteristic
279C2×C4×C71#12411yescharacteristic
280C2×C4×C71#12611yescharacteristic
281C2×C4×C71#12611yescharacteristic
282C8×C71—42no—
283C8×C71—42no—
284C22×C71—21no—
285C22×C71—21no—
286C22×C71—21no—
287C22×C71—21no—
288C22×C71—21no—
289C22×C71—21no—
290C22×C71—41no—
291C22×C71—41no—
292C22×C71—41no—
293C22×C71—41no—
294C22×C71—42no—
295C22×C71—82no—
296C22×C71—82no—
297C22×C71—81no—
298C22×C71—81no—
299C22×C71#38011yescharacteristic
300C22×C71#38611yescharacteristic
301C22×C71#39011yescharacteristic
302C22×C71#39211yescharacteristic
303C22×C71#39311yescharacteristic
304C22×C71#39711yescharacteristic
305C22×C71#39911yescharacteristic
306C4×C71—44no—
307C4×C71—42no—
308C4×C71—82no—
309C4×C71—81no—
310C4×C71—82no—
311C4×C71—81no—
312C4×C71#38511yescharacteristic
313C4×C71#38511yescharacteristic
314C4×C71#38811yescharacteristic
315C4×C71#38811yescharacteristic
316C4×C71#39411yescharacteristic
317C4×C71#39411yescharacteristic
318C4×C71#39611yescharacteristic
319C4×C71#39611yescharacteristic
320#11093C7111yesmaximal, characteristic, direct product factor, Sylow
321C2×C71—21no—
322C2×C71—21no—
323C2×C71—21no—
324C2×C71—21no—
325C2×C71—21no—
326C2×C71—21no—
327C2×C71—82no—
328C2×C71#143911yescharacteristic
329C2×C71#200811yescharacteristic
330C2×C71#200911yescharacteristic
331#2896C2×C7111yescharacteristic
332C2×#417C2×C7111yescharacteristic
333C2×#410C2×C7111yescharacteristic
334#2317C2×C7111yescharacteristic, complemented
335C2×#344C2×C7111yescharacteristic, complemented
336#1736C2×C7111yescharacteristic, complemented
337#1733C2×C7111yescharacteristic, complemented
338C71#1109311yescharacteristic, direct product factor, Sylow
339C22×#41C22×C7111yescharacteristic
340C22×M16C22×C7111yescharacteristic
341C22×M16C22×C7111yescharacteristic
342C2×#117C22×C7111yescharacteristic
343C2×#117C22×C7111yescharacteristic
344C2×#117C22×C7111yescharacteristic
345C22×C4⋊C4C22×C7111yescharacteristic
346#419—21no—
347#418—21no—
348#417—22no—
349#417—22no—
350#417—21no—
351#417—21no—
352#416—22no—
353C2×#103—21no—
354C2×#102—21no—
355#410—22no—
356#410C4×C7114yescomplemented
357C2×#99—22no—
358#389—22no—
359#385—22no—
360#384—21no—
361#382—21no—
362#344—22no—
363#340—22no—
364#339—22no—
365C2×#41—21no—
366C2×#41—21no—
367C2×#41—21no—
368C2×#41—21no—
369C2×#41—21no—
370C2×#41—21no—
371C2×#41—21no—
372C2×#41—21no—
373C2×#41—21no—
374C2×#41—21no—
375C2×#41C2×C4×C7112yes—
376C2×#41Q8×C7112yes—
377C22×Q8D4×C7111yescharacteristic
378C22×D4—21no—
379C22×D4D4×C7111yescharacteristic
380C23×C4—21no—
381C23×C4—21no—
382C23×C4C23×C7111yescharacteristic
383C2×M16—21no—
384C2×M16—21no—
385C2×M16—21no—
386C2×M16—21no—
387C2×M16—21no—
388C2×M16—21no—
389C2×M16—21no—
390C2×M16—21no—
391C2×M16—21no—
392C2×M16—21no—
393C2×M16—21no—
394C2×M16D4×C7112yes—
395C22×C8—21no—
396C22×C8D4×C7112yes—
397#117—21no—
398#117—21no—
399#117—21no—
400#117—21no—
401#117—21no—
402#117—21no—
403#117—21no—
404#117—21no—
405#117—21no—
406#117C2×C4×C7112yes—
407#117C2×C4×C7112yes—
408#117D4×C7112yes—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—24no—
417C2×C4⋊C4D4×C7112yes—
418C2×C4⋊C4D4×C7111yescharacteristic
419C2×C4⋊C4D4×C7111yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4D4×C7112yes—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—21no—
426#103—24no—
427#103—22no—
428#103—22no—
429#102—24no—
430#102—22no—
431#102—22no—
432#99—28no—
433#95—24no—
434C24—21no—
435C24C2×D4×C7111yescharacteristic
436#41—22no—
437#41—22no—
438#41—22no—
439#41—22no—
440#41—22no—
441#41—22no—
442#41—22no—
443#41—22no—
444#41—41no—
445#41—41no—
446#41—41no—
447#41—41no—
448#41—41no—
449#41—41no—
450#41—41no—
451#41—41no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8C4⋊C4×C7112yescomplemented
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4C4⋊C4×C7112yescomplemented
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—82no—
485C22×C4C22×C4×C7111yescharacteristic
486C22×C4C2×D4×C7111yescharacteristic
487C22×C4C2×D4×C7111yescharacteristic
488C22×C4C2×D4×C7111yescharacteristic
489C22×C4C2×D4×C7111yescharacteristic
490C22×C4C2×D4×C7111yescharacteristic
491C22×C4C2×D4×C7111yescharacteristic
492C22×C4C2×Q8×C7111yescharacteristic
493C22×C4#41×C7111yescharacteristic
494C22×C4#41×C7111yescharacteristic
495C22×C4#41×C7111yescharacteristic
496C22×C4#41×C7111yescharacteristic
497C22×C4#41×C7111yescharacteristic
498C22×C4#41×C7111yescharacteristic
499M16—22no—
500M16—22no—
501M16—41no—
502M16—41no—
503M16—41no—
504M16—41no—
505M16—41no—
506M16—41no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—41no—
512C2×C8—41no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—84no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4C22⋊C4×C7114yes—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—41no—
536C42—41no—
537C42—41no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—81no—
544C23—81no—
545C23C2×C22⋊C4×C7111yescharacteristic
546C23#120×C7111yescharacteristic
547C23#121×C7111yescharacteristic
548C23C22⋊Q8×C7111yescharacteristic
549C23#123×C7111yescharacteristic
550C23#124×C7111yescharacteristic
551C23#127×C7111yescharacteristic
552Q8—22no—
553Q8—22no—
554Q8—41no—
555Q8—41no—
556D4—22no—
557D4—22no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—41no—
563D4—41no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4C2×C4⋊C4×C7111yescharacteristic
586C2×C4#117×C7111yescharacteristic
587C2×C4C4×D4×C7111yescharacteristic
588C2×C4C4×D4×C7111yescharacteristic
589C2×C4C4×D4×C7111yescharacteristic
590C2×C4C4×D4×C7111yescharacteristic
591C2×C4#120×C7111yescharacteristic
592C2×C4#120×C7111yescharacteristic
593C2×C4#121×C7111yescharacteristic
594C2×C4#121×C7111yescharacteristic
595C2×C4#121×C7111yescharacteristic
596C2×C4#121×C7111yescharacteristic
597C2×C4#121×C7111yescharacteristic
598C2×C4#121×C7111yescharacteristic
599C2×C4#121×C7111yescharacteristic
600C2×C4C22⋊Q8×C7111yescharacteristic
601C2×C4C22⋊Q8×C7111yescharacteristic
602C2×C4C22⋊Q8×C7111yescharacteristic
603C2×C4C22⋊Q8×C7111yescharacteristic
604C2×C4C22⋊Q8×C7111yescharacteristic
605C2×C4#123×C7111yescharacteristic
606C2×C4#123×C7111yescharacteristic
607C2×C4#123×C7111yescharacteristic
608C2×C4#123×C7111yescharacteristic
609C2×C4#124×C7111yescharacteristic
610C2×C4#124×C7111yescharacteristic
611C2×C4#126×C7111yescharacteristic
612C2×C4#126×C7111yescharacteristic
613C8—42no—
614C8—42no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—42no—
626C22—82no—
627C22—82no—
628C22—81no—
629C22—81no—
630C22#380×C7111yescharacteristic
631C22#386×C7111yescharacteristic
632C22#390×C7111yescharacteristic
633C22#392×C7111yescharacteristic
634C22#393×C7111yescharacteristic
635C22#397×C7111yescharacteristic
636C22#399×C7111yescharacteristic
637C4—44no—
638C4—42no—
639C4—82no—
640C4—82no—
641C4—81no—
642C4—81no—
643C4#385×C7111yescharacteristic
644C4#385×C7111yescharacteristic
645C4#388×C7111yescharacteristic
646C4#388×C7111yescharacteristic
647C4#394×C7111yescharacteristic
648C4#394×C7111yescharacteristic
649C4#396×C7111yescharacteristic
650C4#396×C7111yescharacteristic
651C2—21no—
652C2—21no—
653C2—21no—
654C2—21no—
655C2—21no—
656C2—21no—
657C2—82no—
658C2C71×#143911yescharacteristic
659C2C71×#200811yescharacteristic
660C2C71×#200911yescharacteristic
subgroupG/H
center, socleC22×C71#392
commutatorC22×C4C22×C4×C71
Frattini, cosocleC23×C4C23×C71
prime radicalC22×#41×C71C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C23×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
4. C71×#2317C2semidirect
5. C2×#344×C71C2semidirect
6. C71×#1736C2semidirect
7. C71×#1733C2semidirect
25. #410×C71C4semidirect
126. C2×Q8×C71C4⋊C4semidirect
138. C2×D4×C71C4⋊C4semidirect
320. #11093C71direct (N × H)
334. #2317C2×C71semidirect
335. C2×#344C2×C71semidirect
336. #1736C2×C71semidirect
337. #1733C2×C71semidirect
338. C71#11093direct (N × H)
356. #410C4×C71semidirect
457. C2×Q8C4⋊C4×C71semidirect
469. C2×D4C4⋊C4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#28962
C2×#417×C712
C2×#410×C712
C71×#23172
C2×#344×C712
C71×#17362
C71×#17332
#1109371