Finite Group Databasebeta

← C71×#24261 #31845133

Subgroups — C71×#24261 #31845133

Order 18176.

1546 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3496C211yesmaximal, characteristic
2C2×#458×C71C212yesmaximal, complemented
3C71×#2854C212yesmaximal, complemented
4C4×#123×C71C211yesmaximal, characteristic, complemented
5C71×#2044C212yesmaximal, complemented
6C71×#2022C212yesmaximal, complemented
7C71×#1964C212yesmaximal, complemented
8C71×#1799C211yesmaximal, characteristic, complemented
9C71×#1790C212yesmaximal, complemented
10C2×H(5,𝔽2)×C71C2212yes—
11#547×C71—22no—
12#546×C71—24no—
13D42×C71—22no—
14#537×C71—21no—
15#534×C71—21no—
16#534×C71C2212yes—
17#532×C71—21no—
18C2×#123×C71C2211yescharacteristic
19C2×#123×C71C2214yescomplemented
20C2×#121×C71—21no—
21C2×#121×C71C2212yes—
22C2×C4×D4×C71C2211yescharacteristic
23C22×C42×C71C2211yescharacteristic, complemented
24#458×C71—28no—
25C2×#99×C71C2212yes—
26C2×#99×C71C2214yescomplemented
27C2×#99×C71C2214yescomplemented
28#390×C71—22no—
29#389×C71—22no—
30#388×C71—21no—
31#388×C71—24no—
32#387×C71—21no—
33#386×C71C2211yescharacteristic
34#386×C71C2214yescomplemented
35#385×C71C2211yescharacteristic, complemented
36#381×C71C2211yescharacteristic, complemented
37#380×C71C2212yescomplemented
38C4×C4⋊C4×C71—21no—
39C4×C22⋊C4×C71—21no—
40C4×C22⋊C4×C71C2211yescharacteristic, complemented
41#342×C71C2212yescomplemented
42H(5,𝔽2)×C71—24no—
43H(5,𝔽2)×C71—24no—
44H(5,𝔽2)×C71—44no—
45C2×#41×C71—22no—
46C2×#41×C71—24no—
47C22×D4×C71—22no—
48C22×D4×C71—24no—
49C22×D4×C71—21no—
50C22×D4×C71D412yes—
51C22×D4×C71C2314yes—
52C22×D4×C71C2311yescharacteristic
53C23×C4×C71D411yescharacteristic
54C23×C4×C71C2311yescharacteristic
55C23×C4×C71C2311yescharacteristic
56C23×C4×C71C2311yescharacteristic
57C4⋊Q8×C71—21no—
58#127×C71—21no—
59#124×C71—22no—
60#123×C71—24no—
61#123×C71—22no—
62#123×C71—24no—
63#123×C71—28no—
64#123×C71—44no—
65#123×C71D414yescomplemented
66C22⋊Q8×C71—24no—
67#121×C71—22no—
68#121×C71—22no—
69#121×C71—24no—
70#121×C71—22no—
71#121×C71—22no—
72#121×C71—22no—
73#121×C71—22no—
74#121×C71—22no—
75#121×C71—22no—
76#120×C71—24no—
77#120×C71—24no—
78C4×D4×C71—22no—
79C4×D4×C71—22no—
80C4×D4×C71—22no—
81C4×D4×C71—22no—
82C2×C4⋊C4×C71—21no—
83C2×C4⋊C4×C71—21no—
84C2×C4⋊C4×C71—24no—
85C2×C4⋊C4×C71—21no—
86C2×C4⋊C4×C71—22no—
87C2×C4⋊C4×C71D411yescharacteristic
88C2×C4⋊C4×C71D412yes—
89C2×C22⋊C4×C71—22no—
90C2×C22⋊C4×C71—21no—
91C2×C22⋊C4×C71—24no—
92C2×C22⋊C4×C71—21no—
93C2×C22⋊C4×C71—24no—
94C2×C22⋊C4×C71D411yescharacteristic
95C2×C22⋊C4×C71D414yes—
96C2×C22⋊C4×C71C2314yes—
97C2×C22⋊C4×C71C2311yescharacteristic
98C2×C22⋊C4×C71C2311yescharacteristic
99C2×C22⋊C4×C71C2311yescharacteristic
100C2×C42×C71—21no—
101C2×C42×C71—21no—
102C2×C42×C71—22no—
103C2×C42×C71—22no—
104C2×C42×C71—21no—
105C2×C42×C71D411yescharacteristic
106#99×C71—24no—
107#99×C71—28no—
108#99×C71—44no—
109#95×C71—21no—
110#95×C71—21no—
111#95×C71—21no—
112#95×C71—21no—
113#95×C71—21no—
114#95×C71—22no—
115#95×C71—24no—
116C24×C71—22no—
117C24×C71C2×D414yes—
118C24×C71C2×D411yescharacteristic
119C24×C71C2411yescharacteristic
120#41×C71—48no—
121#41×C71—48no—
122#41×C71—44no—
123#41×C71—44no—
124C2×Q8×C71—22no—
125C2×D4×C71—22no—
126C2×D4×C71—22no—
127C2×D4×C71—24no—
128C2×D4×C71—22no—
129C2×D4×C71—24no—
130C2×D4×C71—24no—
131C2×D4×C71—24no—
132C2×D4×C71—24no—
133C2×D4×C71—24no—
134C2×D4×C71—24no—
135C2×D4×C71—22no—
136C2×D4×C71—22no—
137C2×D4×C71—22no—
138C2×D4×C71—22no—
139C2×D4×C71—44no—
140C2×D4×C71—48no—
141C2×D4×C71—48no—
142C2×D4×C71—44no—
143C2×D4×C71—44no—
144C2×D4×C71—82no—
145C2×D4×C71C2×D412yes—
146C2×D4×C71#4112yes—
147C22×C4×C71—22no—
148C22×C4×C71—24no—
149C22×C4×C71—22no—
150C22×C4×C71—21no—
151C22×C4×C71—21no—
152C22×C4×C71—21no—
153C22×C4×C71—22no—
154C22×C4×C71—21no—
155C22×C4×C71—22no—
156C22×C4×C71—21no—
157C22×C4×C71—21no—
158C22×C4×C71—21no—
159C22×C4×C71—24no—
160C22×C4×C71—42no—
161C22×C4×C71—42no—
162C22×C4×C71—42no—
163C22×C4×C71C2×D414yes—
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71C2×D412yes—
166C22×C4×C71C2×D411yescharacteristic
167C22×C4×C71C2×D411yescharacteristic
168C22×C4×C71C2×D411yescharacteristic
169C22×C4×C71C2×D412yes—
170C22×C4×C71C2×D411yescharacteristic
171C22×C4×C71C2×D412yescomplemented
172C4⋊C4×C71—22no—
173C4⋊C4×C71—22no—
174C4⋊C4×C71—22no—
175C4⋊C4×C71—22no—
176C4⋊C4×C71—22no—
177C4⋊C4×C71—24no—
178C4⋊C4×C71—48no—
179C4⋊C4×C71—44no—
180C4⋊C4×C71—42no—
181C22⋊C4×C71—24no—
182C22⋊C4×C71—24no—
183C22⋊C4×C71—24no—
184C22⋊C4×C71—22no—
185C22⋊C4×C71—21no—
186C22⋊C4×C71—24no—
187C22⋊C4×C71—22no—
188C22⋊C4×C71—24no—
189C22⋊C4×C71—24no—
190C22⋊C4×C71—22no—
191C22⋊C4×C71—24no—
192C22⋊C4×C71—48no—
193C22⋊C4×C71—48no—
194C22⋊C4×C71—42no—
195C22⋊C4×C71C2×D412yescomplemented
196C42×C71—21no—
197C42×C71—21no—
198C42×C71—42no—
199C42×C71—41no—
200C42×C71—42no—
201C42×C71—41no—
202C23×C71—21no—
203C23×C71—21no—
204C23×C71—24no—
205C23×C71—24no—
206C23×C71—24no—
207C23×C71—41no—
208C23×C71—41no—
209C23×C71—41no—
210C23×C71—41no—
211C23×C71—44no—
212C23×C71—44no—
213C23×C71—44no—
214C23×C71—82no—
215C23×C71—82no—
216C23×C71#12112yes—
217C23×C71C22×D411yescharacteristic
218C23×C71C22×D412yes—
219C23×C71C22×D411yescharacteristic
220C23×C71C2×#4111yescharacteristic
221C23×C71H(5,𝔽2)12yes—
222Q8×C71—44no—
223D4×C71—24no—
224D4×C71—24no—
225D4×C71—44no—
226D4×C71—48no—
227D4×C71—44no—
228D4×C71—44no—
229D4×C71—44no—
230D4×C71—44no—
231D4×C71—44no—
232D4×C71—82no—
233D4×C71—82no—
234D4×C71—82no—
235C2×C4×C71—21no—
236C2×C4×C71—21no—
237C2×C4×C71—22no—
238C2×C4×C71—24no—
239C2×C4×C71—24no—
240C2×C4×C71—22no—
241C2×C4×C71—21no—
242C2×C4×C71—21no—
243C2×C4×C71—24no—
244C2×C4×C71—22no—
245C2×C4×C71—22no—
246C2×C4×C71—24no—
247C2×C4×C71—24no—
248C2×C4×C71—24no—
249C2×C4×C71—42no—
250C2×C4×C71—41no—
251C2×C4×C71—48no—
252C2×C4×C71—41no—
253C2×C4×C71—41no—
254C2×C4×C71—41no—
255C2×C4×C71—42no—
256C2×C4×C71—44no—
257C2×C4×C71—41no—
258C2×C4×C71—41no—
259C2×C4×C71—41no—
260C2×C4×C71—42no—
261C2×C4×C71—42no—
262C2×C4×C71—41no—
263C2×C4×C71—41no—
264C2×C4×C71—41no—
265C2×C4×C71—41no—
266C2×C4×C71—44no—
267C2×C4×C71—44no—
268C2×C4×C71—41no—
269C2×C4×C71—44no—
270C2×C4×C71—81no—
271C2×C4×C71#12011yescharacteristic
272C2×C4×C71#12012yes—
273C2×C4×C71#12011yescharacteristic
274C2×C4×C71#12112yes—
275C22×C71—21no—
276C22×C71—22no—
277C22×C71—21no—
278C22×C71—22no—
279C22×C71—22no—
280C22×C71—41no—
281C22×C71—41no—
282C22×C71—42no—
283C22×C71—42no—
284C22×C71—41no—
285C22×C71—41no—
286C22×C71—44no—
287C22×C71—44no—
288C22×C71—44no—
289C22×C71—81no—
290C22×C71—84no—
291C22×C71—84no—
292C22×C71—164no—
293C22×C71C2×#12011yescharacteristic
294C22×C71C2×#12111yescharacteristic
295C22×C71#53511yescharacteristic
296C22×C71D4212yes—
297C22×C71#54612yes—
298C4×C71—22no—
299C4×C71—21no—
300C4×C71—22no—
301C4×C71—42no—
302C4×C71—42no—
303C4×C71—44no—
304C4×C71—41no—
305C4×C71—42no—
306C4×C71—41no—
307C4×C71—84no—
308C4×C71—81no—
309C4×C71#45812yescomplemented
310#24261C7111yesmaximal, characteristic, direct product factor, Sylow
311C2×C71—21no—
312C2×C71—22no—
313C2×C71—21no—
314C2×C71—22no—
315C2×C71—42no—
316C2×C71—84no—
317C2×C71#240811yescharacteristic
318C2×C71C2×#45811yescharacteristic
319C2×C71#302511yescharacteristic
320#3496C2×C7111yescharacteristic
321C2×#458C2×C7112yescomplemented
322#2854C2×C7112yescomplemented
323C4×#123C2×C7111yescharacteristic, complemented
324#2044C2×C7112yescomplemented
325#2022C2×C7112yescomplemented
326#1964C2×C7112yescomplemented
327#1799C2×C7111yescharacteristic, complemented
328#1790C2×C7112yescomplemented
329C71#2426111yescharacteristic, direct product factor, Sylow
330C2×H(5,𝔽2)C22×C7112yes—
331#547—22no—
332#546—24no—
333D42—22no—
334#537—21no—
335#534—21no—
336#534C22×C7112yes—
337#532—21no—
338C2×#123C22×C7111yescharacteristic
339C2×#123C22×C7114yescomplemented
340C2×#121—21no—
341C2×#121C22×C7112yes—
342C2×C4×D4C22×C7111yescharacteristic
343C22×C42C22×C7111yescharacteristic, complemented
344#458—28no—
345C2×#99C22×C7112yes—
346C2×#99C22×C7114yescomplemented
347C2×#99C22×C7114yescomplemented
348#390—22no—
349#389—22no—
350#388—24no—
351#388—21no—
352#387—21no—
353#386C22×C7111yescharacteristic
354#386C22×C7114yescomplemented
355#385C22×C7111yescharacteristic, complemented
356#381C22×C7111yescharacteristic, complemented
357#380C22×C7112yescomplemented
358C4×C4⋊C4—21no—
359C4×C22⋊C4—21no—
360C4×C22⋊C4C22×C7111yescharacteristic, complemented
361#342C22×C7112yescomplemented
362H(5,𝔽2)—24no—
363H(5,𝔽2)—24no—
364H(5,𝔽2)—44no—
365C2×#41—22no—
366C2×#41—24no—
367C22×D4—22no—
368C22×D4—24no—
369C22×D4—21no—
370C22×D4D4×C7112yes—
371C22×D4C23×C7114yes—
372C22×D4C23×C7111yescharacteristic
373C23×C4D4×C7111yescharacteristic
374C23×C4C23×C7111yescharacteristic
375C23×C4C23×C7111yescharacteristic
376C23×C4C23×C7111yescharacteristic
377C4⋊Q8—21no—
378#127—21no—
379#124—22no—
380#123—24no—
381#123—22no—
382#123—24no—
383#123—28no—
384#123—44no—
385#123D4×C7114yescomplemented
386C22⋊Q8—24no—
387#121—22no—
388#121—22no—
389#121—24no—
390#121—22no—
391#121—22no—
392#121—22no—
393#121—22no—
394#121—22no—
395#121—22no—
396#120—24no—
397#120—24no—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4D4×C7111yescharacteristic
408C2×C4⋊C4D4×C7112yes—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—24no—
414C2×C22⋊C4D4×C7111yescharacteristic
415C2×C22⋊C4D4×C7114yes—
416C2×C22⋊C4C23×C7114yes—
417C2×C22⋊C4C23×C7111yescharacteristic
418C2×C22⋊C4C23×C7111yescharacteristic
419C2×C22⋊C4C23×C7111yescharacteristic
420C2×C42—21no—
421C2×C42—21no—
422C2×C42—21no—
423C2×C42—22no—
424C2×C42—22no—
425C2×C42D4×C7111yescharacteristic
426#99—24no—
427#99—28no—
428#99—44no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—22no—
435#95—24no—
436C24—22no—
437C24C2×D4×C7114yes—
438C24C2×D4×C7111yescharacteristic
439C24C24×C7111yescharacteristic
440#41—48no—
441#41—48no—
442#41—44no—
443#41—44no—
444C2×Q8—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—44no—
460C2×D4—48no—
461C2×D4—48no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—82no—
465C2×D4C2×D4×C7112yes—
466C2×D4#41×C7112yes—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—24no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4C2×D4×C7114yes—
484C22×C4C2×D4×C7111yescharacteristic
485C22×C4C2×D4×C7112yes—
486C22×C4C2×D4×C7111yescharacteristic
487C22×C4C2×D4×C7111yescharacteristic
488C22×C4C2×D4×C7111yescharacteristic
489C22×C4C2×D4×C7112yes—
490C22×C4C2×D4×C7111yescharacteristic
491C22×C4C2×D4×C7112yescomplemented
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—24no—
507C22⋊C4—22no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—22no—
511C22⋊C4—24no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—42no—
515C22⋊C4C2×D4×C7112yescomplemented
516C42—21no—
517C42—21no—
518C42—42no—
519C42—41no—
520C42—42no—
521C42—41no—
522C23—21no—
523C23—21no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—82no—
535C23—82no—
536C23#121×C7112yes—
537C23C22×D4×C7111yescharacteristic
538C23C22×D4×C7112yes—
539C23C22×D4×C7111yescharacteristic
540C23C2×#41×C7111yescharacteristic
541C23H(5,𝔽2)×C7112yes—
542Q8—44no—
543D4—24no—
544D4—24no—
545D4—44no—
546D4—48no—
547D4—44no—
548D4—44no—
549D4—44no—
550D4—44no—
551D4—44no—
552D4—82no—
553D4—82no—
554D4—82no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—48no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—81no—
591C2×C4#120×C7111yescharacteristic
592C2×C4#120×C7112yes—
593C2×C4#120×C7111yescharacteristic
594C2×C4#121×C7112yes—
595C22—21no—
596C22—22no—
597C22—21no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—41no—
605C22—41no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—81no—
610C22—84no—
611C22—84no—
612C22—164no—
613C22C2×#120×C7111yescharacteristic
614C22C2×#121×C7111yescharacteristic
615C22#535×C7111yescharacteristic
616C22D42×C7112yes—
617C22#546×C7112yes—
618C4—22no—
619C4—21no—
620C4—22no—
621C4—42no—
622C4—44no—
623C4—42no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—84no—
628C4—81no—
629C4#458×C7112yescomplemented
630C2—21no—
631C2—22no—
632C2—21no—
633C2—22no—
634C2—42no—
635C2—84no—
636C2C71×#240811yescharacteristic
637C2C2×#458×C7111yescharacteristic
638C2C71×#302511yescharacteristic
subgroupG/H
center, socleC22×C71C2×#120
commutator, Frattini, cosocleC24C24×C71
prime radicalC71×#3496C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C23×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#458×C71C2semidirect
3. C71×#2854C2semidirect
4. C4×#123×C71C2semidirect
5. C71×#2044C2semidirect
6. C71×#2022C2semidirect
7. C71×#1964C2semidirect
8. C71×#1799C2semidirect
9. C71×#1790C2semidirect
19. C2×#123×C71C22semidirect
23. C22×C42×C71C22semidirect
26. C2×#99×C71C22semidirect
27. C2×#99×C71C22semidirect
34. #386×C71C22semidirect
35. #385×C71C22semidirect
36. #381×C71C22semidirect
37. #380×C71C22semidirect
40. C4×C22⋊C4×C71C22semidirect
41. #342×C71C22semidirect
65. #123×C71D4semidirect
171. C22×C4×C71C2×D4semidirect
195. C22⋊C4×C71C2×D4semidirect
309. C4×C71#458semidirect
310. #24261C71direct (N × H)
321. C2×#458C2×C71semidirect
322. #2854C2×C71semidirect
323. C4×#123C2×C71semidirect
324. #2044C2×C71semidirect
325. #2022C2×C71semidirect
326. #1964C2×C71semidirect
327. #1799C2×C71semidirect
328. #1790C2×C71semidirect
329. C71#24261direct (N × H)
339. C2×#123C22×C71semidirect
343. C22×C42C22×C71semidirect
346. C2×#99C22×C71semidirect
347. C2×#99C22×C71semidirect
354. #386C22×C71semidirect
355. #385C22×C71semidirect
356. #381C22×C71semidirect
357. #380C22×C71semidirect
360. C4×C22⋊C4C22×C71semidirect
361. #342C22×C71semidirect
385. #123D4×C71semidirect
491. C22×C4C2×D4×C71semidirect
515. C22⋊C4C2×D4×C71semidirect
629. C4#458×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#34962
C2×#458×C712
C71×#28542
C4×#123×C712
C71×#20442
C71×#20222
C71×#19642
C71×#17992
C71×#17902
#2426171