Finite Group Databasebeta

← C71×#24464 #31845336

Subgroups — C71×#24464 #31845336

Order 18176.

1600 conjugacy classes of subgroups.

The subgroup lattice has 1100 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3488C211yesmaximal, characteristic, complemented
2C71×#3055C211yesmaximal, characteristic, complemented
3C2×#453×C71C211yesmaximal, characteristic, complemented
4C71×#3006C211yesmaximal, characteristic, complemented
5C22×#104×C71C211yesmaximal, characteristic, complemented
6C71×#2856C211yesmaximal, characteristic, complemented
7C71×#2039C211yesmaximal, characteristic, complemented
8C71×#2037C211yesmaximal, characteristic, complemented
9C71×#2006C211yesmaximal, characteristic, complemented
10C71×#2002C211yesmaximal, characteristic, complemented
11C71×#1900C211yesmaximal, characteristic, complemented
12C71×#1896C211yesmaximal, characteristic, complemented
13C71×#1883C211yesmaximal, characteristic, complemented
14C71×#1859C211yesmaximal, characteristic, complemented
15C71×#1792C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)×C71C2211yescharacteristic
17C22×#41×C71C2211yescharacteristic
18C2×AGL(1,ℤ/8)×C71C2211yescharacteristic
19C2×AGL(1,ℤ/8)×C71C2211yescharacteristic, complemented
20C2×AGL(1,ℤ/8)×C71C2211yescharacteristic, complemented
21C22×M16×C71C2211yescharacteristic, complemented
22#550×C71—21no—
23#548×C71—21no—
24#546×C71—21no—
25#546×C71—22no—
26D42×C71—22no—
27D42×C71—21no—
28#537×C71—21no—
29#537×C71C2211yescharacteristic
30#535×C71—21no—
31#535×C71C2211yescharacteristic
32C2×#127×C71C2211yescharacteristic, complemented
33#525×C71—21no—
34#525×C71C2211yescharacteristic
35C2×#121×C71—21no—
36C2×#121×C71—22no—
37C2×#121×C71C2211yescharacteristic
38C2×#121×C71C2211yescharacteristic, complemented
39#520×C71C2211yescharacteristic
40C22×C42×C71C2211yescharacteristic, complemented
41#469×C71—21no—
42#468×C71—21no—
43#466×C71—21no—
44#465×C71—21no—
45#453×C71—24no—
46#449×C71—21no—
47#449×C71—21no—
48#448×C71—21no—
49#447×C71—21no—
50#428×C71C2211yescharacteristic, complemented
51C2×#104×C71—24no—
52C2×#104×C71C2211yescharacteristic
53C2×#104×C71C2211yescharacteristic
54C2×#104×C71C2211yescharacteristic, complemented
55C2×#104×C71C2211yescharacteristic, complemented
56C2×#104×C71C2211yescharacteristic, complemented
57C2×#104×C71C2211yescharacteristic, complemented
58#418×C71C2211yescharacteristic, complemented
59#417×C71C2211yescharacteristic
60#417×C71C2211yescharacteristic, complemented
61#417×C71C2211yescharacteristic, complemented
62C2×#100×C71C2211yescharacteristic, complemented
63C2×#100×C71C2211yescharacteristic, complemented
64#407×C71C2211yescharacteristic, complemented
65#390×C71—22no—
66#390×C71C2211yescharacteristic, complemented
67#390×C71C2211yescharacteristic, complemented
68#384×C71C2211yescharacteristic, complemented
69#343×C71C2211yescharacteristic, complemented
70#343×C71C2211yescharacteristic, complemented
71#339×C71C2211yescharacteristic, complemented
72#339×C71C2211yescharacteristic, complemented
73H(5,𝔽2)×C71—22no—
74H(5,𝔽2)×C71—22no—
75H(5,𝔽2)×C71—41no—
76H(5,𝔽2)×C71—41no—
77C2×#41×C71—21no—
78C2×#41×C71—21no—
79C2×#41×C71—21no—
80C2×#41×C71—21no—
81C2×#41×C71—22no—
82C2×#41×C71—22no—
83C2×#41×C71—41no—
84C2×#41×C71—41no—
85C2×#41×C71D411yescharacteristic, complemented
86C2×#41×C71D411yescharacteristic, complemented
87C2×#41×C71C2311yescharacteristic
88C2×#41×C71C2311yescharacteristic
89C2×#41×C71C2311yescharacteristic
90C22×Q8×C71D411yescharacteristic, complemented
91C22×D4×C71—21no—
92C22×D4×C71—22no—
93C22×D4×C71—22no—
94C22×D4×C71—21no—
95C22×D4×C71—21no—
96C22×D4×C71—21no—
97C22×D4×C71—22no—
98C22×D4×C71D412yes—
99C22×D4×C71D411yescharacteristic, complemented
100C22×D4×C71C2311yescharacteristic
101C22×D4×C71C2311yescharacteristic
102C23×C4×C71—21no—
103C23×C4×C71—21no—
104C23×C4×C71C2311yescharacteristic
105AGL(1,ℤ/8)×C71—22no—
106AGL(1,ℤ/8)×C71—22no—
107AGL(1,ℤ/8)×C71—22no—
108AGL(1,ℤ/8)×C71—41no—
109AGL(1,ℤ/8)×C71—41no—
110AGL(1,ℤ/8)×C71—41no—
111C2×SD16×C71—21no—
112C2×SD16×C71—21no—
113C2×SD16×C71—21no—
114C2×D8×C71—21no—
115C2×D8×C71—21no—
116C2×D8×C71—21no—
117C2×M16×C71—21no—
118C2×M16×C71—22no—
119C2×M16×C71—41no—
120C2×M16×C71C2311yescharacteristic
121C2×M16×C71C2311yescharacteristic
122C2×M16×C71C2311yescharacteristic
123C2×M16×C71C2311yescharacteristic
124C22×C8×C71—21no—
125#127×C71—22no—
126#127×C71—21no—
127#127×C71—22no—
128#127×C71—41no—
129#124×C71—22no—
130#124×C71—21no—
131#123×C71—22no—
132#123×C71—21no—
133C22⋊Q8×C71—22no—
134C22⋊Q8×C71—21no—
135C22⋊Q8×C71—21no—
136C22⋊Q8×C71D411yescharacteristic, complemented
137C22⋊Q8×C71D411yescharacteristic, complemented
138#121×C71—22no—
139#121×C71—22no—
140#121×C71—22no—
141#121×C71—22no—
142#121×C71—22no—
143#121×C71—21no—
144#121×C71—21no—
145#121×C71—21no—
146#121×C71—21no—
147#121×C71—21no—
148#121×C71—21no—
149#121×C71—22no—
150#121×C71—21no—
151#121×C71—21no—
152#121×C71—21no—
153#121×C71—21no—
154#121×C71—21no—
155#121×C71—21no—
156#121×C71—48no—
157#121×C71D411yescharacteristic, complemented
158#121×C71D411yescharacteristic, complemented
159#120×C71—22no—
160#120×C71—22no—
161#120×C71—22no—
162#120×C71—21no—
163#120×C71—22no—
164C4×Q8×C71—21no—
165C4×D4×C71—22no—
166C4×D4×C71—22no—
167C4×D4×C71—21no—
168C4×D4×C71—22no—
169C4×D4×C71—21no—
170C4×D4×C71—21no—
171#117×C71—21no—
172#117×C71C2311yescharacteristic
173C2×C4⋊C4×C71—21no—
174C2×C4⋊C4×C71—22no—
175C2×C4⋊C4×C71C2311yescharacteristic
176C2×C4⋊C4×C71C2311yescharacteristic
177C2×C22⋊C4×C71—22no—
178C2×C22⋊C4×C71—22no—
179C2×C22⋊C4×C71—22no—
180C2×C22⋊C4×C71—21no—
181C2×C22⋊C4×C71D412yes—
182C2×C42×C71—22no—
183C2×C42×C71—21no—
184C2×C42×C71—41no—
185C2×C42×C71C2311yescharacteristic
186C2×C42×C71C2311yescharacteristic
187#107×C71—21no—
188#106×C71—21no—
189#104×C71—22no—
190#104×C71—22no—
191#104×C71—22no—
192#104×C71—22no—
193#104×C71—41no—
194#104×C71D414yescomplemented
195#103×C71—21no—
196#103×C71—21no—
197#103×C71—21no—
198#102×C71—21no—
199#102×C71—21no—
200#102×C71—21no—
201#102×C71—21no—
202#102×C71—21no—
203#100×C71—22no—
204#100×C71—41no—
205C22⋊C8×C71—21no—
206C22⋊C8×C71—21no—
207C24×C71—22no—
208C24×C71—21no—
209C24×C71—21no—
210C24×C71C2×D411yescharacteristic
211C24×C71C2×D412yes—
212#41×C71—21no—
213#41×C71—21no—
214#41×C71—22no—
215#41×C71—22no—
216#41×C71—21no—
217#41×C71—21no—
218#41×C71—21no—
219#41×C71—21no—
220#41×C71—21no—
221#41×C71—21no—
222#41×C71—21no—
223#41×C71—21no—
224#41×C71—42no—
225#41×C71—42no—
226#41×C71—41no—
227#41×C71—41no—
228#41×C71—42no—
229#41×C71—42no—
230#41×C71—42no—
231#41×C71—42no—
232#41×C71—42no—
233#41×C71—42no—
234#41×C71C2×D412yes—
235#41×C71#4112yes—
236C2×Q8×C71—21no—
237C2×Q8×C71—22no—
238C2×Q8×C71—41no—
239C2×Q8×C71C2×D411yescharacteristic
240C2×Q8×C71C2×D411yescharacteristic
241C2×Q8×C71C2×D411yescharacteristic
242C2×D4×C71—21no—
243C2×D4×C71—22no—
244C2×D4×C71—22no—
245C2×D4×C71—22no—
246C2×D4×C71—22no—
247C2×D4×C71—21no—
248C2×D4×C71—21no—
249C2×D4×C71—21no—
250C2×D4×C71—21no—
251C2×D4×C71—21no—
252C2×D4×C71—22no—
253C2×D4×C71—22no—
254C2×D4×C71—22no—
255C2×D4×C71—21no—
256C2×D4×C71—22no—
257C2×D4×C71—21no—
258C2×D4×C71—21no—
259C2×D4×C71—21no—
260C2×D4×C71—21no—
261C2×D4×C71—21no—
262C2×D4×C71—21no—
263C2×D4×C71—21no—
264C2×D4×C71—21no—
265C2×D4×C71—21no—
266C2×D4×C71—21no—
267C2×D4×C71—21no—
268C2×D4×C71—21no—
269C2×D4×C71—21no—
270C2×D4×C71—41no—
271C2×D4×C71—42no—
272C2×D4×C71—42no—
273C2×D4×C71—44no—
274C2×D4×C71—44no—
275C2×D4×C71—44no—
276C2×D4×C71—42no—
277C2×D4×C71—42no—
278C2×D4×C71—44no—
279C2×D4×C71—44no—
280C2×D4×C71—42no—
281C2×D4×C71—42no—
282C2×D4×C71—44no—
283C2×D4×C71—44no—
284C2×D4×C71—41no—
285C2×D4×C71—41no—
286C2×D4×C71—84no—
287C2×D4×C71C2×D411yescharacteristic
288C2×D4×C71C2×D411yescharacteristic
289C2×D4×C71C2×D411yescharacteristic
290C22×C4×C71—21no—
291C22×C4×C71—21no—
292C22×C4×C71—21no—
293C22×C4×C71—21no—
294C22×C4×C71—21no—
295C22×C4×C71—21no—
296C22×C4×C71—22no—
297C22×C4×C71—24no—
298C22×C4×C71—21no—
299C22×C4×C71—21no—
300C22×C4×C71—22no—
301C22×C4×C71—21no—
302C22×C4×C71—21no—
303C22×C4×C71—21no—
304C22×C4×C71—21no—
305C22×C4×C71—21no—
306C22×C4×C71—42no—
307C22×C4×C71—42no—
308C22×C4×C71—42no—
309C22×C4×C71—42no—
310C22×C4×C71C2×D411yescharacteristic
311C22×C4×C71C2×D411yescharacteristic
312C22×C4×C71C2×D411yescharacteristic
313C22×C4×C71C2×D411yescharacteristic
314C22×C4×C71C2×D411yescharacteristic
315C22×C4×C71C2×D412yes—
316C22×C4×C71C2411yescharacteristic
317SD16×C71—42no—
318SD16×C71—42no—
319SD16×C71—42no—
320D8×C71—42no—
321D8×C71—42no—
322D8×C71—42no—
323M16×C71—21no—
324M16×C71—21no—
325M16×C71—21no—
326M16×C71—21no—
327M16×C71—21no—
328M16×C71—21no—
329M16×C71—21no—
330M16×C71—42no—
331M16×C71C2×D412yescomplemented
332C2×C8×C71—21no—
333C2×C8×C71—21no—
334C2×C8×C71—21no—
335C2×C8×C71—21no—
336C2×C8×C71—42no—
337C4⋊C4×C71—22no—
338C4⋊C4×C71—22no—
339C4⋊C4×C71—21no—
340C4⋊C4×C71—21no—
341C4⋊C4×C71—21no—
342C4⋊C4×C71—21no—
343C4⋊C4×C71—21no—
344C4⋊C4×C71—21no—
345C4⋊C4×C71—44no—
346C4⋊C4×C71C2×D411yescharacteristic
347C4⋊C4×C71C2×D411yescharacteristic
348C22⋊C4×C71—22no—
349C22⋊C4×C71—22no—
350C22⋊C4×C71—22no—
351C22⋊C4×C71—22no—
352C22⋊C4×C71—22no—
353C22⋊C4×C71—21no—
354C22⋊C4×C71—21no—
355C22⋊C4×C71—21no—
356C22⋊C4×C71—21no—
357C22⋊C4×C71—22no—
358C22⋊C4×C71—22no—
359C22⋊C4×C71—21no—
360C22⋊C4×C71—44no—
361C22⋊C4×C71—44no—
362C22⋊C4×C71—41no—
363C22⋊C4×C71—41no—
364C42×C71—22no—
365C42×C71—21no—
366C42×C71—21no—
367C42×C71—21no—
368C42×C71—21no—
369C42×C71—42no—
370C42×C71C2×D412yescomplemented
371C23×C71—21no—
372C23×C71—21no—
373C23×C71—21no—
374C23×C71—22no—
375C23×C71—22no—
376C23×C71—21no—
377C23×C71—21no—
378C23×C71—22no—
379C23×C71—22no—
380C23×C71—41no—
381C23×C71—42no—
382C23×C71—44no—
383C23×C71—42no—
384C23×C71—42no—
385C23×C71—42no—
386C23×C71—42no—
387C23×C71—42no—
388C23×C71—41no—
389C23×C71—41no—
390C23×C71—41no—
391C23×C71—81no—
392C23×C71—82no—
393C23×C71—82no—
394C23×C71#12011yescharacteristic
395C23×C71#12011yescharacteristic
396C23×C71C22×D411yescharacteristic
397Q8×C71—21no—
398Q8×C71—21no—
399Q8×C71—21no—
400Q8×C71—21no—
401Q8×C71—21no—
402Q8×C71—42no—
403Q8×C71#12112yescomplemented
404D4×C71—22no—
405D4×C71—22no—
406D4×C71—21no—
407D4×C71—21no—
408D4×C71—21no—
409D4×C71—21no—
410D4×C71—21no—
411D4×C71—42no—
412D4×C71—42no—
413D4×C71—44no—
414D4×C71—42no—
415D4×C71—44no—
416D4×C71—41no—
417D4×C71—41no—
418D4×C71—41no—
419D4×C71—41no—
420D4×C71—44no—
421D4×C71—42no—
422D4×C71—41no—
423D4×C71—41no—
424D4×C71—41no—
425D4×C71—41no—
426D4×C71—41no—
427D4×C71—41no—
428D4×C71—41no—
429D4×C71—41no—
430D4×C71—88no—
431D4×C71—81no—
432D4×C71—82no—
433D4×C71—82no—
434D4×C71—82no—
435D4×C71—82no—
436D4×C71—82no—
437D4×C71—82no—
438D4×C71#12112yescomplemented
439C2×C4×C71—22no—
440C2×C4×C71—22no—
441C2×C4×C71—21no—
442C2×C4×C71—22no—
443C2×C4×C71—21no—
444C2×C4×C71—21no—
445C2×C4×C71—21no—
446C2×C4×C71—21no—
447C2×C4×C71—21no—
448C2×C4×C71—21no—
449C2×C4×C71—21no—
450C2×C4×C71—21no—
451C2×C4×C71—22no—
452C2×C4×C71—21no—
453C2×C4×C71—21no—
454C2×C4×C71—22no—
455C2×C4×C71—22no—
456C2×C4×C71—21no—
457C2×C4×C71—22no—
458C2×C4×C71—21no—
459C2×C4×C71—21no—
460C2×C4×C71—21no—
461C2×C4×C71—42no—
462C2×C4×C71—42no—
463C2×C4×C71—42no—
464C2×C4×C71—48no—
465C2×C4×C71—44no—
466C2×C4×C71—41no—
467C2×C4×C71—41no—
468C2×C4×C71—41no—
469C2×C4×C71—41no—
470C2×C4×C71—41no—
471C2×C4×C71—41no—
472C2×C4×C71—42no—
473C2×C4×C71—42no—
474C2×C4×C71—41no—
475C2×C4×C71—82no—
476C2×C4×C71#12011yescharacteristic
477C2×C4×C71#12011yescharacteristic
478C2×C4×C71C22×D411yescharacteristic
479C2×C4×C71C22×D411yescharacteristic
480C2×C4×C71C22×D411yescharacteristic
481C2×C4×C71C2×#4111yescharacteristic
482C2×C4×C71H(5,𝔽2)11yescharacteristic
483C2×C4×C71H(5,𝔽2)11yescharacteristic
484C8×C71—22no—
485C8×C71—41no—
486C8×C71—41no—
487C8×C71—41no—
488C22×C71—21no—
489C22×C71—21no—
490C22×C71—21no—
491C22×C71—21no—
492C22×C71—21no—
493C22×C71—22no—
494C22×C71—21no—
495C22×C71—21no—
496C22×C71—42no—
497C22×C71—42no—
498C22×C71—41no—
499C22×C71—41no—
500C22×C71—41no—
501C22×C71—41no—
502C22×C71—42no—
503C22×C71—42no—
504C22×C71—41no—
505C22×C71—41no—
506C22×C71—41no—
507C22×C71—42no—
508C22×C71—41no—
509C22×C71—84no—
510C22×C71—82no—
511C22×C71—82no—
512C22×C71—82no—
513C22×C71—82no—
514C22×C71—81no—
515C22×C71—81no—
516C22×C71—81no—
517C22×C71—164no—
518C22×C71#45312yescomplemented
519C22×C71C2×#12011yescharacteristic
520C22×C71D4211yescharacteristic
521C22×C71#54611yescharacteristic
522C4×C71—24no—
523C4×C71—21no—
524C4×C71—21no—
525C4×C71—22no—
526C4×C71—42no—
527C4×C71—41no—
528C4×C71—41no—
529C4×C71—42no—
530C4×C71—41no—
531C4×C71—41no—
532C4×C71—81no—
533C4×C71C2×#12111yescharacteristic
534C4×C71#53411yescharacteristic
535C4×C71D4211yescharacteristic
536C4×C71#54611yescharacteristic
537#24464C7111yesmaximal, characteristic, direct product factor, Sylow
538C2×C71—21no—
539C2×C71—21no—
540C2×C71—22no—
541C2×C71—41no—
542C2×C71—42no—
543C2×C71—81no—
544C2×C71—82no—
545C2×C71—81no—
546C2×C71—161no—
547C2×C71#241011yescharacteristic
548C2×C71C2×#45311yescharacteristic
549C2×C71#301811yescharacteristic
550#3488C2×C7111yescharacteristic, complemented
551#3055C2×C7111yescharacteristic, complemented
552C2×#453C2×C7111yescharacteristic, complemented
553#3006C2×C7111yescharacteristic, complemented
554C22×#104C2×C7111yescharacteristic, complemented
555#2856C2×C7111yescharacteristic, complemented
556#2039C2×C7111yescharacteristic, complemented
557#2037C2×C7111yescharacteristic, complemented
558#2006C2×C7111yescharacteristic, complemented
559#2002C2×C7111yescharacteristic, complemented
560#1900C2×C7111yescharacteristic, complemented
561#1896C2×C7111yescharacteristic, complemented
562#1883C2×C7111yescharacteristic, complemented
563#1859C2×C7111yescharacteristic, complemented
564#1792C2×C7111yescharacteristic, complemented
565C71#2446411yescharacteristic, direct product factor, Sylow
566C2×H(5,𝔽2)C22×C7111yescharacteristic
567C22×#41C22×C7111yescharacteristic
568C2×AGL(1,ℤ/8)C22×C7111yescharacteristic
569C2×AGL(1,ℤ/8)C22×C7111yescharacteristic, complemented
570C2×AGL(1,ℤ/8)C22×C7111yescharacteristic, complemented
571C22×M16C22×C7111yescharacteristic, complemented
572#550—21no—
573#548—21no—
574#546—21no—
575#546—22no—
576D42—22no—
577D42—21no—
578#537—21no—
579#537C22×C7111yescharacteristic
580#535—21no—
581#535C22×C7111yescharacteristic
582C2×#127C22×C7111yescharacteristic, complemented
583#525—21no—
584#525C22×C7111yescharacteristic
585C2×#121—21no—
586C2×#121—22no—
587C2×#121C22×C7111yescharacteristic
588C2×#121C22×C7111yescharacteristic, complemented
589#520C22×C7111yescharacteristic
590C22×C42C22×C7111yescharacteristic, complemented
591#469—21no—
592#468—21no—
593#466—21no—
594#465—21no—
595#453—24no—
596#449—21no—
597#449—21no—
598#448—21no—
599#447—21no—
600#428C22×C7111yescharacteristic, complemented
601C2×#104—24no—
602C2×#104C22×C7111yescharacteristic
603C2×#104C22×C7111yescharacteristic
604C2×#104C22×C7111yescharacteristic, complemented
605C2×#104C22×C7111yescharacteristic, complemented
606C2×#104C22×C7111yescharacteristic, complemented
607C2×#104C22×C7111yescharacteristic, complemented
608#418C22×C7111yescharacteristic, complemented
609#417C22×C7111yescharacteristic
610#417C22×C7111yescharacteristic, complemented
611#417C22×C7111yescharacteristic, complemented
612C2×#100C22×C7111yescharacteristic, complemented
613C2×#100C22×C7111yescharacteristic, complemented
614#407C22×C7111yescharacteristic, complemented
615#390—22no—
616#390C22×C7111yescharacteristic, complemented
617#390C22×C7111yescharacteristic, complemented
618#384C22×C7111yescharacteristic, complemented
619#343C22×C7111yescharacteristic, complemented
620#343C22×C7111yescharacteristic, complemented
621#339C22×C7111yescharacteristic, complemented
622#339C22×C7111yescharacteristic, complemented
623H(5,𝔽2)—22no—
624H(5,𝔽2)—22no—
625H(5,𝔽2)—41no—
626H(5,𝔽2)—41no—
627C2×#41—21no—
628C2×#41—21no—
629C2×#41—21no—
630C2×#41—21no—
631C2×#41—22no—
632C2×#41—22no—
633C2×#41—41no—
634C2×#41—41no—
635C2×#41D4×C7111yescharacteristic, complemented
636C2×#41D4×C7111yescharacteristic, complemented
637C2×#41C23×C7111yescharacteristic
638C2×#41C23×C7111yescharacteristic
639C2×#41C23×C7111yescharacteristic
640C22×Q8D4×C7111yescharacteristic, complemented
641C22×D4—21no—
642C22×D4—22no—
643C22×D4—22no—
644C22×D4—21no—
645C22×D4—21no—
646C22×D4—21no—
647C22×D4—22no—
648C22×D4D4×C7112yes—
649C22×D4D4×C7111yescharacteristic, complemented
650C22×D4C23×C7111yescharacteristic
651C22×D4C23×C7111yescharacteristic
652C23×C4—21no—
653C23×C4—21no—
654C23×C4C23×C7111yescharacteristic
655AGL(1,ℤ/8)—22no—
656AGL(1,ℤ/8)—22no—
657AGL(1,ℤ/8)—22no—
658AGL(1,ℤ/8)—41no—
659AGL(1,ℤ/8)—41no—
660AGL(1,ℤ/8)—41no—
661C2×SD16—21no—
662C2×SD16—21no—
663C2×SD16—21no—
664C2×D8—21no—
665C2×D8—21no—
666C2×D8—21no—
667C2×M16—21no—
668C2×M16—22no—
669C2×M16—41no—
670C2×M16C23×C7111yescharacteristic
671C2×M16C23×C7111yescharacteristic
672C2×M16C23×C7111yescharacteristic
673C2×M16C23×C7111yescharacteristic
674C22×C8—21no—
675#127—22no—
676#127—21no—
677#127—22no—
678#127—41no—
679#124—22no—
680#124—21no—
681#123—22no—
682#123—21no—
683C22⋊Q8—22no—
684C22⋊Q8—21no—
685C22⋊Q8—21no—
686C22⋊Q8D4×C7111yescharacteristic, complemented
687C22⋊Q8D4×C7111yescharacteristic, complemented
688#121—22no—
689#121—22no—
690#121—22no—
691#121—22no—
692#121—22no—
693#121—21no—
694#121—21no—
695#121—21no—
696#121—21no—
697#121—21no—
698#121—21no—
699#121—22no—
700#121—21no—
701#121—21no—
702#121—21no—
703#121—21no—
704#121—21no—
705#121—21no—
706#121—48no—
707#121D4×C7111yescharacteristic, complemented
708#121D4×C7111yescharacteristic, complemented
709#120—22no—
710#120—22no—
711#120—22no—
712#120—21no—
713#120—22no—
714C4×Q8—21no—
715C4×D4—22no—
716C4×D4—22no—
717C4×D4—21no—
718C4×D4—22no—
719C4×D4—21no—
720C4×D4—21no—
721#117—21no—
722#117C23×C7111yescharacteristic
723C2×C4⋊C4—21no—
724C2×C4⋊C4—22no—
725C2×C4⋊C4C23×C7111yescharacteristic
726C2×C4⋊C4C23×C7111yescharacteristic
727C2×C22⋊C4—22no—
728C2×C22⋊C4—22no—
729C2×C22⋊C4—22no—
730C2×C22⋊C4—21no—
731C2×C22⋊C4D4×C7112yes—
732C2×C42—22no—
733C2×C42—21no—
734C2×C42—41no—
735C2×C42C23×C7111yescharacteristic
736C2×C42C23×C7111yescharacteristic
737#107—21no—
738#106—21no—
739#104—22no—
740#104—22no—
741#104—22no—
742#104—22no—
743#104—41no—
744#104D4×C7114yescomplemented
745#103—21no—
746#103—21no—
747#103—21no—
748#102—21no—
749#102—21no—
750#102—21no—
751#102—21no—
752#102—21no—
753#100—22no—
754#100—41no—
755C22⋊C8—21no—
756C22⋊C8—21no—
757C24—22no—
758C24—21no—
759C24—21no—
760C24C2×D4×C7111yescharacteristic
761C24C2×D4×C7112yes—
762#41—21no—
763#41—21no—
764#41—22no—
765#41—22no—
766#41—21no—
767#41—21no—
768#41—21no—
769#41—21no—
770#41—21no—
771#41—21no—
772#41—21no—
773#41—21no—
774#41—42no—
775#41—42no—
776#41—41no—
777#41—41no—
778#41—42no—
779#41—42no—
780#41—42no—
781#41—42no—
782#41—42no—
783#41—42no—
784#41C2×D4×C7112yes—
785#41#41×C7112yes—
786C2×Q8—21no—
787C2×Q8—22no—
788C2×Q8—41no—
789C2×Q8C2×D4×C7111yescharacteristic
790C2×Q8C2×D4×C7111yescharacteristic
791C2×Q8C2×D4×C7111yescharacteristic
792C2×D4—21no—
793C2×D4—22no—
794C2×D4—22no—
795C2×D4—22no—
796C2×D4—22no—
797C2×D4—21no—
798C2×D4—21no—
799C2×D4—21no—
800C2×D4—21no—
801C2×D4—21no—
802C2×D4—22no—
803C2×D4—22no—
804C2×D4—22no—
805C2×D4—21no—
806C2×D4—22no—
807C2×D4—21no—
808C2×D4—21no—
809C2×D4—21no—
810C2×D4—21no—
811C2×D4—21no—
812C2×D4—21no—
813C2×D4—21no—
814C2×D4—21no—
815C2×D4—21no—
816C2×D4—21no—
817C2×D4—21no—
818C2×D4—21no—
819C2×D4—21no—
820C2×D4—41no—
821C2×D4—42no—
822C2×D4—42no—
823C2×D4—44no—
824C2×D4—44no—
825C2×D4—44no—
826C2×D4—42no—
827C2×D4—42no—
828C2×D4—44no—
829C2×D4—44no—
830C2×D4—42no—
831C2×D4—42no—
832C2×D4—44no—
833C2×D4—44no—
834C2×D4—41no—
835C2×D4—41no—
836C2×D4—84no—
837C2×D4C2×D4×C7111yescharacteristic
838C2×D4C2×D4×C7111yescharacteristic
839C2×D4C2×D4×C7111yescharacteristic
840C22×C4—21no—
841C22×C4—21no—
842C22×C4—21no—
843C22×C4—21no—
844C22×C4—21no—
845C22×C4—21no—
846C22×C4—22no—
847C22×C4—24no—
848C22×C4—21no—
849C22×C4—21no—
850C22×C4—22no—
851C22×C4—21no—
852C22×C4—21no—
853C22×C4—21no—
854C22×C4—21no—
855C22×C4—21no—
856C22×C4—42no—
857C22×C4—42no—
858C22×C4—42no—
859C22×C4—42no—
860C22×C4C2×D4×C7111yescharacteristic
861C22×C4C2×D4×C7111yescharacteristic
862C22×C4C2×D4×C7111yescharacteristic
863C22×C4C2×D4×C7111yescharacteristic
864C22×C4C2×D4×C7111yescharacteristic
865C22×C4C2×D4×C7112yes—
866C22×C4C24×C7111yescharacteristic
867SD16—42no—
868SD16—42no—
869SD16—42no—
870D8—42no—
871D8—42no—
872D8—42no—
873M16—21no—
874M16—21no—
875M16—21no—
876M16—21no—
877M16—21no—
878M16—21no—
879M16—21no—
880M16—42no—
881M16C2×D4×C7112yescomplemented
882C2×C8—21no—
883C2×C8—21no—
884C2×C8—21no—
885C2×C8—21no—
886C2×C8—42no—
887C4⋊C4—22no—
888C4⋊C4—22no—
889C4⋊C4—21no—
890C4⋊C4—21no—
891C4⋊C4—21no—
892C4⋊C4—21no—
893C4⋊C4—21no—
894C4⋊C4—21no—
895C4⋊C4—44no—
896C4⋊C4C2×D4×C7111yescharacteristic
897C4⋊C4C2×D4×C7111yescharacteristic
898C22⋊C4—22no—
899C22⋊C4—22no—
900C22⋊C4—22no—
901C22⋊C4—22no—
902C22⋊C4—22no—
903C22⋊C4—21no—
904C22⋊C4—21no—
905C22⋊C4—21no—
906C22⋊C4—21no—
907C22⋊C4—22no—
908C22⋊C4—22no—
909C22⋊C4—21no—
910C22⋊C4—44no—
911C22⋊C4—44no—
912C22⋊C4—41no—
913C22⋊C4—41no—
914C42—22no—
915C42—21no—
916C42—21no—
917C42—21no—
918C42—21no—
919C42—42no—
920C42C2×D4×C7112yescomplemented
921C23—22no—
922C23—21no—
923C23—21no—
924C23—21no—
925C23—22no—
926C23—21no—
927C23—21no—
928C23—22no—
929C23—22no—
930C23—41no—
931C23—42no—
932C23—44no—
933C23—42no—
934C23—42no—
935C23—42no—
936C23—42no—
937C23—42no—
938C23—41no—
939C23—41no—
940C23—41no—
941C23—81no—
942C23—82no—
943C23—82no—
944C23#120×C7111yescharacteristic
945C23#120×C7111yescharacteristic
946C23C22×D4×C7111yescharacteristic
947Q8—21no—
948Q8—21no—
949Q8—21no—
950Q8—21no—
951Q8—21no—
952Q8—42no—
953Q8#121×C7112yescomplemented
954D4—22no—
955D4—22no—
956D4—21no—
957D4—21no—
958D4—21no—
959D4—21no—
960D4—21no—
961D4—42no—
962D4—42no—
963D4—44no—
964D4—42no—
965D4—44no—
966D4—44no—
967D4—42no—
968D4—41no—
969D4—41no—
970D4—41no—
971D4—41no—
972D4—41no—
973D4—41no—
974D4—41no—
975D4—41no—
976D4—41no—
977D4—41no—
978D4—41no—
979D4—41no—
980D4—88no—
981D4—82no—
982D4—81no—
983D4—82no—
984D4—82no—
985D4—82no—
986D4—82no—
987D4—82no—
988D4#121×C7112yescomplemented
989C2×C4—22no—
990C2×C4—22no—
991C2×C4—21no—
992C2×C4—22no—
993C2×C4—21no—
994C2×C4—21no—
995C2×C4—21no—
996C2×C4—21no—
997C2×C4—21no—
998C2×C4—21no—
999C2×C4—21no—
1000C2×C4—21no—
1001C2×C4—22no—
1002C2×C4—21no—
1003C2×C4—21no—
1004C2×C4—22no—
1005C2×C4—22no—
1006C2×C4—21no—
1007C2×C4—22no—
1008C2×C4—21no—
1009C2×C4—21no—
1010C2×C4—21no—
1011C2×C4—42no—
1012C2×C4—42no—
1013C2×C4—42no—
1014C2×C4—48no—
1015C2×C4—44no—
1016C2×C4—41no—
1017C2×C4—41no—
1018C2×C4—41no—
1019C2×C4—41no—
1020C2×C4—41no—
1021C2×C4—41no—
1022C2×C4—42no—
1023C2×C4—42no—
1024C2×C4—41no—
1025C2×C4—82no—
1026C2×C4#120×C7111yescharacteristic
1027C2×C4#120×C7111yescharacteristic
1028C2×C4C22×D4×C7111yescharacteristic
1029C2×C4C22×D4×C7111yescharacteristic
1030C2×C4C22×D4×C7111yescharacteristic
1031C2×C4C2×#41×C7111yescharacteristic
1032C2×C4H(5,𝔽2)×C7111yescharacteristic
1033C2×C4H(5,𝔽2)×C7111yescharacteristic
1034C8—22no—
1035C8—41no—
1036C8—41no—
1037C8—41no—
1038C22—21no—
1039C22—21no—
1040C22—21no—
1041C22—21no—
1042C22—21no—
1043C22—22no—
1044C22—21no—
1045C22—21no—
1046C22—42no—
1047C22—42no—
1048C22—41no—
1049C22—41no—
1050C22—41no—
1051C22—41no—
1052C22—42no—
1053C22—42no—
1054C22—41no—
1055C22—41no—
1056C22—41no—
1057C22—42no—
1058C22—41no—
1059C22—84no—
1060C22—82no—
1061C22—82no—
1062C22—82no—
1063C22—82no—
1064C22—81no—
1065C22—81no—
1066C22—81no—
1067C22—164no—
1068C22#453×C7112yescomplemented
1069C22C2×#120×C7111yescharacteristic
1070C22D42×C7111yescharacteristic
1071C22#546×C7111yescharacteristic
1072C4—24no—
1073C4—21no—
1074C4—21no—
1075C4—22no—
1076C4—42no—
1077C4—41no—
1078C4—41no—
1079C4—42no—
1080C4—41no—
1081C4—41no—
1082C4—81no—
1083C4C2×#121×C7111yescharacteristic
1084C4#534×C7111yescharacteristic
1085C4D42×C7111yescharacteristic
1086C4#546×C7111yescharacteristic
1087C2—21no—
1088C2—21no—
1089C2—22no—
1090C2—41no—
1091C2—42no—
1092C2—81no—
1093C2—82no—
1094C2—81no—
1095C2—161no—
1096C2C71×#241011yescharacteristic
1097C2C2×#453×C7111yescharacteristic
1098C2C71×#301811yescharacteristic
subgroupG/H
center, socleC22×C71C2×#120
commutator, Frattini, cosocleC22×C4C24×C71
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
1. C71×#3488C2semidirect
2. C71×#3055C2semidirect
3. C2×#453×C71C2semidirect
4. C71×#3006C2semidirect
5. C22×#104×C71C2semidirect
6. C71×#2856C2semidirect
7. C71×#2039C2semidirect
8. C71×#2037C2semidirect
9. C71×#2006C2semidirect
10. C71×#2002C2semidirect
11. C71×#1900C2semidirect
12. C71×#1896C2semidirect
13. C71×#1883C2semidirect
14. C71×#1859C2semidirect
15. C71×#1792C2semidirect
19. C2×AGL(1,ℤ/8)×C71C22semidirect
20. C2×AGL(1,ℤ/8)×C71C22semidirect
21. C22×M16×C71C22semidirect
32. C2×#127×C71C22semidirect
38. C2×#121×C71C22semidirect
40. C22×C42×C71C22semidirect
50. #428×C71C22semidirect
54. C2×#104×C71C22semidirect
55. C2×#104×C71C22semidirect
56. C2×#104×C71C22semidirect
57. C2×#104×C71C22semidirect
58. #418×C71C22semidirect
60. #417×C71C22semidirect
61. #417×C71C22semidirect
62. C2×#100×C71C22semidirect
63. C2×#100×C71C22semidirect
64. #407×C71C22semidirect
66. #390×C71C22semidirect
67. #390×C71C22semidirect
68. #384×C71C22semidirect
69. #343×C71C22semidirect
70. #343×C71C22semidirect
71. #339×C71C22semidirect
72. #339×C71C22semidirect
85. C2×#41×C71D4semidirect
86. C2×#41×C71D4semidirect
90. C22×Q8×C71D4semidirect
99. C22×D4×C71D4semidirect
136. C22⋊Q8×C71D4semidirect
137. C22⋊Q8×C71D4semidirect
157. #121×C71D4semidirect
158. #121×C71D4semidirect
194. #104×C71D4semidirect
331. M16×C71C2×D4semidirect
370. C42×C71C2×D4semidirect
403. Q8×C71#121semidirect
438. D4×C71#121semidirect
518. C22×C71#453semidirect
537. #24464C71direct (N × H)
550. #3488C2×C71semidirect
551. #3055C2×C71semidirect
552. C2×#453C2×C71semidirect
553. #3006C2×C71semidirect
554. C22×#104C2×C71semidirect
555. #2856C2×C71semidirect
556. #2039C2×C71semidirect
557. #2037C2×C71semidirect
558. #2006C2×C71semidirect
559. #2002C2×C71semidirect
560. #1900C2×C71semidirect
561. #1896C2×C71semidirect
562. #1883C2×C71semidirect
563. #1859C2×C71semidirect
564. #1792C2×C71semidirect
565. C71#24464direct (N × H)
569. C2×AGL(1,ℤ/8)C22×C71semidirect
570. C2×AGL(1,ℤ/8)C22×C71semidirect
571. C22×M16C22×C71semidirect
582. C2×#127C22×C71semidirect
588. C2×#121C22×C71semidirect
590. C22×C42C22×C71semidirect
600. #428C22×C71semidirect
604. C2×#104C22×C71semidirect
605. C2×#104C22×C71semidirect
606. C2×#104C22×C71semidirect
607. C2×#104C22×C71semidirect
608. #418C22×C71semidirect
610. #417C22×C71semidirect
611. #417C22×C71semidirect
612. C2×#100C22×C71semidirect
613. C2×#100C22×C71semidirect
614. #407C22×C71semidirect
616. #390C22×C71semidirect
617. #390C22×C71semidirect
618. #384C22×C71semidirect
619. #343C22×C71semidirect
620. #343C22×C71semidirect
621. #339C22×C71semidirect
622. #339C22×C71semidirect
635. C2×#41D4×C71semidirect
636. C2×#41D4×C71semidirect
640. C22×Q8D4×C71semidirect
649. C22×D4D4×C71semidirect
686. C22⋊Q8D4×C71semidirect
687. C22⋊Q8D4×C71semidirect
707. #121D4×C71semidirect
708. #121D4×C71semidirect
744. #104D4×C71semidirect
881. M16C2×D4×C71semidirect
920. C42C2×D4×C71semidirect
953. Q8#121×C71semidirect
988. D4#121×C71semidirect
1068. C22#453×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#34882
C71×#30552
C2×#453×C712
C71×#30062
C22×#104×C712
C71×#28562
C71×#20392
C71×#20372
C71×#20062
C71×#20022
C71×#19002
C71×#18962
C71×#18832
C71×#18592
C71×#17922
#2446471