Finite Group Databasebeta

← C71×#24468 #31845340

Subgroups — C71×#24468 #31845340

Order 18176.

1502 conjugacy classes of subgroups.

The subgroup lattice has 1090 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
5C2×#421×C71C211yesmaximal, characteristic, complemented
6C71×#2619C211yesmaximal, characteristic, complemented
7C71×#2040C211yesmaximal, characteristic, complemented
8C71×#2037C211yesmaximal, characteristic, complemented
9C71×#2006C211yesmaximal, characteristic, complemented
10C71×#2002C211yesmaximal, characteristic, complemented
11C71×#1900C211yesmaximal, characteristic, complemented
12C71×#1897C211yesmaximal, characteristic, complemented
13C71×#1883C211yesmaximal, characteristic, complemented
14C71×#1860C211yesmaximal, 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×C71C2211yescharacteristic
37C2×#121×C71C2211yescharacteristic, complemented
38#520×C71C2211yescharacteristic
39C2×#117×C71C2211yescharacteristic, complemented
40#469×C71—21no—
41#468×C71—21no—
42#466×C71—21no—
43#465×C71—21no—
44#453×C71—24no—
45#449×C71—21no—
46#449×C71—21no—
47#448×C71—21no—
48#447×C71—21no—
49#428×C71C2211yescharacteristic, complemented
50#421×C71—24no—
51C2×#104×C71C2211yescharacteristic
52C2×#104×C71C2211yescharacteristic
53C2×#104×C71C2211yescharacteristic, complemented
54C2×#104×C71C2211yescharacteristic, complemented
55C2×#104×C71C2211yescharacteristic, complemented
56C2×#104×C71C2211yescharacteristic, complemented
57#418×C71C2211yescharacteristic, complemented
58#417×C71C2211yescharacteristic
59#417×C71C2211yescharacteristic, complemented
60#417×C71C2211yescharacteristic, complemented
61C2×#100×C71C2211yescharacteristic, complemented
62C2×#100×C71C2211yescharacteristic, complemented
63#407×C71C2211yescharacteristic, complemented
64#394×C71—22no—
65#392×C71—22no—
66#390×C71C2211yescharacteristic, complemented
67#390×C71C2211yescharacteristic, complemented
68#382×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×C71C2311yescharacteristic
104AGL(1,ℤ/8)×C71—22no—
105AGL(1,ℤ/8)×C71—22no—
106AGL(1,ℤ/8)×C71—22no—
107AGL(1,ℤ/8)×C71—41no—
108AGL(1,ℤ/8)×C71—41no—
109AGL(1,ℤ/8)×C71—41no—
110C2×SD16×C71—21no—
111C2×SD16×C71—21no—
112C2×SD16×C71—21no—
113C2×D8×C71—21no—
114C2×D8×C71—21no—
115C2×D8×C71—21no—
116C2×M16×C71—21no—
117C2×M16×C71—22no—
118C2×M16×C71—41no—
119C2×M16×C71C2311yescharacteristic
120C2×M16×C71C2311yescharacteristic
121C2×M16×C71C2311yescharacteristic
122C2×M16×C71C2311yescharacteristic
123C22×C8×C71—21no—
124#127×C71—22no—
125#127×C71—21no—
126#127×C71—22no—
127#127×C71—41no—
128#124×C71—22no—
129#124×C71—21no—
130#123×C71—22no—
131#123×C71—21no—
132C22⋊Q8×C71—22no—
133C22⋊Q8×C71—21no—
134C22⋊Q8×C71—21no—
135C22⋊Q8×C71D411yescharacteristic, complemented
136C22⋊Q8×C71D411yescharacteristic, complemented
137#121×C71—22no—
138#121×C71—22no—
139#121×C71—22no—
140#121×C71—22no—
141#121×C71—22no—
142#121×C71—21no—
143#121×C71—21no—
144#121×C71—21no—
145#121×C71—21no—
146#121×C71—21no—
147#121×C71—21no—
148#121×C71—22no—
149#121×C71—21no—
150#121×C71—21no—
151#121×C71—21no—
152#121×C71—21no—
153#121×C71—21no—
154#121×C71—21no—
155#121×C71D411yescharacteristic, complemented
156#121×C71D411yescharacteristic, complemented
157#120×C71—22no—
158#120×C71—22no—
159#120×C71—22no—
160#120×C71—21no—
161#120×C71—22no—
162C4×Q8×C71—21no—
163C4×D4×C71—22no—
164C4×D4×C71—22no—
165C4×D4×C71—21no—
166C4×D4×C71—22no—
167C4×D4×C71—21no—
168C4×D4×C71—21no—
169#117×C71—21no—
170#117×C71—22no—
171#117×C71—41no—
172#117×C71C2311yescharacteristic
173C2×C4⋊C4×C71—21no—
174C2×C4⋊C4×C71—21no—
175C2×C4⋊C4×C71C2311yescharacteristic
176C2×C4⋊C4×C71C2311yescharacteristic
177C2×C22⋊C4×C71—22no—
178C2×C22⋊C4×C71—21no—
179C2×C22⋊C4×C71—21no—
180C2×C22⋊C4×C71—22no—
181C2×C22⋊C4×C71—22no—
182C2×C22⋊C4×C71D412yes—
183C2×C42×C71C2311yescharacteristic
184C2×C42×C71C2311yescharacteristic
185#107×C71—21no—
186#106×C71—21no—
187#104×C71—22no—
188#104×C71—22no—
189#104×C71—22no—
190#104×C71—22no—
191#104×C71—41no—
192#104×C71D414yescomplemented
193#103×C71—21no—
194#103×C71—21no—
195#103×C71—21no—
196#102×C71—21no—
197#102×C71—21no—
198#102×C71—21no—
199#102×C71—21no—
200#102×C71—21no—
201#100×C71—22no—
202#100×C71—41no—
203C22⋊C8×C71—21no—
204C22⋊C8×C71—21no—
205#95×C71—22no—
206C24×C71—22no—
207C24×C71—21no—
208C24×C71—21no—
209C24×C71C2×D412yes—
210C24×C71C2×D411yescharacteristic
211#41×C71—21no—
212#41×C71—21no—
213#41×C71—22no—
214#41×C71—22no—
215#41×C71—21no—
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—42no—
224#41×C71—42no—
225#41×C71—41no—
226#41×C71—41no—
227#41×C71—42no—
228#41×C71—42no—
229#41×C71—42no—
230#41×C71—42no—
231#41×C71—42no—
232#41×C71—42no—
233#41×C71C2×D412yes—
234#41×C71#4112yes—
235C2×Q8×C71—21no—
236C2×Q8×C71—22no—
237C2×Q8×C71—41no—
238C2×Q8×C71C2×D411yescharacteristic
239C2×Q8×C71C2×D411yescharacteristic
240C2×Q8×C71C2×D411yescharacteristic
241C2×D4×C71—21no—
242C2×D4×C71—22no—
243C2×D4×C71—22no—
244C2×D4×C71—22no—
245C2×D4×C71—22no—
246C2×D4×C71—21no—
247C2×D4×C71—21no—
248C2×D4×C71—21no—
249C2×D4×C71—21no—
250C2×D4×C71—21no—
251C2×D4×C71—22no—
252C2×D4×C71—22no—
253C2×D4×C71—22no—
254C2×D4×C71—21no—
255C2×D4×C71—22no—
256C2×D4×C71—21no—
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—41no—
270C2×D4×C71—42no—
271C2×D4×C71—42no—
272C2×D4×C71—44no—
273C2×D4×C71—42no—
274C2×D4×C71—42no—
275C2×D4×C71—44no—
276C2×D4×C71—44no—
277C2×D4×C71—42no—
278C2×D4×C71—42no—
279C2×D4×C71—41no—
280C2×D4×C71—41no—
281C2×D4×C71—82no—
282C2×D4×C71—82no—
283C2×D4×C71—82no—
284C2×D4×C71—82no—
285C2×D4×C71—84no—
286C2×D4×C71C2×D411yescharacteristic
287C2×D4×C71C2×D411yescharacteristic
288C2×D4×C71C2×D411yescharacteristic
289C22×C4×C71—21no—
290C22×C4×C71—21no—
291C22×C4×C71—21no—
292C22×C4×C71—21no—
293C22×C4×C71—21no—
294C22×C4×C71—22no—
295C22×C4×C71—21no—
296C22×C4×C71—21no—
297C22×C4×C71—22no—
298C22×C4×C71—21no—
299C22×C4×C71—21no—
300C22×C4×C71—21no—
301C22×C4×C71—21no—
302C22×C4×C71—21no—
303C22×C4×C71—21no—
304C22×C4×C71—42no—
305C22×C4×C71—42no—
306C22×C4×C71C2×D411yescharacteristic
307C22×C4×C71C2×D411yescharacteristic
308C22×C4×C71C2×D411yescharacteristic
309C22×C4×C71C2×D411yescharacteristic
310C22×C4×C71C2×D411yescharacteristic
311C22×C4×C71C2×D412yes—
312C22×C4×C71C2411yescharacteristic
313SD16×C71—42no—
314SD16×C71—42no—
315SD16×C71—42no—
316D8×C71—42no—
317D8×C71—42no—
318D8×C71—42no—
319M16×C71—21no—
320M16×C71—21no—
321M16×C71—21no—
322M16×C71—21no—
323M16×C71—21no—
324M16×C71—21no—
325M16×C71—21no—
326M16×C71—42no—
327M16×C71C2×D412yescomplemented
328C2×C8×C71—21no—
329C2×C8×C71—21no—
330C2×C8×C71—21no—
331C2×C8×C71—21no—
332C2×C8×C71—42no—
333C4⋊C4×C71—22no—
334C4⋊C4×C71—22no—
335C4⋊C4×C71—21no—
336C4⋊C4×C71—21no—
337C4⋊C4×C71—21no—
338C4⋊C4×C71—21no—
339C4⋊C4×C71—21no—
340C4⋊C4×C71—21no—
341C4⋊C4×C71—42no—
342C4⋊C4×C71C2×D411yescharacteristic
343C4⋊C4×C71C2×D411yescharacteristic
344C22⋊C4×C71—22no—
345C22⋊C4×C71—22no—
346C22⋊C4×C71—22no—
347C22⋊C4×C71—22no—
348C22⋊C4×C71—22no—
349C22⋊C4×C71—21no—
350C22⋊C4×C71—21no—
351C22⋊C4×C71—21no—
352C22⋊C4×C71—21no—
353C22⋊C4×C71—22no—
354C22⋊C4×C71—22no—
355C22⋊C4×C71—21no—
356C22⋊C4×C71—42no—
357C22⋊C4×C71—41no—
358C22⋊C4×C71—41no—
359C22⋊C4×C71—82no—
360C22⋊C4×C71—82no—
361C42×C71—22no—
362C42×C71—21no—
363C42×C71—21no—
364C42×C71—21no—
365C42×C71—21no—
366C42×C71C2×D412yescomplemented
367C23×C71—21no—
368C23×C71—21no—
369C23×C71—21no—
370C23×C71—22no—
371C23×C71—21no—
372C23×C71—21no—
373C23×C71—22no—
374C23×C71—22no—
375C23×C71—41no—
376C23×C71—41no—
377C23×C71—42no—
378C23×C71—42no—
379C23×C71—44no—
380C23×C71—42no—
381C23×C71—42no—
382C23×C71—42no—
383C23×C71—42no—
384C23×C71—41no—
385C23×C71—41no—
386C23×C71—41no—
387C23×C71—81no—
388C23×C71—82no—
389C23×C71—82no—
390C23×C71#12011yescharacteristic
391C23×C71#12011yescharacteristic
392C23×C71C22×D411yescharacteristic
393Q8×C71—21no—
394Q8×C71—21no—
395Q8×C71—21no—
396Q8×C71—21no—
397Q8×C71—21no—
398Q8×C71—42no—
399Q8×C71#12112yes—
400D4×C71—22no—
401D4×C71—22no—
402D4×C71—21no—
403D4×C71—21no—
404D4×C71—21no—
405D4×C71—21no—
406D4×C71—21no—
407D4×C71—42no—
408D4×C71—42no—
409D4×C71—44no—
410D4×C71—42no—
411D4×C71—44no—
412D4×C71—44no—
413D4×C71—42no—
414D4×C71—41no—
415D4×C71—41no—
416D4×C71—41no—
417D4×C71—41no—
418D4×C71—41no—
419D4×C71—41no—
420D4×C71—41no—
421D4×C71—41no—
422D4×C71—41no—
423D4×C71—41no—
424D4×C71—41no—
425D4×C71—41no—
426D4×C71—81no—
427D4×C71—82no—
428D4×C71—82no—
429D4×C71—82no—
430D4×C71—82no—
431D4×C71—88no—
432D4×C71—82no—
433D4×C71—82no—
434D4×C71#12112yes—
435C2×C4×C71—21no—
436C2×C4×C71—21no—
437C2×C4×C71—21no—
438C2×C4×C71—21no—
439C2×C4×C71—21no—
440C2×C4×C71—21no—
441C2×C4×C71—21no—
442C2×C4×C71—21no—
443C2×C4×C71—21no—
444C2×C4×C71—22no—
445C2×C4×C71—21no—
446C2×C4×C71—21no—
447C2×C4×C71—22no—
448C2×C4×C71—21no—
449C2×C4×C71—22no—
450C2×C4×C71—21no—
451C2×C4×C71—21no—
452C2×C4×C71—21no—
453C2×C4×C71—22no—
454C2×C4×C71—42no—
455C2×C4×C71—42no—
456C2×C4×C71—42no—
457C2×C4×C71—44no—
458C2×C4×C71—41no—
459C2×C4×C71—41no—
460C2×C4×C71—41no—
461C2×C4×C71—41no—
462C2×C4×C71—41no—
463C2×C4×C71—41no—
464C2×C4×C71—42no—
465C2×C4×C71—41no—
466C2×C4×C71—41no—
467C2×C4×C71—42no—
468C2×C4×C71—41no—
469C2×C4×C71—41no—
470C2×C4×C71—82no—
471C2×C4×C71#12011yescharacteristic
472C2×C4×C71#12011yescharacteristic
473C2×C4×C71C22×D411yescharacteristic
474C2×C4×C71C22×D411yescharacteristic
475C2×C4×C71C22×D411yescharacteristic
476C2×C4×C71C2×#4111yescharacteristic
477C2×C4×C71H(5,𝔽2)11yescharacteristic
478C2×C4×C71H(5,𝔽2)11yescharacteristic
479C8×C71—22no—
480C8×C71—41no—
481C8×C71—41no—
482C8×C71—41no—
483C22×C71—21no—
484C22×C71—21no—
485C22×C71—21no—
486C22×C71—21no—
487C22×C71—22no—
488C22×C71—21no—
489C22×C71—21no—
490C22×C71—21no—
491C22×C71—21no—
492C22×C71—41no—
493C22×C71—41no—
494C22×C71—41no—
495C22×C71—41no—
496C22×C71—42no—
497C22×C71—42no—
498C22×C71—41no—
499C22×C71—41no—
500C22×C71—41no—
501C22×C71—42no—
502C22×C71—41no—
503C22×C71—81no—
504C22×C71—81no—
505C22×C71—82no—
506C22×C71—82no—
507C22×C71—82no—
508C22×C71—82no—
509C22×C71—84no—
510C22×C71—81no—
511C22×C71—81no—
512C22×C71—81no—
513C22×C71—164no—
514C22×C71C2×#12011yescharacteristic
515C22×C71D4211yescharacteristic
516C22×C71#54611yescharacteristic
517C4×C71—21no—
518C4×C71—21no—
519C4×C71—22no—
520C4×C71—42no—
521C4×C71—41no—
522C4×C71—41no—
523C4×C71—42no—
524C4×C71—41no—
525C4×C71—41no—
526C4×C71—81no—
527C4×C71—81no—
528C4×C71C2×#12111yescharacteristic
529C4×C71#53411yescharacteristic
530C4×C71D4211yescharacteristic
531C4×C71#54611yescharacteristic
532#24468C7111yesmaximal, characteristic, direct product factor, Sylow
533C2×C71—21no—
534C2×C71—21no—
535C2×C71—41no—
536C2×C71—41no—
537C2×C71—42no—
538C2×C71—81no—
539C2×C71—82no—
540C2×C71—81no—
541C2×C71—161no—
542C2×C71#241011yescharacteristic
543C2×C71#301911yescharacteristic
544C2×C71#302111yescharacteristic
545#3488C2×C7111yescharacteristic, complemented
546#3055C2×C7111yescharacteristic, complemented
547C2×#453C2×C7111yescharacteristic, complemented
548#3006C2×C7111yescharacteristic, complemented
549C2×#421C2×C7111yescharacteristic, complemented
550#2619C2×C7111yescharacteristic, complemented
551#2040C2×C7111yescharacteristic, complemented
552#2037C2×C7111yescharacteristic, complemented
553#2006C2×C7111yescharacteristic, complemented
554#2002C2×C7111yescharacteristic, complemented
555#1900C2×C7111yescharacteristic, complemented
556#1897C2×C7111yescharacteristic, complemented
557#1883C2×C7111yescharacteristic, complemented
558#1860C2×C7111yescharacteristic, complemented
559#1792C2×C7111yescharacteristic, complemented
560C71#2446811yescharacteristic, direct product factor, Sylow
561C2×H(5,𝔽2)C22×C7111yescharacteristic
562C22×#41C22×C7111yescharacteristic
563C2×AGL(1,ℤ/8)C22×C7111yescharacteristic
564C2×AGL(1,ℤ/8)C22×C7111yescharacteristic, complemented
565C2×AGL(1,ℤ/8)C22×C7111yescharacteristic, complemented
566C22×M16C22×C7111yescharacteristic, complemented
567#550—21no—
568#548—21no—
569#546—21no—
570#546—22no—
571D42—22no—
572D42—21no—
573#537—21no—
574#537C22×C7111yescharacteristic
575#535—21no—
576#535C22×C7111yescharacteristic
577C2×#127C22×C7111yescharacteristic, complemented
578#525—21no—
579#525C22×C7111yescharacteristic
580C2×#121—21no—
581C2×#121C22×C7111yescharacteristic
582C2×#121C22×C7111yescharacteristic, complemented
583#520C22×C7111yescharacteristic
584C2×#117C22×C7111yescharacteristic, complemented
585#469—21no—
586#468—21no—
587#466—21no—
588#465—21no—
589#453—24no—
590#449—21no—
591#449—21no—
592#448—21no—
593#447—21no—
594#428C22×C7111yescharacteristic, complemented
595#421—24no—
596C2×#104C22×C7111yescharacteristic
597C2×#104C22×C7111yescharacteristic
598C2×#104C22×C7111yescharacteristic, complemented
599C2×#104C22×C7111yescharacteristic, complemented
600C2×#104C22×C7111yescharacteristic, complemented
601C2×#104C22×C7111yescharacteristic, complemented
602#418C22×C7111yescharacteristic, complemented
603#417C22×C7111yescharacteristic
604#417C22×C7111yescharacteristic, complemented
605#417C22×C7111yescharacteristic, complemented
606C2×#100C22×C7111yescharacteristic, complemented
607C2×#100C22×C7111yescharacteristic, complemented
608#407C22×C7111yescharacteristic, complemented
609#394—22no—
610#392—22no—
611#390C22×C7111yescharacteristic, complemented
612#390C22×C7111yescharacteristic, complemented
613#382C22×C7111yescharacteristic, complemented
614#343C22×C7111yescharacteristic, complemented
615#343C22×C7111yescharacteristic, complemented
616#339C22×C7111yescharacteristic, complemented
617#339C22×C7111yescharacteristic, complemented
618H(5,𝔽2)—22no—
619H(5,𝔽2)—22no—
620H(5,𝔽2)—41no—
621H(5,𝔽2)—41no—
622C2×#41—21no—
623C2×#41—21no—
624C2×#41—21no—
625C2×#41—21no—
626C2×#41—22no—
627C2×#41—22no—
628C2×#41—41no—
629C2×#41—41no—
630C2×#41D4×C7111yescharacteristic, complemented
631C2×#41D4×C7111yescharacteristic, complemented
632C2×#41C23×C7111yescharacteristic
633C2×#41C23×C7111yescharacteristic
634C2×#41C23×C7111yescharacteristic
635C22×Q8D4×C7111yescharacteristic, complemented
636C22×D4—21no—
637C22×D4—22no—
638C22×D4—22no—
639C22×D4—21no—
640C22×D4—21no—
641C22×D4—21no—
642C22×D4—22no—
643C22×D4D4×C7112yes—
644C22×D4D4×C7111yescharacteristic, complemented
645C22×D4C23×C7111yescharacteristic
646C22×D4C23×C7111yescharacteristic
647C23×C4—21no—
648C23×C4C23×C7111yescharacteristic
649AGL(1,ℤ/8)—22no—
650AGL(1,ℤ/8)—22no—
651AGL(1,ℤ/8)—22no—
652AGL(1,ℤ/8)—41no—
653AGL(1,ℤ/8)—41no—
654AGL(1,ℤ/8)—41no—
655C2×SD16—21no—
656C2×SD16—21no—
657C2×SD16—21no—
658C2×D8—21no—
659C2×D8—21no—
660C2×D8—21no—
661C2×M16—21no—
662C2×M16—22no—
663C2×M16—41no—
664C2×M16C23×C7111yescharacteristic
665C2×M16C23×C7111yescharacteristic
666C2×M16C23×C7111yescharacteristic
667C2×M16C23×C7111yescharacteristic
668C22×C8—21no—
669#127—22no—
670#127—21no—
671#127—22no—
672#127—41no—
673#124—22no—
674#124—21no—
675#123—22no—
676#123—21no—
677C22⋊Q8—22no—
678C22⋊Q8—21no—
679C22⋊Q8—21no—
680C22⋊Q8D4×C7111yescharacteristic, complemented
681C22⋊Q8D4×C7111yescharacteristic, complemented
682#121—22no—
683#121—22no—
684#121—22no—
685#121—22no—
686#121—22no—
687#121—21no—
688#121—21no—
689#121—21no—
690#121—21no—
691#121—21no—
692#121—21no—
693#121—22no—
694#121—21no—
695#121—21no—
696#121—21no—
697#121—21no—
698#121—21no—
699#121—21no—
700#121D4×C7111yescharacteristic, complemented
701#121D4×C7111yescharacteristic, complemented
702#120—22no—
703#120—22no—
704#120—22no—
705#120—21no—
706#120—22no—
707C4×Q8—21no—
708C4×D4—22no—
709C4×D4—22no—
710C4×D4—21no—
711C4×D4—22no—
712C4×D4—21no—
713C4×D4—21no—
714#117—21no—
715#117—22no—
716#117—41no—
717#117C23×C7111yescharacteristic
718C2×C4⋊C4—21no—
719C2×C4⋊C4—21no—
720C2×C4⋊C4C23×C7111yescharacteristic
721C2×C4⋊C4C23×C7111yescharacteristic
722C2×C22⋊C4—22no—
723C2×C22⋊C4—21no—
724C2×C22⋊C4—21no—
725C2×C22⋊C4—22no—
726C2×C22⋊C4—22no—
727C2×C22⋊C4D4×C7112yes—
728C2×C42C23×C7111yescharacteristic
729C2×C42C23×C7111yescharacteristic
730#107—21no—
731#106—21no—
732#104—22no—
733#104—22no—
734#104—22no—
735#104—22no—
736#104—41no—
737#104D4×C7114yescomplemented
738#103—21no—
739#103—21no—
740#103—21no—
741#102—21no—
742#102—21no—
743#102—21no—
744#102—21no—
745#102—21no—
746#100—22no—
747#100—41no—
748C22⋊C8—21no—
749C22⋊C8—21no—
750#95—22no—
751C24—22no—
752C24—21no—
753C24—21no—
754C24C2×D4×C7112yes—
755C24C2×D4×C7111yescharacteristic
756#41—21no—
757#41—21no—
758#41—22no—
759#41—22no—
760#41—21no—
761#41—21no—
762#41—21no—
763#41—21no—
764#41—21no—
765#41—21no—
766#41—21no—
767#41—21no—
768#41—42no—
769#41—42no—
770#41—41no—
771#41—41no—
772#41—42no—
773#41—42no—
774#41—42no—
775#41—42no—
776#41—42no—
777#41—42no—
778#41C2×D4×C7112yes—
779#41#41×C7112yes—
780C2×Q8—21no—
781C2×Q8—22no—
782C2×Q8—41no—
783C2×Q8C2×D4×C7111yescharacteristic
784C2×Q8C2×D4×C7111yescharacteristic
785C2×Q8C2×D4×C7111yescharacteristic
786C2×D4—21no—
787C2×D4—22no—
788C2×D4—22no—
789C2×D4—22no—
790C2×D4—22no—
791C2×D4—21no—
792C2×D4—21no—
793C2×D4—21no—
794C2×D4—21no—
795C2×D4—21no—
796C2×D4—22no—
797C2×D4—22no—
798C2×D4—22no—
799C2×D4—21no—
800C2×D4—22no—
801C2×D4—21no—
802C2×D4—21no—
803C2×D4—21no—
804C2×D4—21no—
805C2×D4—21no—
806C2×D4—21no—
807C2×D4—21no—
808C2×D4—21no—
809C2×D4—21no—
810C2×D4—21no—
811C2×D4—21no—
812C2×D4—21no—
813C2×D4—21no—
814C2×D4—41no—
815C2×D4—42no—
816C2×D4—42no—
817C2×D4—44no—
818C2×D4—42no—
819C2×D4—42no—
820C2×D4—44no—
821C2×D4—44no—
822C2×D4—42no—
823C2×D4—42no—
824C2×D4—41no—
825C2×D4—41no—
826C2×D4—82no—
827C2×D4—82no—
828C2×D4—82no—
829C2×D4—82no—
830C2×D4—84no—
831C2×D4C2×D4×C7111yescharacteristic
832C2×D4C2×D4×C7111yescharacteristic
833C2×D4C2×D4×C7111yescharacteristic
834C22×C4—21no—
835C22×C4—21no—
836C22×C4—21no—
837C22×C4—21no—
838C22×C4—21no—
839C22×C4—22no—
840C22×C4—21no—
841C22×C4—21no—
842C22×C4—22no—
843C22×C4—21no—
844C22×C4—21no—
845C22×C4—21no—
846C22×C4—21no—
847C22×C4—21no—
848C22×C4—21no—
849C22×C4—42no—
850C22×C4—42no—
851C22×C4C2×D4×C7111yescharacteristic
852C22×C4C2×D4×C7111yescharacteristic
853C22×C4C2×D4×C7111yescharacteristic
854C22×C4C2×D4×C7111yescharacteristic
855C22×C4C2×D4×C7111yescharacteristic
856C22×C4C2×D4×C7112yes—
857C22×C4C24×C7111yescharacteristic
858SD16—42no—
859SD16—42no—
860SD16—42no—
861D8—42no—
862D8—42no—
863D8—42no—
864M16—21no—
865M16—21no—
866M16—21no—
867M16—21no—
868M16—21no—
869M16—21no—
870M16—21no—
871M16—42no—
872M16C2×D4×C7112yescomplemented
873C2×C8—21no—
874C2×C8—21no—
875C2×C8—21no—
876C2×C8—21no—
877C2×C8—42no—
878C4⋊C4—22no—
879C4⋊C4—22no—
880C4⋊C4—21no—
881C4⋊C4—21no—
882C4⋊C4—21no—
883C4⋊C4—21no—
884C4⋊C4—21no—
885C4⋊C4—21no—
886C4⋊C4—42no—
887C4⋊C4C2×D4×C7111yescharacteristic
888C4⋊C4C2×D4×C7111yescharacteristic
889C22⋊C4—22no—
890C22⋊C4—22no—
891C22⋊C4—22no—
892C22⋊C4—22no—
893C22⋊C4—22no—
894C22⋊C4—21no—
895C22⋊C4—21no—
896C22⋊C4—21no—
897C22⋊C4—21no—
898C22⋊C4—22no—
899C22⋊C4—22no—
900C22⋊C4—21no—
901C22⋊C4—42no—
902C22⋊C4—41no—
903C22⋊C4—41no—
904C22⋊C4—82no—
905C22⋊C4—82no—
906C42—22no—
907C42—21no—
908C42—21no—
909C42—21no—
910C42—21no—
911C42C2×D4×C7112yescomplemented
912C23—21no—
913C23—21no—
914C23—21no—
915C23—22no—
916C23—21no—
917C23—21no—
918C23—22no—
919C23—22no—
920C23—41no—
921C23—41no—
922C23—42no—
923C23—42no—
924C23—44no—
925C23—42no—
926C23—42no—
927C23—42no—
928C23—42no—
929C23—41no—
930C23—41no—
931C23—41no—
932C23—81no—
933C23—82no—
934C23—82no—
935C23#120×C7111yescharacteristic
936C23#120×C7111yescharacteristic
937C23C22×D4×C7111yescharacteristic
938Q8—21no—
939Q8—21no—
940Q8—21no—
941Q8—21no—
942Q8—21no—
943Q8—42no—
944Q8#121×C7112yes—
945D4—22no—
946D4—22no—
947D4—21no—
948D4—21no—
949D4—21no—
950D4—21no—
951D4—21no—
952D4—42no—
953D4—42no—
954D4—44no—
955D4—42no—
956D4—44no—
957D4—44no—
958D4—42no—
959D4—41no—
960D4—41no—
961D4—41no—
962D4—41no—
963D4—41no—
964D4—41no—
965D4—41no—
966D4—41no—
967D4—41no—
968D4—41no—
969D4—41no—
970D4—41no—
971D4—82no—
972D4—81no—
973D4—82no—
974D4—82no—
975D4—82no—
976D4—88no—
977D4—82no—
978D4—82no—
979D4#121×C7112yes—
980C2×C4—21no—
981C2×C4—21no—
982C2×C4—21no—
983C2×C4—21no—
984C2×C4—21no—
985C2×C4—21no—
986C2×C4—21no—
987C2×C4—21no—
988C2×C4—21no—
989C2×C4—22no—
990C2×C4—21no—
991C2×C4—21no—
992C2×C4—22no—
993C2×C4—21no—
994C2×C4—22no—
995C2×C4—21no—
996C2×C4—21no—
997C2×C4—21no—
998C2×C4—22no—
999C2×C4—42no—
1000C2×C4—42no—
1001C2×C4—42no—
1002C2×C4—44no—
1003C2×C4—41no—
1004C2×C4—41no—
1005C2×C4—41no—
1006C2×C4—41no—
1007C2×C4—41no—
1008C2×C4—41no—
1009C2×C4—42no—
1010C2×C4—41no—
1011C2×C4—41no—
1012C2×C4—42no—
1013C2×C4—41no—
1014C2×C4—41no—
1015C2×C4—82no—
1016C2×C4#120×C7111yescharacteristic
1017C2×C4#120×C7111yescharacteristic
1018C2×C4C22×D4×C7111yescharacteristic
1019C2×C4C22×D4×C7111yescharacteristic
1020C2×C4C22×D4×C7111yescharacteristic
1021C2×C4C2×#41×C7111yescharacteristic
1022C2×C4H(5,𝔽2)×C7111yescharacteristic
1023C2×C4H(5,𝔽2)×C7111yescharacteristic
1024C8—22no—
1025C8—41no—
1026C8—41no—
1027C8—41no—
1028C22—21no—
1029C22—21no—
1030C22—21no—
1031C22—21no—
1032C22—22no—
1033C22—21no—
1034C22—21no—
1035C22—21no—
1036C22—21no—
1037C22—41no—
1038C22—41no—
1039C22—41no—
1040C22—41no—
1041C22—42no—
1042C22—42no—
1043C22—41no—
1044C22—41no—
1045C22—41no—
1046C22—42no—
1047C22—41no—
1048C22—81no—
1049C22—81no—
1050C22—82no—
1051C22—82no—
1052C22—82no—
1053C22—82no—
1054C22—84no—
1055C22—81no—
1056C22—81no—
1057C22—81no—
1058C22—164no—
1059C22C2×#120×C7111yescharacteristic
1060C22D42×C7111yescharacteristic
1061C22#546×C7111yescharacteristic
1062C4—21no—
1063C4—21no—
1064C4—22no—
1065C4—42no—
1066C4—41no—
1067C4—41no—
1068C4—42no—
1069C4—41no—
1070C4—41no—
1071C4—81no—
1072C4—81no—
1073C4C2×#121×C7111yescharacteristic
1074C4#534×C7111yescharacteristic
1075C4D42×C7111yescharacteristic
1076C4#546×C7111yescharacteristic
1077C2—21no—
1078C2—21no—
1079C2—41no—
1080C2—41no—
1081C2—42no—
1082C2—81no—
1083C2—82no—
1084C2—81no—
1085C2—161no—
1086C2C71×#241011yescharacteristic
1087C2C71×#301911yescharacteristic
1088C2C71×#302111yescharacteristic
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. C2×#421×C71C2semidirect
6. C71×#2619C2semidirect
7. C71×#2040C2semidirect
8. C71×#2037C2semidirect
9. C71×#2006C2semidirect
10. C71×#2002C2semidirect
11. C71×#1900C2semidirect
12. C71×#1897C2semidirect
13. C71×#1883C2semidirect
14. C71×#1860C2semidirect
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
37. C2×#121×C71C22semidirect
39. C2×#117×C71C22semidirect
49. #428×C71C22semidirect
53. C2×#104×C71C22semidirect
54. C2×#104×C71C22semidirect
55. C2×#104×C71C22semidirect
56. C2×#104×C71C22semidirect
57. #418×C71C22semidirect
59. #417×C71C22semidirect
60. #417×C71C22semidirect
61. C2×#100×C71C22semidirect
62. C2×#100×C71C22semidirect
63. #407×C71C22semidirect
66. #390×C71C22semidirect
67. #390×C71C22semidirect
68. #382×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
135. C22⋊Q8×C71D4semidirect
136. C22⋊Q8×C71D4semidirect
155. #121×C71D4semidirect
156. #121×C71D4semidirect
192. #104×C71D4semidirect
327. M16×C71C2×D4semidirect
366. C42×C71C2×D4semidirect
532. #24468C71direct (N × H)
545. #3488C2×C71semidirect
546. #3055C2×C71semidirect
547. C2×#453C2×C71semidirect
548. #3006C2×C71semidirect
549. C2×#421C2×C71semidirect
550. #2619C2×C71semidirect
551. #2040C2×C71semidirect
552. #2037C2×C71semidirect
553. #2006C2×C71semidirect
554. #2002C2×C71semidirect
555. #1900C2×C71semidirect
556. #1897C2×C71semidirect
557. #1883C2×C71semidirect
558. #1860C2×C71semidirect
559. #1792C2×C71semidirect
560. C71#24468direct (N × H)
564. C2×AGL(1,ℤ/8)C22×C71semidirect
565. C2×AGL(1,ℤ/8)C22×C71semidirect
566. C22×M16C22×C71semidirect
577. C2×#127C22×C71semidirect
582. C2×#121C22×C71semidirect
584. C2×#117C22×C71semidirect
594. #428C22×C71semidirect
598. C2×#104C22×C71semidirect
599. C2×#104C22×C71semidirect
600. C2×#104C22×C71semidirect
601. C2×#104C22×C71semidirect
602. #418C22×C71semidirect
604. #417C22×C71semidirect
605. #417C22×C71semidirect
606. C2×#100C22×C71semidirect
607. C2×#100C22×C71semidirect
608. #407C22×C71semidirect
611. #390C22×C71semidirect
612. #390C22×C71semidirect
613. #382C22×C71semidirect
614. #343C22×C71semidirect
615. #343C22×C71semidirect
616. #339C22×C71semidirect
617. #339C22×C71semidirect
630. C2×#41D4×C71semidirect
631. C2×#41D4×C71semidirect
635. C22×Q8D4×C71semidirect
644. C22×D4D4×C71semidirect
680. C22⋊Q8D4×C71semidirect
681. C22⋊Q8D4×C71semidirect
700. #121D4×C71semidirect
701. #121D4×C71semidirect
737. #104D4×C71semidirect
872. M16C2×D4×C71semidirect
911. C42C2×D4×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
C2×#421×C712
C71×#26192
C71×#20402
C71×#20372
C71×#20062
C71×#20022
C71×#19002
C71×#18972
C71×#18832
C71×#18602
C71×#17922
#2446871