Finite Group Databasebeta

← C71×#11146 #31832018

Subgroups — C71×#11146 #31832018

Order 18176.

894 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#430×C71C211yesmaximal, characteristic, complemented
2C2×#421×C71C211yesmaximal, characteristic
3C22×#104×C71C211yesmaximal, characteristic
4C2×#413×C71C211yesmaximal, characteristic
5C2×#344×C71C211yesmaximal, characteristic, complemented
6C2×#339×C71C211yesmaximal, characteristic, complemented
7C71×#1728C211yesmaximal, characteristic, complemented
8C22×#41×C71C2211yescharacteristic
9C22×M16×C71C2211yescharacteristic
10C22×M16×C71C2211yescharacteristic
11C22×M16×C71C2211yescharacteristic
12C22×M16×C71C2211yescharacteristic
13C2×#117×C71C2211yescharacteristic
14C22×C42×C71C2211yescharacteristic
15#430×C71—22no—
16#430×C71—22no—
17C2×#108×C71—22no—
18#421×C71—22no—
19#421×C71—22no—
20C2×#104×C71—22no—
21C2×#104×C71—22no—
22C2×#104×C71—22no—
23C2×#104×C71—22no—
24#413×C71—22no—
25#413×C71—22no—
26C2×#101×C71C412yescomplemented
27C2×#100×C71C412yescomplemented
28#344×C71—22no—
29#339×C71—22no—
30#336×C71—22no—
31C2×#41×C71—21no—
32C2×#41×C71—21no—
33C2×#41×C71—21no—
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×C71D412yes—
42C2×#41×C71Q812yes—
43C22×Q8×C71C2×C411yescharacteristic
44C22×D4×C71—21no—
45C22×D4×C71C2×C411yescharacteristic
46C23×C4×C71—21no—
47C23×C4×C71—21no—
48C23×C4×C71C2311yescharacteristic
49C2×M16×C71—21no—
50C2×M16×C71—21no—
51C2×M16×C71—21no—
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×C71—21no—
64C2×M16×C71—21no—
65C2×M16×C71—21no—
66C2×M16×C71—21no—
67C2×M16×C71C2×C412yes—
68C2×M16×C71C2×C412yes—
69C2×M16×C71D412yes—
70C2×M16×C71D412yes—
71C2×M16×C71D412yes—
72C2×M16×C71D412yes—
73C22×C8×C71—21no—
74C22×C8×C71—21no—
75C22×C8×C71—21no—
76C22×C8×C71—21no—
77#117×C71—21no—
78#117×C71—21no—
79#117×C71—21no—
80#117×C71D412yes—
81C2×C4⋊C4×C71—21no—
82C2×C22⋊C4×C71—21no—
83C2×C42×C71—21no—
84C2×C42×C71—21no—
85C2×C42×C71—21no—
86C2×C42×C71—21no—
87C2×C42×C71—21no—
88C2×C42×C71—21no—
89C2×C42×C71D412yes—
90#108×C71—28no—
91#104×C71—28no—
92#104×C71—28no—
93#101×C71—24no—
94#100×C71—24no—
95C24×C71—21no—
96C24×C71#4111yescharacteristic
97#41×C71—22no—
98#41×C71—22no—
99#41×C71—22no—
100#41×C71—22no—
101#41×C71—22no—
102#41×C71—22no—
103#41×C71—22no—
104#41×C71—22no—
105#41×C71—41no—
106#41×C71—41no—
107#41×C71—41no—
108#41×C71—41no—
109#41×C71—41no—
110#41×C71—41no—
111#41×C71—41no—
112#41×C71—41no—
113C2×Q8×C71—21no—
114C2×Q8×C71—21no—
115C2×Q8×C71—21no—
116C2×Q8×C71—21no—
117C2×Q8×C71—21no—
118C2×Q8×C71C4⋊C412yes—
119C2×D4×C71—22no—
120C2×D4×C71—22no—
121C2×D4×C71—21no—
122C2×D4×C71—21no—
123C2×D4×C71—21no—
124C2×D4×C71—21no—
125C2×D4×C71—21no—
126C2×D4×C71—41no—
127C2×D4×C71—41no—
128C2×D4×C71—41no—
129C2×D4×C71—41no—
130C2×D4×C71C4⋊C412yes—
131C22×C4×C71—21no—
132C22×C4×C71—22no—
133C22×C4×C71—22no—
134C22×C4×C71—22no—
135C22×C4×C71—44no—
136C22×C4×C71—41no—
137C22×C4×C71—41no—
138C22×C4×C71—41no—
139C22×C4×C71—41no—
140C22×C4×C71—41no—
141C22×C4×C71—41no—
142C22×C4×C71—41no—
143C22×C4×C71C22×C411yescharacteristic
144C22×C4×C71C2×D411yescharacteristic
145C22×C4×C71C2×D411yescharacteristic
146C22×C4×C71C2×D411yescharacteristic
147C22×C4×C71C2×D411yescharacteristic
148C22×C4×C71C2×D411yescharacteristic
149C22×C4×C71C2×D411yescharacteristic
150C22×C4×C71C2×D411yescharacteristic
151C22×C4×C71C2×Q811yescharacteristic
152C22×C4×C71#4111yescharacteristic
153C22×C4×C71#4111yescharacteristic
154C22×C4×C71#4111yescharacteristic
155C22×C4×C71#4111yescharacteristic
156C22×C4×C71#4111yescharacteristic
157M16×C71—22no—
158M16×C71—22no—
159M16×C71—22no—
160M16×C71—22no—
161M16×C71—22no—
162M16×C71—22no—
163M16×C71—22no—
164M16×C71—22no—
165M16×C71—22no—
166M16×C71—41no—
167M16×C71—41no—
168M16×C71—41no—
169M16×C71—41no—
170M16×C71—41no—
171M16×C71—41no—
172M16×C71C22⋊C414yes—
173C2×C8×C71—22no—
174C2×C8×C71—22no—
175C2×C8×C71—22no—
176C2×C8×C71—22no—
177C2×C8×C71—22no—
178C2×C8×C71—22no—
179C2×C8×C71—41no—
180C2×C8×C71—41no—
181C2×C8×C71—41no—
182C2×C8×C71—41no—
183C2×C8×C71—41no—
184C2×C8×C71—41no—
185C4⋊C4×C71—22no—
186C4⋊C4×C71—41no—
187C22⋊C4×C71—22no—
188C22⋊C4×C71—41no—
189C42×C71—22no—
190C42×C71—22no—
191C42×C71—22no—
192C42×C71—41no—
193C42×C71—41no—
194C42×C71—41no—
195C23×C71—22no—
196C23×C71—42no—
197C23×C71—41no—
198C23×C71—41no—
199C23×C71—82no—
200C23×C71#11711yescharacteristic
201C23×C71#12111yescharacteristic
202C23×C71C22⋊Q811yescharacteristic
203C23×C71#12311yescharacteristic
204C23×C71#12311yescharacteristic
205C23×C71#12411yescharacteristic
206C23×C71#12411yescharacteristic
207Q8×C71—22no—
208Q8×C71—22no—
209Q8×C71—41no—
210Q8×C71—41no—
211D4×C71—22no—
212D4×C71—22no—
213D4×C71—42no—
214D4×C71—42no—
215D4×C71—42no—
216D4×C71—42no—
217D4×C71—41no—
218D4×C71—41no—
219C2×C4×C71—24no—
220C2×C4×C71—22no—
221C2×C4×C71—42no—
222C2×C4×C71—44no—
223C2×C4×C71—44no—
224C2×C4×C71—41no—
225C2×C4×C71—41no—
226C2×C4×C71—42no—
227C2×C4×C71—42no—
228C2×C4×C71—42no—
229C2×C4×C71—42no—
230C2×C4×C71—42no—
231C2×C4×C71—42no—
232C2×C4×C71—82no—
233C2×C4×C71C2×C22⋊C411yescharacteristic
234C2×C4×C71C2×C4⋊C411yescharacteristic
235C2×C4×C71C4×D411yescharacteristic
236C2×C4×C71C4×D411yescharacteristic
237C2×C4×C71C4×D411yescharacteristic
238C2×C4×C71C4×D411yescharacteristic
239C2×C4×C71#12011yescharacteristic
240C2×C4×C71#12011yescharacteristic
241C2×C4×C71#12011yescharacteristic
242C2×C4×C71#12111yescharacteristic
243C2×C4×C71#12111yescharacteristic
244C2×C4×C71#12111yescharacteristic
245C2×C4×C71#12111yescharacteristic
246C2×C4×C71#12111yescharacteristic
247C2×C4×C71#12111yescharacteristic
248C2×C4×C71#12111yescharacteristic
249C2×C4×C71C22⋊Q811yescharacteristic
250C2×C4×C71C22⋊Q811yescharacteristic
251C2×C4×C71C22⋊Q811yescharacteristic
252C2×C4×C71C22⋊Q811yescharacteristic
253C2×C4×C71C22⋊Q811yescharacteristic
254C2×C4×C71#12311yescharacteristic
255C2×C4×C71#12311yescharacteristic
256C2×C4×C71#12311yescharacteristic
257C2×C4×C71#12411yescharacteristic
258C2×C4×C71#12611yescharacteristic
259C2×C4×C71#12611yescharacteristic
260C2×C4×C71#12711yescharacteristic
261C8×C71—24no—
262C8×C71—42no—
263C8×C71—42no—
264C8×C71—42no—
265C22×C71—22no—
266C22×C71—22no—
267C22×C71—22no—
268C22×C71—41no—
269C22×C71—41no—
270C22×C71—41no—
271C22×C71—41no—
272C22×C71—42no—
273C22×C71—82no—
274C22×C71—82no—
275C22×C71—82no—
276C22×C71#38011yescharacteristic
277C22×C71#38811yescharacteristic
278C22×C71#38811yescharacteristic
279C22×C71#39311yescharacteristic
280C22×C71#39411yescharacteristic
281C22×C71#39411yescharacteristic
282C22×C71#39911yescharacteristic
283C4×C71—44no—
284C4×C71—42no—
285C4×C71—82no—
286C4×C71#38511yescharacteristic
287C4×C71#38511yescharacteristic
288C4×C71#38611yescharacteristic
289C4×C71#39011yescharacteristic
290C4×C71#39211yescharacteristic
291C4×C71#39611yescharacteristic
292C4×C71#39611yescharacteristic
293C4×C71#39711yescharacteristic
294#11146C7111yesmaximal, characteristic, direct product factor, Sylow
295C2×C71—21no—
296C2×C71—21no—
297C2×C71—21no—
298C2×C71—21no—
299C2×C71—21no—
300C2×C71—21no—
301C2×C71—82no—
302C2×C71#143911yescharacteristic
303C2×C71#193212yes—
304C2×#430C2×C7111yescharacteristic, complemented
305C2×#421C2×C7111yescharacteristic
306C22×#104C2×C7111yescharacteristic
307C2×#413C2×C7111yescharacteristic
308C2×#344C2×C7111yescharacteristic, complemented
309C2×#339C2×C7111yescharacteristic, complemented
310#1728C2×C7111yescharacteristic, complemented
311C71#1114611yescharacteristic, direct product factor, Sylow
312C22×#41C22×C7111yescharacteristic
313C22×M16C22×C7111yescharacteristic
314C22×M16C22×C7111yescharacteristic
315C22×M16C22×C7111yescharacteristic
316C22×M16C22×C7111yescharacteristic
317C2×#117C22×C7111yescharacteristic
318C22×C42C22×C7111yescharacteristic
319#430—22no—
320#430—22no—
321C2×#108—22no—
322#421—22no—
323#421—22no—
324C2×#104—22no—
325C2×#104—22no—
326C2×#104—22no—
327C2×#104—22no—
328#413—22no—
329#413—22no—
330C2×#101C4×C7112yescomplemented
331C2×#100C4×C7112yescomplemented
332#344—22no—
333#339—22no—
334#336—22no—
335C2×#41—21no—
336C2×#41—21no—
337C2×#41—21no—
338C2×#41—21no—
339C2×#41—21no—
340C2×#41—21no—
341C2×#41—21no—
342C2×#41—21no—
343C2×#41—21no—
344C2×#41—21no—
345C2×#41D4×C7112yes—
346C2×#41Q8×C7112yes—
347C22×Q8C2×C4×C7111yescharacteristic
348C22×D4—21no—
349C22×D4C2×C4×C7111yescharacteristic
350C23×C4—21no—
351C23×C4—21no—
352C23×C4C23×C7111yescharacteristic
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—21no—
370C2×M16—21no—
371C2×M16C2×C4×C7112yes—
372C2×M16C2×C4×C7112yes—
373C2×M16D4×C7112yes—
374C2×M16D4×C7112yes—
375C2×M16D4×C7112yes—
376C2×M16D4×C7112yes—
377C22×C8—21no—
378C22×C8—21no—
379C22×C8—21no—
380C22×C8—21no—
381#117—21no—
382#117—21no—
383#117—21no—
384#117D4×C7112yes—
385C2×C4⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42D4×C7112yes—
394#108—28no—
395#104—28no—
396#104—28no—
397#101—24no—
398#100—24no—
399C24—21no—
400C24#41×C7111yescharacteristic
401#41—22no—
402#41—22no—
403#41—22no—
404#41—22no—
405#41—22no—
406#41—22no—
407#41—22no—
408#41—22no—
409#41—41no—
410#41—41no—
411#41—41no—
412#41—41no—
413#41—41no—
414#41—41no—
415#41—41no—
416#41—41no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8C4⋊C4×C7112yes—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—41no—
433C2×D4—41no—
434C2×D4C4⋊C4×C7112yes—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—44no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4C22×C4×C7111yescharacteristic
448C22×C4C2×D4×C7111yescharacteristic
449C22×C4C2×D4×C7111yescharacteristic
450C22×C4C2×D4×C7111yescharacteristic
451C22×C4C2×D4×C7111yescharacteristic
452C22×C4C2×D4×C7111yescharacteristic
453C22×C4C2×D4×C7111yescharacteristic
454C22×C4C2×D4×C7111yescharacteristic
455C22×C4C2×Q8×C7111yescharacteristic
456C22×C4#41×C7111yescharacteristic
457C22×C4#41×C7111yescharacteristic
458C22×C4#41×C7111yescharacteristic
459C22×C4#41×C7111yescharacteristic
460C22×C4#41×C7111yescharacteristic
461M16—22no—
462M16—22no—
463M16—22no—
464M16—22no—
465M16—22no—
466M16—22no—
467M16—22no—
468M16—22no—
469M16—22no—
470M16—41no—
471M16—41no—
472M16—41no—
473M16—41no—
474M16—41no—
475M16—41no—
476M16C22⋊C4×C7114yes—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C4⋊C4—22no—
490C4⋊C4—41no—
491C22⋊C4—22no—
492C22⋊C4—41no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—41no—
497C42—41no—
498C42—41no—
499C23—22no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—82no—
504C23#117×C7111yescharacteristic
505C23#121×C7111yescharacteristic
506C23C22⋊Q8×C7111yescharacteristic
507C23#123×C7111yescharacteristic
508C23#123×C7111yescharacteristic
509C23#124×C7111yescharacteristic
510C23#124×C7111yescharacteristic
511Q8—22no—
512Q8—22no—
513Q8—41no—
514Q8—41no—
515D4—22no—
516D4—22no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—41no—
522D4—41no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—82no—
537C2×C4C2×C22⋊C4×C7111yescharacteristic
538C2×C4C2×C4⋊C4×C7111yescharacteristic
539C2×C4C4×D4×C7111yescharacteristic
540C2×C4C4×D4×C7111yescharacteristic
541C2×C4C4×D4×C7111yescharacteristic
542C2×C4C4×D4×C7111yescharacteristic
543C2×C4#120×C7111yescharacteristic
544C2×C4#120×C7111yescharacteristic
545C2×C4#120×C7111yescharacteristic
546C2×C4#121×C7111yescharacteristic
547C2×C4#121×C7111yescharacteristic
548C2×C4#121×C7111yescharacteristic
549C2×C4#121×C7111yescharacteristic
550C2×C4#121×C7111yescharacteristic
551C2×C4#121×C7111yescharacteristic
552C2×C4#121×C7111yescharacteristic
553C2×C4C22⋊Q8×C7111yescharacteristic
554C2×C4C22⋊Q8×C7111yescharacteristic
555C2×C4C22⋊Q8×C7111yescharacteristic
556C2×C4C22⋊Q8×C7111yescharacteristic
557C2×C4C22⋊Q8×C7111yescharacteristic
558C2×C4#123×C7111yescharacteristic
559C2×C4#123×C7111yescharacteristic
560C2×C4#123×C7111yescharacteristic
561C2×C4#124×C7111yescharacteristic
562C2×C4#126×C7111yescharacteristic
563C2×C4#126×C7111yescharacteristic
564C2×C4#127×C7111yescharacteristic
565C8—24no—
566C8—42no—
567C8—42no—
568C8—42no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22#380×C7111yescharacteristic
581C22#388×C7111yescharacteristic
582C22#388×C7111yescharacteristic
583C22#393×C7111yescharacteristic
584C22#394×C7111yescharacteristic
585C22#394×C7111yescharacteristic
586C22#399×C7111yescharacteristic
587C4—44no—
588C4—42no—
589C4—82no—
590C4#385×C7111yescharacteristic
591C4#385×C7111yescharacteristic
592C4#386×C7111yescharacteristic
593C4#390×C7111yescharacteristic
594C4#392×C7111yescharacteristic
595C4#396×C7111yescharacteristic
596C4#396×C7111yescharacteristic
597C4#397×C7111yescharacteristic
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—21no—
603C2—21no—
604C2—82no—
605C2C71×#143911yescharacteristic
606C2C71×#193212yes—
subgroupG/H
centerC2×C4×C71#121
commutatorC22×C4C22×C4×C71
Frattini, cosocleC23×C4C23×C71
socleC22×C71#388
prime radicalC22×#41×C71C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C71 ◁ C23×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#430×C71C2semidirect
5. C2×#344×C71C2semidirect
6. C2×#339×C71C2semidirect
7. C71×#1728C2semidirect
26. C2×#101×C71C4semidirect
27. C2×#100×C71C4semidirect
294. #11146C71direct (N × H)
304. C2×#430C2×C71semidirect
308. C2×#344C2×C71semidirect
309. C2×#339C2×C71semidirect
310. #1728C2×C71semidirect
311. C71#11146direct (N × H)
330. C2×#101C4×C71semidirect
331. C2×#100C4×C71semidirect

Maximal subgroups

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

subgroupindex
C2×#430×C712
C2×#421×C712
C22×#104×C712
C2×#413×C712
C2×#344×C712
C2×#339×C712
C71×#17282
#1114671