Finite Group Databasebeta

← C71×#19087 #31839959

Subgroups — C71×#19087 #31839959

Order 18176.

930 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3558C211yesmaximal, characteristic
2C71×#3240C211yesmaximal, characteristic, complemented
3C71×#3216C211yesmaximal, characteristic, complemented
4C71×#3215C211yesmaximal, characteristic, complemented
5C71×#1661C212yesmaximal, complemented
6C71×#1661C212yesmaximal, complemented
7C71×#1659C212yesmaximal, complemented
8C71×#1659C212yesmaximal, complemented
9C71×#1536C211yesmaximal, characteristic, complemented
10C71×#1535C211yesmaximal, characteristic, complemented
11C71×#1533C211yesmaximal, characteristic, complemented
12#562×C71C2212yes—
13H(3,𝔽4)×C71C2212yes—
14#559×C71—21no—
15#548×C71—21no—
16#546×C71—22no—
17#546×C71—21no—
18D42×C71—21no—
19#542×C71—21no—
20#538×C71—22no—
21#538×C71—21no—
22#535×C71—21no—
23#534×C71—21no—
24#533×C71C2211yescharacteristic, complemented
25#532×C71C2211yescharacteristic
26#529×C71C2211yescharacteristic
27C2×#124×C71C2211yescharacteristic
28#501×C71C2211yescharacteristic, complemented
29#501×C71C2211yescharacteristic, complemented
30#497×C71C2212yescomplemented
31#497×C71C2212yescomplemented
32#489×C71C2211yescharacteristic, complemented
33#489×C71C2211yescharacteristic, complemented
34#489×C71C2212yescomplemented
35#489×C71C2212yescomplemented
36#485×C71—21no—
37#482×C71—21no—
38#479×C71—21no—
39#478×C71—21no—
40#464×C71—22no—
41#462×C71—21no—
42#460×C71—21no—
43#452×C71—22no—
44#451×C71—21no—
45#450×C71—21no—
46#432×C71C2211yescharacteristic, complemented
47#432×C71C2211yescharacteristic, complemented
48#432×C71C2211yescharacteristic, complemented
49#330×C71C2212yescomplemented
50#329×C71C2212yescomplemented
51#329×C71C2212yescomplemented
52#329×C71C2212yescomplemented
53#329×C71C2212yescomplemented
54#329×C71C2212yescomplemented
55C2×#41×C71—21no—
56C2×#41×C71D412yes—
57C22×Q8×C71D411yescharacteristic
58C22×D4×C71—22no—
59C22×D4×C71—21no—
60C22×D4×C71D411yescharacteristic
61C2×Q16×C71—22no—
62C2×Q16×C71—22no—
63C2×SD16×C71—22no—
64C2×SD16×C71—22no—
65C4⋊Q8×C71D411yescharacteristic
66C4⋊Q8×C71C2311yescharacteristic
67C4⋊Q8×C71C2311yescharacteristic
68#127×C71—21no—
69#127×C71D411yescharacteristic
70#126×C71—22no—
71#126×C71—21no—
72#125×C71—21no—
73#125×C71C2311yescharacteristic
74#124×C71—21no—
75#124×C71—22no—
76#124×C71—22no—
77#124×C71—22no—
78#124×C71—21no—
79#124×C71—21no—
80#124×C71D412yes—
81#124×C71C2311yescharacteristic
82#124×C71C2312yes—
83#124×C71C2312yes—
84#123×C71—22no—
85#123×C71—22no—
86#123×C71—21no—
87#123×C71—22no—
88#123×C71—22no—
89C22⋊Q8×C71—21no—
90C22⋊Q8×C71—22no—
91C22⋊Q8×C71—22no—
92C22⋊Q8×C71—21no—
93C22⋊Q8×C71—21no—
94C22⋊Q8×C71D411yescharacteristic
95#121×C71—22no—
96#121×C71—22no—
97#121×C71—22no—
98#121×C71—21no—
99#121×C71—22no—
100#121×C71—22no—
101#121×C71—21no—
102#121×C71—21no—
103#121×C71D411yescharacteristic
104#120×C71—22no—
105#120×C71—22no—
106#120×C71—22no—
107#120×C71—22no—
108#120×C71—22no—
109C4×Q8×C71—21no—
110C4×Q8×C71—21no—
111C4×Q8×C71—21no—
112C4×Q8×C71D411yescharacteristic
113C4×D4×C71—22no—
114C4×D4×C71—22no—
115C4×D4×C71—21no—
116C4×D4×C71—21no—
117C4×D4×C71—21no—
118C4×D4×C71—21no—
119C4×D4×C71D411yescharacteristic
120#117×C71—21no—
121#117×C71—21no—
122C2×C22⋊C4×C71—22no—
123C2×C22⋊C4×C71—21no—
124C2×C22⋊C4×C71—21no—
125C2×C42×C71C2311yescharacteristic
126#107×C71—21no—
127#107×C71—21no—
128#106×C71—21no—
129#106×C71—21no—
130C4⋊C8×C71—21no—
131C4⋊C8×C71—21no—
132C4⋊C8×C71—21no—
133#103×C71—22no—
134#103×C71—22no—
135#103×C71—21no—
136#103×C71—21no—
137#102×C71—22no—
138#102×C71—22no—
139#102×C71—21no—
140#102×C71—21no—
141C22⋊C8×C71—21no—
142C22⋊C8×C71—21no—
143C22⋊C8×C71—21no—
144#97×C71C2312yes—
145#97×C71C2312yes—
146#97×C71C2311yescharacteristic
147#97×C71C2311yescharacteristic
148C24×C71—22no—
149C24×C71—21no—
150#41×C71—82no—
151#41×C71—82no—
152C2×Q8×C71—21no—
153C2×Q8×C71—21no—
154C2×Q8×C71—21no—
155C2×Q8×C71—81no—
156C2×Q8×C71C2×D411yescharacteristic
157C2×Q8×C71C2×D412yes—
158C2×Q8×C71C2×D412yes—
159C2×D4×C71—22no—
160C2×D4×C71—21no—
161C2×D4×C71—22no—
162C2×D4×C71—22no—
163C2×D4×C71—22no—
164C2×D4×C71—22no—
165C2×D4×C71—22no—
166C2×D4×C71—21no—
167C2×D4×C71—21no—
168C2×D4×C71—21no—
169C2×D4×C71—81no—
170C2×D4×C71—82no—
171C2×D4×C71—84no—
172C2×D4×C71C2×D411yescharacteristic
173C2×D4×C71C2×D412yes—
174C2×D4×C71C2×D412yes—
175C22×C4×C71—22no—
176C22×C4×C71—22no—
177C22×C4×C71—21no—
178C22×C4×C71—21no—
179C22×C4×C71—22no—
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183Q16×C71—82no—
184Q16×C71—82no—
185SD16×C71—82no—
186SD16×C71—82no—
187C2×C8×C71—21no—
188C2×C8×C71—21no—
189C2×C8×C71—22no—
190C2×C8×C71—22no—
191C4⋊C4×C71—22no—
192C4⋊C4×C71—22no—
193C4⋊C4×C71—21no—
194C4⋊C4×C71—21no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71—21no—
197C4⋊C4×C71—21no—
198C4⋊C4×C71—21no—
199C4⋊C4×C71—21no—
200C4⋊C4×C71—21no—
201C4⋊C4×C71—21no—
202C4⋊C4×C71—21no—
203C4⋊C4×C71—41no—
204C4⋊C4×C71—41no—
205C4⋊C4×C71C2×D411yescharacteristic
206C4⋊C4×C71C2×D411yescharacteristic
207C22⋊C4×C71—22no—
208C22⋊C4×C71—22no—
209C22⋊C4×C71—22no—
210C22⋊C4×C71—22no—
211C22⋊C4×C71—22no—
212C22⋊C4×C71—21no—
213C22⋊C4×C71—22no—
214C22⋊C4×C71—21no—
215C22⋊C4×C71—22no—
216C22⋊C4×C71—21no—
217C22⋊C4×C71—22no—
218C22⋊C4×C71—22no—
219C22⋊C4×C71—21no—
220C22⋊C4×C71—21no—
221C22⋊C4×C71—22no—
222C22⋊C4×C71—22no—
223C22⋊C4×C71—41no—
224C42×C71—22no—
225C42×C71—21no—
226C42×C71—41no—
227C42×C71C2×D411yescharacteristic
228C42×C71C2×D411yescharacteristic
229C42×C71C2×D411yescharacteristic
230C42×C71C2411yescharacteristic
231C23×C71—21no—
232C23×C71—22no—
233C23×C71—22no—
234C23×C71—42no—
235C23×C71—81no—
236C23×C71—81no—
237C23×C71—82no—
238C23×C71—82no—
239C23×C71—82no—
240C23×C71#12011yescharacteristic
241Q8×C71—42no—
242Q8×C71—42no—
243Q8×C71—41no—
244Q8×C71—42no—
245Q8×C71—41no—
246Q8×C71—42no—
247Q8×C71—41no—
248D4×C71—42no—
249D4×C71—42no—
250D4×C71—41no—
251D4×C71—42no—
252D4×C71—42no—
253D4×C71—42no—
254D4×C71—44no—
255D4×C71—81no—
256D4×C71—82no—
257D4×C71—82no—
258D4×C71—82no—
259D4×C71—82no—
260D4×C71—82no—
261C2×C4×C71—21no—
262C2×C4×C71—22no—
263C2×C4×C71—22no—
264C2×C4×C71—21no—
265C2×C4×C71—21no—
266C2×C4×C71—41no—
267C2×C4×C71—41no—
268C2×C4×C71—41no—
269C2×C4×C71—42no—
270C2×C4×C71—41no—
271C2×C4×C71—41no—
272C2×C4×C71—82no—
273C2×C4×C71—81no—
274C2×C4×C71—82no—
275C2×C4×C71—82no—
276C2×C4×C71#12011yescharacteristic
277C2×C4×C71#12011yescharacteristic
278C2×C4×C71#12011yescharacteristic
279C2×C4×C71C22×D411yescharacteristic
280C2×C4×C71C22×D411yescharacteristic
281C2×C4×C71C22×D411yescharacteristic
282C8×C71—41no—
283C8×C71—41no—
284C8×C71—42no—
285C8×C71—42no—
286C22×C71—21no—
287C22×C71—21no—
288C22×C71—21no—
289C22×C71—41no—
290C22×C71—42no—
291C22×C71—42no—
292C22×C71—42no—
293C22×C71—42no—
294C22×C71—41no—
295C22×C71—41no—
296C22×C71—42no—
297C22×C71—42no—
298C22×C71—162no—
299C22×C71—164no—
300C22×C71C2×#12011yescharacteristic
301C4×C71—21no—
302C4×C71—21no—
303C4×C71—21no—
304C4×C71—21no—
305C4×C71—21no—
306C4×C71—21no—
307C4×C71—41no—
308C4×C71—41no—
309C4×C71—42no—
310C4×C71—42no—
311C4×C71—41no—
312C4×C71—81no—
313C4×C71—81no—
314#19087C7111yesmaximal, characteristic, direct product factor, Sylow
315C2×C71—41no—
316C2×C71—82no—
317C2×C71—81no—
318C2×C71—82no—
319C2×C71#301911yescharacteristic
320C2×C71#301911yescharacteristic
321C2×C71#302211yescharacteristic
322#3558C2×C7111yescharacteristic
323#3240C2×C7111yescharacteristic, complemented
324#3216C2×C7111yescharacteristic, complemented
325#3215C2×C7111yescharacteristic, complemented
326#1661C2×C7112yescomplemented
327#1661C2×C7112yescomplemented
328#1659C2×C7112yescomplemented
329#1659C2×C7112yescomplemented
330#1536C2×C7111yescharacteristic, complemented
331#1535C2×C7111yescharacteristic, complemented
332#1533C2×C7111yescharacteristic, complemented
333C71#1908711yescharacteristic, direct product factor, Sylow
334#562C22×C7112yes—
335H(3,𝔽4)C22×C7112yes—
336#559—21no—
337#548—21no—
338#546—22no—
339#546—21no—
340D42—21no—
341#542—21no—
342#538—22no—
343#538—21no—
344#535—21no—
345#534—21no—
346#533C22×C7111yescharacteristic, complemented
347#532C22×C7111yescharacteristic
348#529C22×C7111yescharacteristic
349C2×#124C22×C7111yescharacteristic
350#501C22×C7111yescharacteristic, complemented
351#501C22×C7111yescharacteristic, complemented
352#497C22×C7112yescomplemented
353#497C22×C7112yescomplemented
354#489C22×C7111yescharacteristic, complemented
355#489C22×C7111yescharacteristic, complemented
356#489C22×C7112yescomplemented
357#489C22×C7112yescomplemented
358#485—21no—
359#482—21no—
360#479—21no—
361#478—21no—
362#464—22no—
363#462—21no—
364#460—21no—
365#452—22no—
366#451—21no—
367#450—21no—
368#432C22×C7111yescharacteristic, complemented
369#432C22×C7111yescharacteristic, complemented
370#432C22×C7111yescharacteristic, complemented
371#330C22×C7112yescomplemented
372#329C22×C7112yescomplemented
373#329C22×C7112yescomplemented
374#329C22×C7112yescomplemented
375#329C22×C7112yescomplemented
376#329C22×C7112yescomplemented
377C2×#41—21no—
378C2×#41D4×C7112yes—
379C22×Q8D4×C7111yescharacteristic
380C22×D4—22no—
381C22×D4—21no—
382C22×D4D4×C7111yescharacteristic
383C2×Q16—22no—
384C2×Q16—22no—
385C2×SD16—22no—
386C2×SD16—22no—
387C4⋊Q8D4×C7111yescharacteristic
388C4⋊Q8C23×C7111yescharacteristic
389C4⋊Q8C23×C7111yescharacteristic
390#127—21no—
391#127D4×C7111yescharacteristic
392#126—22no—
393#126—21no—
394#125—21no—
395#125C23×C7111yescharacteristic
396#124—21no—
397#124—22no—
398#124—22no—
399#124—22no—
400#124—21no—
401#124—21no—
402#124D4×C7112yes—
403#124C23×C7112yes—
404#124C23×C7111yescharacteristic
405#124C23×C7112yes—
406#123—22no—
407#123—22no—
408#123—21no—
409#123—22no—
410#123—22no—
411C22⋊Q8—21no—
412C22⋊Q8—22no—
413C22⋊Q8—22no—
414C22⋊Q8—21no—
415C22⋊Q8—21no—
416C22⋊Q8D4×C7111yescharacteristic
417#121—22no—
418#121—22no—
419#121—22no—
420#121—21no—
421#121—22no—
422#121—22no—
423#121—21no—
424#121—21no—
425#121D4×C7111yescharacteristic
426#120—22no—
427#120—22no—
428#120—22no—
429#120—22no—
430#120—22no—
431C4×Q8—21no—
432C4×Q8—21no—
433C4×Q8—21no—
434C4×Q8D4×C7111yescharacteristic
435C4×D4—22no—
436C4×D4—22no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4—21no—
441C4×D4D4×C7111yescharacteristic
442#117—21no—
443#117—21no—
444C2×C22⋊C4—22no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C42C23×C7111yescharacteristic
448#107—21no—
449#107—21no—
450#106—21no—
451#106—21no—
452C4⋊C8—21no—
453C4⋊C8—21no—
454C4⋊C8—21no—
455#103—22no—
456#103—22no—
457#103—21no—
458#103—21no—
459#102—22no—
460#102—22no—
461#102—21no—
462#102—21no—
463C22⋊C8—21no—
464C22⋊C8—21no—
465C22⋊C8—21no—
466#97C23×C7111yescharacteristic
467#97C23×C7111yescharacteristic
468#97C23×C7112yes—
469#97C23×C7112yes—
470C24—22no—
471C24—21no—
472#41—82no—
473#41—82no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—81no—
478C2×Q8C2×D4×C7111yescharacteristic
479C2×Q8C2×D4×C7112yes—
480C2×Q8C2×D4×C7112yes—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—81no—
492C2×D4—82no—
493C2×D4—84no—
494C2×D4C2×D4×C7111yescharacteristic
495C2×D4C2×D4×C7112yes—
496C2×D4C2×D4×C7112yes—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4C2×D4×C7111yescharacteristic
503C22×C4C2×D4×C7111yescharacteristic
504C22×C4C2×D4×C7111yescharacteristic
505Q16—82no—
506Q16—82no—
507SD16—82no—
508SD16—82no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—22no—
512C2×C8—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4C2×D4×C7111yescharacteristic
528C4⋊C4C2×D4×C7111yescharacteristic
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—21no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—41no—
546C42—22no—
547C42—21no—
548C42—41no—
549C42C2×D4×C7111yescharacteristic
550C42C2×D4×C7111yescharacteristic
551C42C2×D4×C7111yescharacteristic
552C42C24×C7111yescharacteristic
553C23—21no—
554C23—22no—
555C23—22no—
556C23—42no—
557C23—81no—
558C23—81no—
559C23—82no—
560C23—82no—
561C23—82no—
562C23#120×C7111yescharacteristic
563Q8—42no—
564Q8—42no—
565Q8—41no—
566Q8—42no—
567Q8—41no—
568Q8—42no—
569Q8—41no—
570D4—42no—
571D4—42no—
572D4—41no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—44no—
577D4—81no—
578D4—82no—
579D4—82no—
580D4—82no—
581D4—82no—
582D4—82no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4#120×C7111yescharacteristic
599C2×C4#120×C7111yescharacteristic
600C2×C4#120×C7111yescharacteristic
601C2×C4C22×D4×C7111yescharacteristic
602C2×C4C22×D4×C7111yescharacteristic
603C2×C4C22×D4×C7111yescharacteristic
604C8—42no—
605C8—42no—
606C8—41no—
607C8—41no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—162no—
621C22—164no—
622C22C2×#120×C7111yescharacteristic
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—42no—
633C4—41no—
634C4—81no—
635C4—81no—
636C2—41no—
637C2—81no—
638C2—82no—
639C2—82no—
640C2C71×#301911yescharacteristic
641C2C71×#301911yescharacteristic
642C2C71×#302211yescharacteristic
subgroupG/H
center, socleC22×C71C2×#120
commutator, Frattini, cosocleC42C24×C71
prime radicalC71×#3558C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#3240C2semidirect
3. C71×#3216C2semidirect
4. C71×#3215C2semidirect
5. C71×#1661C2semidirect
6. C71×#1661C2semidirect
7. C71×#1659C2semidirect
8. C71×#1659C2semidirect
9. C71×#1536C2semidirect
10. C71×#1535C2semidirect
11. C71×#1533C2semidirect
24. #533×C71C22semidirect
28. #501×C71C22semidirect
29. #501×C71C22semidirect
30. #497×C71C22semidirect
31. #497×C71C22semidirect
32. #489×C71C22semidirect
33. #489×C71C22semidirect
34. #489×C71C22semidirect
35. #489×C71C22semidirect
46. #432×C71C22semidirect
47. #432×C71C22semidirect
48. #432×C71C22semidirect
49. #330×C71C22semidirect
50. #329×C71C22semidirect
51. #329×C71C22semidirect
52. #329×C71C22semidirect
53. #329×C71C22semidirect
54. #329×C71C22semidirect
314. #19087C71direct (N × H)
323. #3240C2×C71semidirect
324. #3216C2×C71semidirect
325. #3215C2×C71semidirect
326. #1661C2×C71semidirect
327. #1661C2×C71semidirect
328. #1659C2×C71semidirect
329. #1659C2×C71semidirect
330. #1536C2×C71semidirect
331. #1535C2×C71semidirect
332. #1533C2×C71semidirect
333. C71#19087direct (N × H)
346. #533C22×C71semidirect
350. #501C22×C71semidirect
351. #501C22×C71semidirect
352. #497C22×C71semidirect
353. #497C22×C71semidirect
354. #489C22×C71semidirect
355. #489C22×C71semidirect
356. #489C22×C71semidirect
357. #489C22×C71semidirect
368. #432C22×C71semidirect
369. #432C22×C71semidirect
370. #432C22×C71semidirect
371. #330C22×C71semidirect
372. #329C22×C71semidirect
373. #329C22×C71semidirect
374. #329C22×C71semidirect
375. #329C22×C71semidirect
376. #329C22×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#35582
C71×#32402
C71×#32162
C71×#32152
C71×#16612
C71×#16592
C71×#15362
C71×#15352
C71×#15332
#1908771