Finite Group Databasebeta

← C71×#14506 #31835378

Subgroups — C71×#14506 #31835378

Order 18176.

1962 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123×C71C211yesmaximal, characteristic
2C2×#388×C71C212yesmaximal, complemented
3C71×#1445C212yesmaximal, complemented
4C71×#1439C212yesmaximal, complemented
5C71×#1438C211yesmaximal, characteristic
6C71×#1438C211yesmaximal, characteristic
7C71×#1437C212yesmaximal, complemented
8C71×C4⋊#95C212yesmaximal, complemented
9C71×C22⋊#95C212yesmaximal, complemented
10C23×D4×C71C2211yescharacteristic
11C24×C4×C71C2211yescharacteristic
12C2×#123×C71—28no—
13C2×#123×C71C418yescomplemented
14C22×C4⋊C4×C71C2212yes—
15C22×C4⋊C4×C71C2212yes—
16C22×C22⋊C4×C71C2212yes—
17C22×C22⋊C4×C71C2212yes—
18C22×C22⋊C4×C71C2211yescharacteristic
19C22×C22⋊C4×C71C2212yes—
20C22×C22⋊C4×C71C2212yes—
21C22×C22⋊C4×C71C2211yescharacteristic
22C22×C22⋊C4×C71C2211yescharacteristic
23C22×C42×C71C2212yes—
24#388×C71—28no—
25C2×#95×C71C2212yes—
26C2×#95×C71C2212yes—
27C2×#95×C71C2212yes—
28C2×#95×C71C2212yes—
29C2×#95×C71C2212yes—
30C2×#95×C71C2212yes—
31C2×#95×C71C2211yescharacteristic
32C2×#95×C71C2212yes—
33C2×#95×C71C2211yescharacteristic
34C25×C71C2311yescharacteristic
35C25×C71C2311yescharacteristic
36C22×D4×C71—24no—
37C22×D4×C71—42no—
38C22×D4×C71—41no—
39C22×D4×C71—41no—
40C22×D4×C71C2×C414yes—
41C23×C4×C71—24no—
42C23×C4×C71—42no—
43C23×C4×C71—41no—
44C23×C4×C71—41no—
45C23×C4×C71C2×C414yes—
46C23×C4×C71C2312yes—
47C23×C4×C71C2312yes—
48C23×C4×C71C2312yes—
49C23×C4×C71C2311yescharacteristic
50C23×C4×C71C2312yes—
51C23×C4×C71C2312yes—
52C23×C4×C71C2312yes—
53#123×C71—232no—
54C2×C4⋊C4×C71—24no—
55C2×C4⋊C4×C71—24no—
56C2×C4⋊C4×C71—24no—
57C2×C4⋊C4×C71—24no—
58C2×C4⋊C4×C71—42no—
59C2×C4⋊C4×C71C2×C418yes—
60C2×C22⋊C4×C71—24no—
61C2×C22⋊C4×C71—24no—
62C2×C22⋊C4×C71—24no—
63C2×C22⋊C4×C71—24no—
64C2×C22⋊C4×C71—24no—
65C2×C22⋊C4×C71—24no—
66C2×C22⋊C4×C71—24no—
67C2×C22⋊C4×C71—24no—
68C2×C22⋊C4×C71—41no—
69C2×C22⋊C4×C71—42no—
70C2×C22⋊C4×C71—42no—
71C2×C22⋊C4×C71—42no—
72C2×C22⋊C4×C71—42no—
73C2×C22⋊C4×C71—41no—
74C2×C22⋊C4×C71—42no—
75C2×C22⋊C4×C71C2×C418yes—
76C2×C22⋊C4×C71C2×C414yes—
77C2×C22⋊C4×C71D418yes—
78C2×C42×C71—24no—
79C2×C42×C71—24no—
80C2×C42×C71—42no—
81#95×C71—24no—
82#95×C71—24no—
83#95×C71—41no—
84#95×C71—42no—
85#95×C71—42no—
86#95×C71—42no—
87#95×C71—42no—
88#95×C71—42no—
89#95×C71—41no—
90C24×C71—22no—
91C24×C71—22no—
92C24×C71—22no—
93C24×C71—22no—
94C24×C71—21no—
95C24×C71—21no—
96C24×C71—22no—
97C24×C71—22no—
98C24×C71—22no—
99C24×C71—22no—
100C24×C71—22no—
101C24×C71—22no—
102C24×C71—21no—
103C24×C71—21no—
104C24×C71—22no—
105C24×C71C22×C412yes—
106C24×C71C22×C412yes—
107C24×C71C2×D414yes—
108C24×C71C2411yescharacteristic
109C2×D4×C71—28no—
110C2×D4×C71—44no—
111C2×D4×C71—44no—
112C2×D4×C71—48no—
113C2×D4×C71—48no—
114C22×C4×C71—22no—
115C22×C4×C71—22no—
116C22×C4×C71—22no—
117C22×C4×C71—22no—
118C22×C4×C71—28no—
119C22×C4×C71—22no—
120C22×C4×C71—22no—
121C22×C4×C71—22no—
122C22×C4×C71—22no—
123C22×C4×C71—22no—
124C22×C4×C71—22no—
125C22×C4×C71—22no—
126C22×C4×C71—22no—
127C22×C4×C71—22no—
128C22×C4×C71—22no—
129C22×C4×C71—22no—
130C22×C4×C71—22no—
131C22×C4×C71—22no—
132C22×C4×C71—22no—
133C22×C4×C71—22no—
134C22×C4×C71—22no—
135C22×C4×C71—21no—
136C22×C4×C71—21no—
137C22×C4×C71—22no—
138C22×C4×C71—22no—
139C22×C4×C71—22no—
140C22×C4×C71—22no—
141C22×C4×C71—22no—
142C22×C4×C71—22no—
143C22×C4×C71—22no—
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—22no—
147C22×C4×C71—22no—
148C22×C4×C71—22no—
149C22×C4×C71—22no—
150C22×C4×C71—22no—
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—22no—
154C22×C4×C71—22no—
155C22×C4×C71—44no—
156C22×C4×C71—44no—
157C22×C4×C71—48no—
158C22×C4×C71—48no—
159C22×C4×C71C22×C414yes—
160C22×C4×C71C22×C412yes—
161C22×C4×C71C22×C414yes—
162C22×C4×C71C2×D414yes—
163C22×C4×C71C2×D414yes—
164C22×C4×C71#4114yes—
165C22×C4×C71#4114yes—
166C4⋊C4×C71—216no—
167C4⋊C4×C71—48no—
168C22⋊C4×C71—216no—
169C22⋊C4×C71—44no—
170C22⋊C4×C71—48no—
171C22⋊C4×C71—48no—
172C22⋊C4×C71—48no—
173C22⋊C4×C71—44no—
174C22⋊C4×C71C22⋊C4116yescomplemented
175C42×C71—48no—
176C23×C71—24no—
177C23×C71—22no—
178C23×C71—22no—
179C23×C71—24no—
180C23×C71—24no—
181C23×C71—24no—
182C23×C71—24no—
183C23×C71—24no—
184C23×C71—24no—
185C23×C71—24no—
186C23×C71—22no—
187C23×C71—24no—
188C23×C71—22no—
189C23×C71—22no—
190C23×C71—24no—
191C23×C71—42no—
192C23×C71—42no—
193C23×C71—42no—
194C23×C71—42no—
195C23×C71—41no—
196C23×C71—42no—
197C23×C71—41no—
198C23×C71—41no—
199C23×C71—42no—
200C23×C71—41no—
201C23×C71—42no—
202C23×C71—42no—
203C23×C71—42no—
204C23×C71—42no—
205C23×C71—42no—
206C23×C71—42no—
207C23×C71—42no—
208C23×C71—42no—
209C23×C71—42no—
210C23×C71—42no—
211C23×C71—41no—
212C23×C71—42no—
213C23×C71—41no—
214C23×C71—41no—
215C23×C71—42no—
216C23×C71—41no—
217C23×C71C2×C22⋊C414yes—
218C23×C71C23×C411yescharacteristic
219C23×C71C22×D412yes—
220C23×C71C2×#4112yes—
221C23×C71C2×#4112yes—
222C23×C71H(5,𝔽2)12yes—
223C23×C71H(5,𝔽2)11yescharacteristic
224C23×C71H(5,𝔽2)11yescharacteristic
225C23×C71H(5,𝔽2)12yes—
226C23×C712−1+412yes—
227D4×C71—416no—
228C2×C4×C71—24no—
229C2×C4×C71—24no—
230C2×C4×C71—24no—
231C2×C4×C71—24no—
232C2×C4×C71—24no—
233C2×C4×C71—24no—
234C2×C4×C71—24no—
235C2×C4×C71—22no—
236C2×C4×C71—24no—
237C2×C4×C71—24no—
238C2×C4×C71—22no—
239C2×C4×C71—22no—
240C2×C4×C71—24no—
241C2×C4×C71—24no—
242C2×C4×C71—24no—
243C2×C4×C71—24no—
244C2×C4×C71—24no—
245C2×C4×C71—24no—
246C2×C4×C71—416no—
247C2×C4×C71—42no—
248C2×C4×C71—42no—
249C2×C4×C71—42no—
250C2×C4×C71—42no—
251C2×C4×C71—42no—
252C2×C4×C71—42no—
253C2×C4×C71—42no—
254C2×C4×C71—42no—
255C2×C4×C71—42no—
256C2×C4×C71—42no—
257C2×C4×C71—42no—
258C2×C4×C71—42no—
259C2×C4×C71—42no—
260C2×C4×C71—42no—
261C2×C4×C71—42no—
262C2×C4×C71—42no—
263C2×C4×C71—42no—
264C2×C4×C71—42no—
265C2×C4×C71—42no—
266C2×C4×C71—42no—
267C2×C4×C71—42no—
268C2×C4×C71—42no—
269C2×C4×C71—42no—
270C2×C4×C71—42no—
271C2×C4×C71—42no—
272C2×C4×C71—42no—
273C2×C4×C71—42no—
274C2×C4×C71—42no—
275C2×C4×C71C2×C22⋊C418yes—
276C22×C71—24no—
277C22×C71—28no—
278C22×C71—44no—
279C22×C71—42no—
280C22×C71—42no—
281C22×C71—44no—
282C22×C71—44no—
283C22×C71—44no—
284C22×C71—44no—
285C22×C71—42no—
286C22×C71—44no—
287C22×C71—44no—
288C22×C71—42no—
289C22×C71—42no—
290C22×C71—82no—
291C22×C71—82no—
292C22×C71—82no—
293C22×C71—82no—
294C22×C71C22×C22⋊C411yescharacteristic
295C22×C71C4×#4112yes—
296C22×C71#51811yescharacteristic
297C22×C71#51811yescharacteristic
298C22×C71#52012yes—
299C22×C71#53512yes—
300C22×C71#53512yes—
301C22×C71#53812yes—
302C22×C71#53912yes—
303C22×C71#54012yes—
304C22×C71#54212yes—
305C22×C71#54612yes—
306C22×C71#54612yes—
307C22×C71#54712yes—
308C22×C71#54712yes—
309C22×C71#55111yescharacteristic
310C22×C71#55312yes—
311C22×C71#55911yescharacteristic
312C22×C71#56011yescharacteristic
313C22×C71#56212yes—
314C22×C71#56311yescharacteristic
315C4×C71—28no—
316C4×C71—44no—
317C4×C71—42no—
318C4×C71—44no—
319C4×C71—44no—
320C4×C71—82no—
321C4×C71—82no—
322#14506C7111yesmaximal, characteristic, direct product factor, Sylow
323C2×C71—44no—
324C2×C71—82no—
325C2×C71#230911yescharacteristic
326C2×C71#233512yes—
327C2×C71#233912yes—
328C2×C71#237411yescharacteristic
329C2×C71#237611yescharacteristic
330C2×C71#270312yes—
331C2×C71#270612yes—
332C2×C71#271012yes—
333C2×C71#271812yes—
334C22×#123C2×C7111yescharacteristic
335C2×#388C2×C7112yescomplemented
336#1445C2×C7112yescomplemented
337#1439C2×C7112yescomplemented
338#1438C2×C7111yescharacteristic
339#1438C2×C7111yescharacteristic
340#1437C2×C7112yescomplemented
341C4⋊#95C2×C7112yescomplemented
342C22⋊#95C2×C7112yescomplemented
343C71#1450611yescharacteristic, direct product factor, Sylow
344C23×D4C22×C7111yescharacteristic
345C24×C4C22×C7111yescharacteristic
346C2×#123—28no—
347C2×#123C4×C7118yescomplemented
348C22×C4⋊C4C22×C7112yes—
349C22×C4⋊C4C22×C7112yes—
350C22×C22⋊C4C22×C7112yes—
351C22×C22⋊C4C22×C7112yes—
352C22×C22⋊C4C22×C7111yescharacteristic
353C22×C22⋊C4C22×C7112yes—
354C22×C22⋊C4C22×C7112yes—
355C22×C22⋊C4C22×C7111yescharacteristic
356C22×C22⋊C4C22×C7111yescharacteristic
357C22×C42C22×C7112yes—
358#388—28no—
359C2×#95C22×C7112yes—
360C2×#95C22×C7112yes—
361C2×#95C22×C7112yes—
362C2×#95C22×C7112yes—
363C2×#95C22×C7112yes—
364C2×#95C22×C7112yes—
365C2×#95C22×C7111yescharacteristic
366C2×#95C22×C7112yes—
367C2×#95C22×C7111yescharacteristic
368C25C23×C7111yescharacteristic
369C25C23×C7111yescharacteristic
370C22×D4—24no—
371C22×D4—42no—
372C22×D4—41no—
373C22×D4—41no—
374C22×D4C2×C4×C7114yes—
375C23×C4—24no—
376C23×C4—42no—
377C23×C4—41no—
378C23×C4—41no—
379C23×C4C2×C4×C7114yes—
380C23×C4C23×C7112yes—
381C23×C4C23×C7112yes—
382C23×C4C23×C7112yes—
383C23×C4C23×C7111yescharacteristic
384C23×C4C23×C7112yes—
385C23×C4C23×C7112yes—
386C23×C4C23×C7112yes—
387#123—232no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4C2×C4×C7118yes—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—41no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—41no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4C2×C4×C7118yes—
410C2×C22⋊C4C2×C4×C7114yes—
411C2×C22⋊C4D4×C7118yes—
412C2×C42—24no—
413C2×C42—24no—
414C2×C42—42no—
415#95—24no—
416#95—24no—
417#95—41no—
418#95—42no—
419#95—42no—
420#95—42no—
421#95—42no—
422#95—42no—
423#95—41no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—21no—
429C24—21no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—21no—
437C24—21no—
438C24—22no—
439C24C22×C4×C7112yes—
440C24C22×C4×C7112yes—
441C24C2×D4×C7114yes—
442C24C24×C7111yescharacteristic
443C2×D4—28no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—48no—
447C2×D4—48no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4C22×C4×C7114yes—
494C22×C4C22×C4×C7112yes—
495C22×C4C22×C4×C7114yes—
496C22×C4C2×D4×C7114yes—
497C22×C4C2×D4×C7114yes—
498C22×C4#41×C7114yes—
499C22×C4#41×C7114yes—
500C4⋊C4—216no—
501C4⋊C4—48no—
502C22⋊C4—216no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4C22⋊C4×C71116yescomplemented
509C42—48no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23C2×C22⋊C4×C7114yes—
552C23C23×C4×C7111yescharacteristic
553C23C22×D4×C7112yes—
554C23C2×#41×C7112yes—
555C23C2×#41×C7112yes—
556C23H(5,𝔽2)×C7112yes—
557C23H(5,𝔽2)×C7111yescharacteristic
558C23H(5,𝔽2)×C7111yescharacteristic
559C23H(5,𝔽2)×C7112yes—
560C232−1+4×C7112yes—
561D4—416no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4C2×C22⋊C4×C7118yes—
610C22—24no—
611C22—28no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C22×C22⋊C4×C7111yescharacteristic
629C22C4×#41×C7112yes—
630C22#518×C7111yescharacteristic
631C22#518×C7111yescharacteristic
632C22#520×C7112yes—
633C22#535×C7112yes—
634C22#535×C7112yes—
635C22#538×C7112yes—
636C22#539×C7112yes—
637C22#540×C7112yes—
638C22#542×C7112yes—
639C22#546×C7112yes—
640C22#546×C7112yes—
641C22#547×C7112yes—
642C22#547×C7112yes—
643C22#551×C7111yescharacteristic
644C22#553×C7112yes—
645C22#559×C7111yescharacteristic
646C22#560×C7111yescharacteristic
647C22#562×C7112yes—
648C22#563×C7111yescharacteristic
649C4—28no—
650C4—44no—
651C4—42no—
652C4—44no—
653C4—44no—
654C4—82no—
655C4—82no—
656C2—44no—
657C2—82no—
658C2C71×#230911yescharacteristic
659C2C71×#233512yes—
660C2C71×#233912yes—
661C2C71×#237411yescharacteristic
662C2C71×#237611yescharacteristic
663C2C71×#270312yes—
664C2C71×#270612yes—
665C2C71×#271012yes—
666C2C71×#271812yes—
subgroupG/H
center, socleC24×C71C24
commutatorC23C23×C4×C71
Frattini, cosocleC24C24×C71
prime radicalC23×D4×C71C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#388×C71C2semidirect
3. C71×#1445C2semidirect
4. C71×#1439C2semidirect
7. C71×#1437C2semidirect
8. C71×C4⋊#95C2semidirect
9. C71×C22⋊#95C2semidirect
13. C2×#123×C71C4semidirect
174. C22⋊C4×C71C22⋊C4semidirect
322. #14506C71direct (N × H)
335. C2×#388C2×C71semidirect
336. #1445C2×C71semidirect
337. #1439C2×C71semidirect
340. #1437C2×C71semidirect
341. C4⋊#95C2×C71semidirect
342. C22⋊#95C2×C71semidirect
343. C71#14506direct (N × H)
347. C2×#123C4×C71semidirect
508. C22⋊C4C22⋊C4×C71semidirect

Maximal subgroups

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

subgroupindex
C22×#123×C712
C2×#388×C712
C71×#14452
C71×#14392
C71×#14382
C71×#14372
C71×C4⋊#952
C71×C22⋊#952
#1450671