Finite Group Databasebeta

← C71×#13877 #31834749

Subgroups — C71×#13877 #31834749

Order 18176.

4046 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C4×C71C211yesmaximal, characteristic
2C23×C22⋊C4×C71C211yesmaximal, characteristic
3C2×#385×C71C212yesmaximal, complemented
4C2×#380×C71C212yesmaximal, complemented
5C2×#379×C71C211yesmaximal, characteristic
6C71×#1439C214yesmaximal, complemented
7C71×C22⋊#95C212yesmaximal, complemented
8C71×C22⋊#95C212yesmaximal, complemented
9C26×C71C2211yescharacteristic
10C24×C4×C71C2212yes—
11C24×C4×C71C2212yes—
12C22×C4⋊C4×C71C2214yescomplemented
13C22×C22⋊C4×C71—24no—
14C22×C22⋊C4×C71—28no—
15C22×C22⋊C4×C71—28no—
16C22×C22⋊C4×C71C418yescomplemented
17C22×C22⋊C4×C71C2211yescharacteristic
18C22×C22⋊C4×C71C2212yes—
19C22×C22⋊C4×C71C2211yescharacteristic
20C22×C22⋊C4×C71C2212yes—
21C22×C22⋊C4×C71C2211yescharacteristic
22C22×C22⋊C4×C71C2212yes—
23C22×C22⋊C4×C71C2212yes—
24C22×C22⋊C4×C71C2212yes—
25C22×C22⋊C4×C71C2211yescharacteristic
26#385×C71—28no—
27#380×C71—28no—
28#379×C71—24no—
29C2×#95×C71C2214yescomplemented
30C2×#95×C71C2214yescomplemented
31C2×#95×C71C2214yescomplemented
32C25×C71—24no—
33C25×C71—24no—
34C25×C71—24no—
35C25×C71—24no—
36C25×C71—44no—
37C25×C71C2×C414yes—
38C25×C71D414yes—
39C25×C71Q814yes—
40C25×C71C2311yescharacteristic
41C25×C71C2311yescharacteristic
42C25×C71C2311yescharacteristic
43C23×C4×C71—24no—
44C23×C4×C71—24no—
45C23×C4×C71—24no—
46C23×C4×C71—28no—
47C23×C4×C71—28no—
48C23×C4×C71—24no—
49C23×C4×C71—24no—
50C23×C4×C71—44no—
51C23×C4×C71—44no—
52C23×C4×C71C2×C418yescomplemented
53C23×C4×C71C2312yes—
54C23×C4×C71C2312yes—
55C23×C4×C71C2312yes—
56C23×C4×C71C2314yes—
57C23×C4×C71C2312yes—
58C2×C4⋊C4×C71—28no—
59C2×C4⋊C4×C71—28no—
60C2×C4⋊C4×C71—44no—
61C2×C22⋊C4×C71—232no—
62C2×C22⋊C4×C71—24no—
63C2×C22⋊C4×C71—24no—
64C2×C22⋊C4×C71—24no—
65C2×C22⋊C4×C71—24no—
66C2×C22⋊C4×C71—28no—
67C2×C22⋊C4×C71—28no—
68C2×C22⋊C4×C71—232no—
69C2×C22⋊C4×C71—24no—
70C2×C22⋊C4×C71—22no—
71C2×C22⋊C4×C71—24no—
72C2×C22⋊C4×C71—22no—
73C2×C22⋊C4×C71—24no—
74C2×C22⋊C4×C71—24no—
75C2×C22⋊C4×C71—24no—
76C2×C22⋊C4×C71—24no—
77C2×C22⋊C4×C71—44no—
78C2×C22⋊C4×C71C2×C414yes—
79C2×C22⋊C4×C71C2×C414yes—
80C2×C22⋊C4×C71C2×C418yescomplemented
81C2×C22⋊C4×C71D418yes—
82C2×C22⋊C4×C71D418yes—
83#95×C71—28no—
84#95×C71—28no—
85#95×C71—44no—
86C24×C71—22no—
87C24×C71—22no—
88C24×C71—22no—
89C24×C71—22no—
90C24×C71—24no—
91C24×C71—24no—
92C24×C71—28no—
93C24×C71—28no—
94C24×C71—216no—
95C24×C71—28no—
96C24×C71—216no—
97C24×C71—216no—
98C24×C71—216no—
99C24×C71—28no—
100C24×C71—22no—
101C24×C71—22no—
102C24×C71—22no—
103C24×C71—22no—
104C24×C71—24no—
105C24×C71—48no—
106C24×C71—48no—
107C24×C71—48no—
108C24×C71—416no—
109C24×C71—416no—
110C24×C71—416no—
111C24×C71—416no—
112C24×C71C4⋊C4116yescomplemented
113C24×C71C22×C412yes—
114C24×C71C22×C412yes—
115C24×C71C22×C412yes—
116C24×C71C2×D412yes—
117C24×C71C2×D414yes—
118C24×C71C2×D412yes—
119C24×C71C2×D412yes—
120C24×C71C2×D414yes—
121C24×C71C2×Q812yes—
122C24×C71C2×Q812yes—
123C24×C71C2×Q812yes—
124C24×C71#4114yes—
125C24×C71#4114yes—
126C24×C71C2411yescharacteristic
127C22×C4×C71—22no—
128C22×C4×C71—22no—
129C22×C4×C71—24no—
130C22×C4×C71—216no—
131C22×C4×C71—216no—
132C22×C4×C71—22no—
133C22×C4×C71—24no—
134C22×C4×C71—24no—
135C22×C4×C71—24no—
136C22×C4×C71—24no—
137C22×C4×C71—22no—
138C22×C4×C71—24no—
139C22×C4×C71—24no—
140C22×C4×C71—24no—
141C22×C4×C71—24no—
142C22×C4×C71—216no—
143C22×C4×C71—22no—
144C22×C4×C71—24no—
145C22×C4×C71—22no—
146C22×C4×C71—22no—
147C22×C4×C71—22no—
148C22×C4×C71—22no—
149C22×C4×C71—22no—
150C22×C4×C71—24no—
151C22×C4×C71—24no—
152C22×C4×C71—24no—
153C22×C4×C71—48no—
154C22×C4×C71—416no—
155C22×C4×C71—416no—
156C22×C4×C71—416no—
157C22×C4×C71—416no—
158C22×C4×C71—48no—
159C22×C4×C71—48no—
160C22×C4×C71C22×C414yes—
161C22×C4×C71C22×C414yes—
162C22×C4×C71C2×D418yes—
163C22×C4×C71C2×D418yes—
164C4⋊C4×C71—416no—
165C22⋊C4×C71—216no—
166C22⋊C4×C71—28no—
167C22⋊C4×C71—216no—
168C22⋊C4×C71—216no—
169C22⋊C4×C71—48no—
170C22⋊C4×C71—44no—
171C22⋊C4×C71—48no—
172C22⋊C4×C71—44no—
173C22⋊C4×C71C22⋊C4116yescomplemented
174C23×C71—22no—
175C23×C71—22no—
176C23×C71—24no—
177C23×C71—24no—
178C23×C71—24no—
179C23×C71—22no—
180C23×C71—24no—
181C23×C71—24no—
182C23×C71—24no—
183C23×C71—22no—
184C23×C71—24no—
185C23×C71—24no—
186C23×C71—24no—
187C23×C71—24no—
188C23×C71—28no—
189C23×C71—28no—
190C23×C71—28no—
191C23×C71—28no—
192C23×C71—28no—
193C23×C71—28no—
194C23×C71—28no—
195C23×C71—28no—
196C23×C71—22no—
197C23×C71—22no—
198C23×C71—24no—
199C23×C71—24no—
200C23×C71—22no—
201C23×C71—24no—
202C23×C71—24no—
203C23×C71—24no—
204C23×C71—22no—
205C23×C71—24no—
206C23×C71—24no—
207C23×C71—232no—
208C23×C71—232no—
209C23×C71—416no—
210C23×C71—432no—
211C23×C71—432no—
212C23×C71—432no—
213C23×C71—432no—
214C23×C71—464no—
215C23×C71—44no—
216C23×C71—44no—
217C23×C71—44no—
218C23×C71—44no—
219C23×C71C2×C22⋊C414yes—
220C23×C71C2×C4⋊C414yes—
221C23×C71C2×C4⋊C414yes—
222C23×C71C2×C4⋊C414yes—
223C23×C71#11714yes—
224C23×C71C4×D418yes—
225C23×C71#12014yes—
226C23×C71#12118yes—
227C23×C71C22⋊Q818yes—
228C23×C71C22⋊Q818yes—
229C23×C71#12314yes—
230C23×C71C23×C411yescharacteristic
231C23×C71C22×D412yes—
232C23×C71C22×D411yescharacteristic
233C23×C71C22×D412yes—
234C23×C71C22×Q811yescharacteristic
235C23×C71C2×#4112yes—
236C23×C71C2×#4112yes—
237C23×C71H(5,𝔽2)14yes—
238C2×C4×C71—24no—
239C2×C4×C71—24no—
240C2×C4×C71—28no—
241C2×C4×C71—28no—
242C2×C4×C71—24no—
243C2×C4×C71—24no—
244C2×C4×C71—24no—
245C2×C4×C71—28no—
246C2×C4×C71—28no—
247C2×C4×C71—28no—
248C2×C4×C71—432no—
249C2×C4×C71—432no—
250C2×C4×C71—44no—
251C2×C4×C71—44no—
252C2×C4×C71—44no—
253C2×C4×C71—44no—
254C2×C4×C71—44no—
255C2×C4×C71—44no—
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—44no—
263C2×C4×C71—44no—
264C2×C4×C71—44no—
265C2×C4×C71—44no—
266C2×C4×C71C2×C22⋊C418yescomplemented
267C22×C71—24no—
268C22×C71—24no—
269C22×C71—28no—
270C22×C71—28no—
271C22×C71—28no—
272C22×C71—28no—
273C22×C71—28no—
274C22×C71—28no—
275C22×C71—28no—
276C22×C71—28no—
277C22×C71—216no—
278C22×C71—216no—
279C22×C71—24no—
280C22×C71—24no—
281C22×C71—24no—
282C22×C71—24no—
283C22×C71—24no—
284C22×C71—464no—
285C22×C71—44no—
286C22×C71—44no—
287C22×C71—44no—
288C22×C71—44no—
289C22×C71—48no—
290C22×C71#38018yescomplemented
291C22×C71#38518yescomplemented
292C22×C71C22×C22⋊C411yescharacteristic
293C22×C71C22×C4⋊C411yescharacteristic
294C22×C71C2×#11711yescharacteristic
295C22×C71C2×C4×D412yes—
296C22×C71#51812yes—
297C22×C71C2×#12011yescharacteristic
298C22×C71C2×#12112yes—
299C22×C71C2×C22⋊Q812yes—
300C22×C71C2×C22⋊Q812yes—
301C22×C71C2×#12311yescharacteristic
302C22×C71#53412yes—
303C22×C71#54312yes—
304C22×C71D4214yes—
305C22×C71#54614yes—
306C22×C71#54614yes—
307C22×C71#55114yes—
308C4×C71—28no—
309C4×C71—44no—
310C4×C71—48no—
311C4×C71—82no—
312C4×C71—82no—
313#13877C7111yesmaximal, characteristic, direct product factor, Sylow
314C2×C71—28no—
315C2×C71—28no—
316C2×C71—44no—
317C2×C71C2×#38011yescharacteristic
318C2×C71C2×#38511yescharacteristic
319C2×C71#231511yescharacteristic
320C2×C71D4×C22⋊C412yes—
321C2×C71#234212yes—
322C2×C71#241012yes—
323C2×C71#241212yes—
324C2×C71#243714yes—
325C23×C22⋊C4C2×C7111yescharacteristic
326C23×C22⋊C4C2×C7111yescharacteristic
327C2×#385C2×C7112yescomplemented
328C2×#380C2×C7112yescomplemented
329C2×#379C2×C7111yescharacteristic
330#1439C2×C7114yescomplemented
331C22⋊#95C2×C7112yescomplemented
332C22⋊#95C2×C7112yescomplemented
333C71#1387711yescharacteristic, direct product factor, Sylow
334C26C22×C7111yescharacteristic
335C24×C4C22×C7112yes—
336C24×C4C22×C7112yes—
337C22×C4⋊C4C22×C7114yescomplemented
338C22×C22⋊C4—24no—
339C22×C22⋊C4—28no—
340C22×C22⋊C4—28no—
341C22×C22⋊C4C4×C7118yescomplemented
342C22×C22⋊C4C22×C7112yes—
343C22×C22⋊C4C22×C7111yescharacteristic
344C22×C22⋊C4C22×C7112yes—
345C22×C22⋊C4C22×C7111yescharacteristic
346C22×C22⋊C4C22×C7111yescharacteristic
347C22×C22⋊C4C22×C7112yes—
348C22×C22⋊C4C22×C7112yes—
349C22×C22⋊C4C22×C7112yes—
350C22×C22⋊C4C22×C7111yescharacteristic
351#385—28no—
352#380—28no—
353#379—24no—
354C2×#95C22×C7114yescomplemented
355C2×#95C22×C7114yescomplemented
356C2×#95C22×C7114yescomplemented
357C25—24no—
358C25—24no—
359C25—24no—
360C25—24no—
361C25—44no—
362C25C2×C4×C7114yes—
363C25D4×C7114yes—
364C25Q8×C7114yes—
365C25C23×C7111yescharacteristic
366C25C23×C7111yescharacteristic
367C25C23×C7111yescharacteristic
368C23×C4—24no—
369C23×C4—24no—
370C23×C4—24no—
371C23×C4—28no—
372C23×C4—28no—
373C23×C4—24no—
374C23×C4—24no—
375C23×C4—44no—
376C23×C4—44no—
377C23×C4C2×C4×C7118yescomplemented
378C23×C4C23×C7112yes—
379C23×C4C23×C7112yes—
380C23×C4C23×C7112yes—
381C23×C4C23×C7114yes—
382C23×C4C23×C7112yes—
383C2×C4⋊C4—28no—
384C2×C4⋊C4—28no—
385C2×C4⋊C4—44no—
386C2×C22⋊C4—232no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—28no—
392C2×C22⋊C4—28no—
393C2×C22⋊C4—232no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—44no—
403C2×C22⋊C4C2×C4×C7114yes—
404C2×C22⋊C4C2×C4×C7114yes—
405C2×C22⋊C4C2×C4×C7118yescomplemented
406C2×C22⋊C4D4×C7118yes—
407C2×C22⋊C4D4×C7118yes—
408#95—28no—
409#95—28no—
410#95—44no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—24no—
416C24—24no—
417C24—28no—
418C24—28no—
419C24—216no—
420C24—28no—
421C24—216no—
422C24—216no—
423C24—216no—
424C24—28no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—24no—
430C24—48no—
431C24—48no—
432C24—48no—
433C24—416no—
434C24—416no—
435C24—416no—
436C24—416no—
437C24C4⋊C4×C71116yescomplemented
438C24C22×C4×C7112yes—
439C24C22×C4×C7112yes—
440C24C22×C4×C7112yes—
441C24C2×D4×C7112yes—
442C24C2×D4×C7114yes—
443C24C2×D4×C7112yes—
444C24C2×D4×C7112yes—
445C24C2×D4×C7114yes—
446C24C2×Q8×C7112yes—
447C24C2×Q8×C7112yes—
448C24C2×Q8×C7112yes—
449C24#41×C7114yes—
450C24#41×C7114yes—
451C24C24×C7111yescharacteristic
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—216no—
456C22×C4—216no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—216no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—48no—
479C22×C4—416no—
480C22×C4—416no—
481C22×C4—416no—
482C22×C4—416no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4C22×C4×C7114yes—
486C22×C4C22×C4×C7114yes—
487C22×C4C2×D4×C7118yes—
488C22×C4C2×D4×C7118yes—
489C4⋊C4—416no—
490C22⋊C4—216no—
491C22⋊C4—28no—
492C22⋊C4—216no—
493C22⋊C4—216no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4C22⋊C4×C71116yescomplemented
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—232no—
533C23—232no—
534C23—416no—
535C23—432no—
536C23—432no—
537C23—432no—
538C23—432no—
539C23—464no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23C2×C22⋊C4×C7114yes—
545C23C2×C4⋊C4×C7114yes—
546C23C2×C4⋊C4×C7114yes—
547C23C2×C4⋊C4×C7114yes—
548C23#117×C7114yes—
549C23C4×D4×C7118yes—
550C23#120×C7114yes—
551C23#121×C7118yes—
552C23C22⋊Q8×C7118yes—
553C23C22⋊Q8×C7118yes—
554C23#123×C7114yes—
555C23C23×C4×C7111yescharacteristic
556C23C22×D4×C7112yes—
557C23C22×D4×C7111yescharacteristic
558C23C22×D4×C7112yes—
559C23C22×Q8×C7111yescharacteristic
560C23C2×#41×C7112yes—
561C23C2×#41×C7112yes—
562C23H(5,𝔽2)×C7114yes—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—432no—
574C2×C4—432no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4C2×C22⋊C4×C7118yescomplemented
592C22—24no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—28no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—28no—
602C22—28no—
603C22—28no—
604C22—216no—
605C22—216no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—464no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—48no—
615C22#380×C7118yescomplemented
616C22#385×C7118yescomplemented
617C22C22×C22⋊C4×C7111yescharacteristic
618C22C22×C4⋊C4×C7111yescharacteristic
619C22C2×#117×C7111yescharacteristic
620C22C2×C4×D4×C7112yes—
621C22#518×C7112yes—
622C22C2×#120×C7111yescharacteristic
623C22C2×#121×C7112yes—
624C22C2×C22⋊Q8×C7112yes—
625C22C2×C22⋊Q8×C7112yes—
626C22C2×#123×C7111yescharacteristic
627C22#534×C7112yes—
628C22#543×C7112yes—
629C22D42×C7114yes—
630C22#546×C7114yes—
631C22#546×C7114yes—
632C22#551×C7114yes—
633C4—28no—
634C4—44no—
635C4—48no—
636C4—82no—
637C4—82no—
638C2—28no—
639C2—28no—
640C2—44no—
641C2C2×#380×C7111yescharacteristic
642C2C2×#385×C7111yescharacteristic
643C2C71×#231511yescharacteristic
644C2D4×C22⋊C4×C7112yes—
645C2C71×#234212yes—
646C2C71×#241012yes—
647C2C71×#241212yes—
648C2C71×#243714yes—
subgroupG/H
center, socleC24×C71C24
commutatorC23C23×C4×C71
Frattini, cosocleC24C24×C71
prime radicalC26×C71C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C71 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#385×C71C2semidirect
4. C2×#380×C71C2semidirect
6. C71×#1439C2semidirect
7. C71×C22⋊#95C2semidirect
8. C71×C22⋊#95C2semidirect
12. C22×C4⋊C4×C71C22semidirect
16. C22×C22⋊C4×C71C4semidirect
29. C2×#95×C71C22semidirect
30. C2×#95×C71C22semidirect
31. C2×#95×C71C22semidirect
52. C23×C4×C71C2×C4semidirect
80. C2×C22⋊C4×C71C2×C4semidirect
112. C24×C71C4⋊C4semidirect
173. C22⋊C4×C71C22⋊C4semidirect
266. C2×C4×C71C2×C22⋊C4semidirect
290. C22×C71#380semidirect
291. C22×C71#385semidirect
313. #13877C71direct (N × H)
327. C2×#385C2×C71semidirect
328. C2×#380C2×C71semidirect
330. #1439C2×C71semidirect
331. C22⋊#95C2×C71semidirect
332. C22⋊#95C2×C71semidirect
333. C71#13877direct (N × H)
337. C22×C4⋊C4C22×C71semidirect
341. C22×C22⋊C4C4×C71semidirect
354. C2×#95C22×C71semidirect
355. C2×#95C22×C71semidirect
356. C2×#95C22×C71semidirect
377. C23×C4C2×C4×C71semidirect
405. C2×C22⋊C4C2×C4×C71semidirect
437. C24C4⋊C4×C71semidirect
498. C22⋊C4C22⋊C4×C71semidirect
591. C2×C4C2×C22⋊C4×C71semidirect
615. C22#380×C71semidirect
616. C22#385×C71semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊C4×C712
C2×#385×C712
C2×#380×C712
C2×#379×C712
C71×#14392
C71×C22⋊#952
#1387771