Finite Group Databasebeta

← C71×#9113 #31829985

Subgroups — C71×#9113 #31829985

Order 18176.

1062 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊C8×C71C211yesmaximal, characteristic, complemented
2C71×#2847C211yesmaximal, characteristic
3C2×#327×C71C212yesmaximal, complemented
4C71×C22⋊#95C211yesmaximal, characteristic, complemented
5C71×#1294C212yesmaximal, complemented
6C24×C4×C71C2211yescharacteristic
7C23×C8×C71—21no—
8C2×#121×C71C411yescharacteristic, complemented
9C2×#121×C71C411yescharacteristic, complemented
10C2×#121×C71C2211yescharacteristic
11C2×#120×C71—21no—
12C22×C22⋊C4×C71—21no—
13C2×C22⋊C8×C71—21no—
14C2×C22⋊C8×C71—24no—
15C2×C22⋊C8×C71C2212yes—
16#386×C71—21no—
17#386×C71—21no—
18#386×C71—21no—
19#380×C71C411yescharacteristic, complemented
20#380×C71C411yescharacteristic, complemented
21#380×C71C2211yescharacteristic
22C2×#95×C71—21no—
23C2×#95×C71C2212yes—
24#327×C71—24no—
25C25×C71D411yescharacteristic
26C22×D4×C71—21no—
27C22×D4×C71—21no—
28C22×D4×C71—21no—
29C22×D4×C71C2×C411yescharacteristic
30C23×C4×C71—22no—
31C23×C4×C71—22no—
32C23×C4×C71—22no—
33C23×C4×C71—42no—
34C23×C4×C71—42no—
35C23×C4×C71C2×C411yescharacteristic
36C23×C4×C71C2×C411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C23×C4×C71D411yescharacteristic
40C23×C4×C71C2311yescharacteristic
41C22×C8×C71—24no—
42C22×C8×C71—22no—
43C22×C8×C71—42no—
44C22×C8×C71—42no—
45#121×C71—22no—
46#121×C71—22no—
47#121×C71—42no—
48#121×C71—42no—
49#120×C71—42no—
50#120×C71—42no—
51C2×C4⋊C4×C71C2×C411yescharacteristic
52C2×C4⋊C4×C71C2×C411yescharacteristic
53C2×C4⋊C4×C71C2×C411yescharacteristic
54C2×C22⋊C4×C71—24no—
55C2×C22⋊C4×C71—21no—
56C2×C22⋊C4×C71—21no—
57C2×C22⋊C4×C71—21no—
58C2×C22⋊C4×C71—21no—
59C2×C22⋊C4×C71—21no—
60C2×C22⋊C4×C71—21no—
61C2×C22⋊C4×C71—42no—
62C2×C22⋊C4×C71—42no—
63C22⋊C8×C71—24no—
64C22⋊C8×C71—22no—
65C22⋊C8×C71D414yescomplemented
66#95×C71—21no—
67#95×C71—21no—
68#95×C71—21no—
69#95×C71—22no—
70#95×C71—22no—
71#95×C71—42no—
72C24×C71—22no—
73C24×C71—22no—
74C24×C71—22no—
75C24×C71—22no—
76C24×C71—21no—
77C24×C71—41no—
78C24×C71—41no—
79C24×C71C22⋊C411yescharacteristic
80C24×C71C22⋊C411yescharacteristic
81C24×C71D812yes—
82C24×C71SD1612yes—
83C24×C71C2×D411yescharacteristic
84C2×D4×C71—22no—
85C2×D4×C71—41no—
86C2×D4×C71—41no—
87C2×D4×C71—42no—
88C2×D4×C71—42no—
89C2×D4×C71—42no—
90C2×D4×C71—42no—
91C2×D4×C71—42no—
92C2×D4×C71—42no—
93C2×D4×C71—42no—
94C2×D4×C71—41no—
95C2×D4×C71—41no—
96C2×D4×C71—42no—
97C22×C4×C71—22no—
98C22×C4×C71—22no—
99C22×C4×C71—21no—
100C22×C4×C71—21no—
101C22×C4×C71—22no—
102C22×C4×C71—22no—
103C22×C4×C71—22no—
104C22×C4×C71—22no—
105C22×C4×C71—22no—
106C22×C4×C71—22no—
107C22×C4×C71—21no—
108C22×C4×C71—21no—
109C22×C4×C71—24no—
110C22×C4×C71—21no—
111C22×C4×C71—21no—
112C22×C4×C71—21no—
113C22×C4×C71—41no—
114C22×C4×C71—41no—
115C22×C4×C71—41no—
116C22×C4×C71—41no—
117C22×C4×C71—41no—
118C22×C4×C71—41no—
119C22×C4×C71—41no—
120C22×C4×C71—48no—
121C22×C4×C71—48no—
122C22×C4×C71—42no—
123C22×C4×C71—42no—
124C22×C4×C71—42no—
125C22×C4×C71—42no—
126C22×C4×C71—42no—
127C22×C4×C71—42no—
128C22×C4×C71C22⋊C411yescharacteristic
129C22×C4×C71C22⋊C411yescharacteristic
130C22×C4×C71C22×C411yescharacteristic
131C22×C4×C71C2×D411yescharacteristic
132C22×C4×C71C2×D412yes—
133C22×C4×C71#4112yes—
134C2×C8×C71—24no—
135C2×C8×C71—48no—
136C2×C8×C71—42no—
137C2×C8×C71—42no—
138C4⋊C4×C71—22no—
139C4⋊C4×C71—41no—
140C4⋊C4×C71—41no—
141C22⋊C4×C71—42no—
142C22⋊C4×C71—48no—
143C22⋊C4×C71—42no—
144C22⋊C4×C71—42no—
145C22⋊C4×C71—41no—
146C22⋊C4×C71—41no—
147C22⋊C4×C71—42no—
148C22⋊C4×C71—41no—
149C22⋊C4×C71—41no—
150C23×C71—21no—
151C23×C71—21no—
152C23×C71—21no—
153C23×C71—21no—
154C23×C71—21no—
155C23×C71—21no—
156C23×C71—21no—
157C23×C71—21no—
158C23×C71—24no—
159C23×C71—24no—
160C23×C71—21no—
161C23×C71—21no—
162C23×C71—21no—
163C23×C71—21no—
164C23×C71—21no—
165C23×C71—21no—
166C23×C71—24no—
167C23×C71—24no—
168C23×C71—42no—
169C23×C71—42no—
170C23×C71—44no—
171C23×C71—44no—
172C23×C71—44no—
173C23×C71—44no—
174C23×C71—41no—
175C23×C71—41no—
176C23×C71—41no—
177C23×C71—41no—
178C23×C71—41no—
179C23×C71—41no—
180C23×C71—41no—
181C23×C71#9912yes—
182C23×C71#10212yes—
183C23×C71#10212yes—
184C23×C71#10412yes—
185C23×C71C2×C22⋊C411yescharacteristic
186C23×C71#12112yes—
187C23×C71C2×D811yescharacteristic
188C23×C71C2×SD1611yescharacteristic
189C23×C71#13512yes—
190D4×C71—82no—
191D4×C71—84no—
192D4×C71—84no—
193D4×C71—82no—
194D4×C71—82no—
195C2×C4×C71—24no—
196C2×C4×C71—24no—
197C2×C4×C71—21no—
198C2×C4×C71—21no—
199C2×C4×C71—21no—
200C2×C4×C71—21no—
201C2×C4×C71—21no—
202C2×C4×C71—21no—
203C2×C4×C71—21no—
204C2×C4×C71—21no—
205C2×C4×C71—21no—
206C2×C4×C71—42no—
207C2×C4×C71—42no—
208C2×C4×C71—44no—
209C2×C4×C71—44no—
210C2×C4×C71—44no—
211C2×C4×C71—42no—
212C2×C4×C71—42no—
213C2×C4×C71—44no—
214C2×C4×C71—44no—
215C2×C4×C71—44no—
216C2×C4×C71—41no—
217C2×C4×C71—41no—
218C2×C4×C71—41no—
219C2×C4×C71—41no—
220C2×C4×C71—41no—
221C2×C4×C71—41no—
222C2×C4×C71—41no—
223C2×C4×C71—41no—
224C2×C4×C71—41no—
225C2×C4×C71—82no—
226C2×C4×C71—82no—
227C2×C4×C71—82no—
228C2×C4×C71—82no—
229C2×C4×C71#9912yes—
230C2×C4×C71C4×D412yes—
231C2×C4×C71#12011yescharacteristic
232C2×C4×C71#12311yescharacteristic
233C8×C71—44no—
234C22×C71—22no—
235C22×C71—22no—
236C22×C71—22no—
237C22×C71—22no—
238C22×C71—22no—
239C22×C71—21no—
240C22×C71—21no—
241C22×C71—22no—
242C22×C71—22no—
243C22×C71—22no—
244C22×C71—22no—
245C22×C71—21no—
246C22×C71—22no—
247C22×C71—21no—
248C22×C71—22no—
249C22×C71—22no—
250C22×C71—22no—
251C22×C71—48no—
252C22×C71—48no—
253C22×C71—41no—
254C22×C71—41no—
255C22×C71—41no—
256C22×C71—41no—
257C22×C71—81no—
258C22×C71—81no—
259C22×C71—81no—
260C22×C71—81no—
261C22×C71—81no—
262C22×C71—81no—
263C22×C71—81no—
264C22×C71#32714yescomplemented
265C22×C71#38611yescharacteristic
266C22×C71C2×#9911yescharacteristic
267C22×C71C2×#9911yescharacteristic
268C22×C71C2×#10211yescharacteristic
269C22×C71#41611yescharacteristic
270C22×C71C2×#10411yescharacteristic
271C22×C71#42111yescharacteristic
272C22×C71#46512yes—
273C22×C71#46612yes—
274C4×C71—21no—
275C4×C71—22no—
276C4×C71—21no—
277C4×C71—22no—
278C4×C71—41no—
279C4×C71—84no—
280C4×C71—81no—
281C4×C71—81no—
282C4×C71—81no—
283#9113C7111yesmaximal, characteristic, direct product factor, Sylow
284C2×C71—22no—
285C2×C71—24no—
286C2×C71—22no—
287C2×C71—42no—
288C2×C71—161no—
289C2×C71C2×#32711yescharacteristic
290C2×C71#150111yescharacteristic
291C2×C71#151611yescharacteristic
292C2×C71#151911yescharacteristic
293C2×C71#187111yescharacteristic
294C2×C71#189311yescharacteristic
295C2×C71#189811yescharacteristic
296C22×C22⋊C8C2×C7111yescharacteristic, complemented
297#2847C2×C7111yescharacteristic
298C2×#327C2×C7112yescomplemented
299C22⋊#95C2×C7111yescharacteristic, complemented
300#1294C2×C7112yescomplemented
301C71#911311yescharacteristic, direct product factor, Sylow
302C24×C4C22×C7111yescharacteristic
303C23×C8—21no—
304C2×#121C4×C7111yescharacteristic, complemented
305C2×#121C4×C7111yescharacteristic, complemented
306C2×#121C22×C7111yescharacteristic
307C2×#120—21no—
308C22×C22⋊C4—21no—
309C2×C22⋊C8—21no—
310C2×C22⋊C8—24no—
311C2×C22⋊C8C22×C7112yes—
312#386—21no—
313#386—21no—
314#386—21no—
315#380C4×C7111yescharacteristic, complemented
316#380C4×C7111yescharacteristic, complemented
317#380C22×C7111yescharacteristic
318C2×#95—21no—
319C2×#95C22×C7112yes—
320#327—24no—
321C25D4×C7111yescharacteristic
322C22×D4—21no—
323C22×D4—21no—
324C22×D4—21no—
325C22×D4C2×C4×C7111yescharacteristic
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—42no—
330C23×C4—42no—
331C23×C4C2×C4×C7111yescharacteristic
332C23×C4C2×C4×C7111yescharacteristic
333C23×C4D4×C7111yescharacteristic
334C23×C4D4×C7111yescharacteristic
335C23×C4D4×C7111yescharacteristic
336C23×C4C23×C7111yescharacteristic
337C22×C8—24no—
338C22×C8—22no—
339C22×C8—42no—
340C22×C8—42no—
341#121—22no—
342#121—22no—
343#121—42no—
344#121—42no—
345#120—42no—
346#120—42no—
347C2×C4⋊C4C2×C4×C7111yescharacteristic
348C2×C4⋊C4C2×C4×C7111yescharacteristic
349C2×C4⋊C4C2×C4×C7111yescharacteristic
350C2×C22⋊C4—24no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4—42no—
359C22⋊C8—24no—
360C22⋊C8—22no—
361C22⋊C8D4×C7114yescomplemented
362#95—21no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—42no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—41no—
374C24—41no—
375C24C22⋊C4×C7111yescharacteristic
376C24C22⋊C4×C7111yescharacteristic
377C24D8×C7112yes—
378C24SD16×C7112yes—
379C24C2×D4×C7111yescharacteristic
380C2×D4—22no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—42no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4C22⋊C4×C7111yescharacteristic
425C22×C4C22⋊C4×C7111yescharacteristic
426C22×C4C22×C4×C7111yescharacteristic
427C22×C4C2×D4×C7111yescharacteristic
428C22×C4C2×D4×C7112yes—
429C22×C4#41×C7112yes—
430C2×C8—24no—
431C2×C8—48no—
432C2×C8—42no—
433C2×C8—42no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—48no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—42no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—21no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23#99×C7112yes—
478C23#102×C7112yes—
479C23#102×C7112yes—
480C23#104×C7112yes—
481C23C2×C22⋊C4×C7111yescharacteristic
482C23#121×C7112yes—
483C23C2×D8×C7111yescharacteristic
484C23C2×SD16×C7111yescharacteristic
485C23#135×C7112yes—
486D4—82no—
487D4—84no—
488D4—82no—
489D4—82no—
490D4—84no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#99×C7112yes—
526C2×C4C4×D4×C7112yes—
527C2×C4#120×C7111yescharacteristic
528C2×C4#123×C7111yescharacteristic
529C8—44no—
530C22—22no—
531C22—22no—
532C22—21no—
533C22—22no—
534C22—21no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—48no—
548C22—48no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22#327×C7114yescomplemented
561C22#386×C7111yescharacteristic
562C22C2×#99×C7111yescharacteristic
563C22C2×#99×C7111yescharacteristic
564C22C2×#102×C7111yescharacteristic
565C22#416×C7111yescharacteristic
566C22C2×#104×C7111yescharacteristic
567C22#421×C7111yescharacteristic
568C22#465×C7112yes—
569C22#466×C7112yes—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—84no—
576C4—81no—
577C4—81no—
578C4—81no—
579C2—22no—
580C2—22no—
581C2—24no—
582C2—42no—
583C2—161no—
584C2C2×#327×C7111yescharacteristic
585C2C71×#150111yescharacteristic
586C2C71×#151611yescharacteristic
587C2C71×#151911yescharacteristic
588C2C71×#187111yescharacteristic
589C2C71×#189311yescharacteristic
590C2C71×#189811yescharacteristic
subgroupG/H
center, socleC23×C71C2×C22⋊C4
commutatorC22×C4C22×C4×C71
Frattini, cosocleC23×C4C23×C71
prime radicalC71×#2847C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C71 ◁ C24×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊C8×C71C2semidirect
3. C2×#327×C71C2semidirect
4. C71×C22⋊#95C2semidirect
5. C71×#1294C2semidirect
8. C2×#121×C71C4semidirect
9. C2×#121×C71C4semidirect
19. #380×C71C4semidirect
20. #380×C71C4semidirect
65. C22⋊C8×C71D4semidirect
264. C22×C71#327semidirect
283. #9113C71direct (N × H)
296. C22×C22⋊C8C2×C71semidirect
298. C2×#327C2×C71semidirect
299. C22⋊#95C2×C71semidirect
300. #1294C2×C71semidirect
301. C71#9113direct (N × H)
304. C2×#121C4×C71semidirect
305. C2×#121C4×C71semidirect
315. #380C4×C71semidirect
316. #380C4×C71semidirect
361. C22⋊C8D4×C71semidirect
560. C22#327×C71semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊C8×C712
C71×#28472
C2×#327×C712
C71×C22⋊#952
C71×#12942
#911371