Finite Group Databasebeta

← C71×#9083 #31829955

Subgroups — C71×#9083 #31829955

Order 18176.

1100 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#407×C71C211yesmaximal, characteristic, complemented
2C2×#386×C71C211yesmaximal, characteristic
3C71×#1779C211yesmaximal, characteristic, complemented
4C2×#323×C71C212yesmaximal, complemented
5C71×#1280C212yesmaximal, complemented
6C23×D4×C71C411yescharacteristic
7C24×C4×C71C2211yescharacteristic
8C22×M16×C71—21no—
9C22×C22⋊C4×C71C411yescharacteristic
10C22×C22⋊C4×C71C411yescharacteristic
11C22×C22⋊C4×C71C2211yescharacteristic
12#407×C71—24no—
13C2×C22⋊C8×C71—21no—
14C2×C22⋊C8×C71—21no—
15C2×C22⋊C8×C71—21no—
16C2×C22⋊C8×C71C2212yes—
17C2×C22⋊C8×C71C2212yes—
18#386×C71—21no—
19#386×C71—21no—
20#386×C71—21no—
21#386×C71—21no—
22C2×#95×C71C411yescharacteristic
23C2×#95×C71C2211yescharacteristic
24#323×C71—24no—
25C25×C71C2×C411yescharacteristic
26C25×C71C2×C411yescharacteristic
27C22×D4×C71—21no—
28C22×D4×C71—21no—
29C22×D4×C71—22no—
30C22×D4×C71—22no—
31C22×D4×C71—42no—
32C22×D4×C71—42no—
33C23×C4×C71—22no—
34C23×C4×C71—22no—
35C23×C4×C71—42no—
36C23×C4×C71—42no—
37C23×C4×C71C2×C411yescharacteristic
38C23×C4×C71C2×C411yescharacteristic
39C23×C4×C71C2×C411yescharacteristic
40C23×C4×C71C2×C411yescharacteristic
41C23×C4×C71D411yescharacteristic
42C23×C4×C71D411yescharacteristic
43C23×C4×C71D411yescharacteristic
44C23×C4×C71D411yescharacteristic
45C23×C4×C71C2311yescharacteristic
46C2×M16×C71—24no—
47C2×M16×C71—42no—
48C2×M16×C71—42no—
49C22×C8×C71—22no—
50C22×C8×C71—22no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71—21no—
53C2×C22⋊C4×C71—21no—
54C2×C22⋊C4×C71—21no—
55C2×C22⋊C4×C71—22no—
56C2×C22⋊C4×C71—21no—
57C2×C22⋊C4×C71—21no—
58C2×C22⋊C4×C71—22no—
59C2×C22⋊C4×C71—42no—
60C2×C22⋊C4×C71—42no—
61C22⋊C8×C71—24no—
62C22⋊C8×C71—22no—
63C22⋊C8×C71—22no—
64C22⋊C8×C71—22no—
65C22⋊C8×C71—42no—
66C22⋊C8×C71—42no—
67C22⋊C8×C71D414yescomplemented
68#95×C71—21no—
69#95×C71—21no—
70#95×C71—21no—
71#95×C71—21no—
72C24×C71—22no—
73C24×C71—22no—
74C24×C71—21no—
75C24×C71—21no—
76C24×C71—21no—
77C24×C71—22no—
78C24×C71—21no—
79C24×C71—22no—
80C24×C71—22no—
81C24×C71—22no—
82C24×C71—41no—
83C24×C71—41no—
84C24×C71—41no—
85C24×C71—41no—
86C24×C71—41no—
87C24×C71—41no—
88C24×C71C22⋊C411yescharacteristic
89C24×C71C22⋊C411yescharacteristic
90C24×C71M1612yescomplemented
91C24×C71C22×C411yescharacteristic
92C2×D4×C71—24no—
93C2×D4×C71—24no—
94C2×D4×C71—48no—
95C2×D4×C71—42no—
96C2×D4×C71—42no—
97C2×D4×C71—48no—
98C2×D4×C71—42no—
99C2×D4×C71—42no—
100C22×C4×C71—22no—
101C22×C4×C71—22no—
102C22×C4×C71—21no—
103C22×C4×C71—21no—
104C22×C4×C71—22no—
105C22×C4×C71—21no—
106C22×C4×C71—22no—
107C22×C4×C71—21no—
108C22×C4×C71—22no—
109C22×C4×C71—22no—
110C22×C4×C71—21no—
111C22×C4×C71—22no—
112C22×C4×C71—22no—
113C22×C4×C71—21no—
114C22×C4×C71—21no—
115C22×C4×C71—21no—
116C22×C4×C71—22no—
117C22×C4×C71—22no—
118C22×C4×C71—41no—
119C22×C4×C71—41no—
120C22×C4×C71—41no—
121C22×C4×C71—41no—
122C22×C4×C71—41no—
123C22×C4×C71—41no—
124C22×C4×C71—41no—
125C22×C4×C71—48no—
126C22×C4×C71—48no—
127C22×C4×C71—41no—
128C22×C4×C71—41no—
129C22×C4×C71—41no—
130C22×C4×C71—41no—
131C22×C4×C71—41no—
132C22×C4×C71—41no—
133C22×C4×C71C22⋊C411yescharacteristic
134C22×C4×C71C22⋊C411yescharacteristic
135C22×C4×C71M1612yescomplemented
136C22×C4×C71C2×D411yescharacteristic
137C22×C4×C71C2×D411yescharacteristic
138C22×C4×C71C2×D412yes—
139C22×C4×C71#4112yes—
140M16×C71—48no—
141C2×C8×C71—24no—
142C2×C8×C71—42no—
143C2×C8×C71—42no—
144C2×C8×C71—42no—
145C2×C8×C71—42no—
146C2×C8×C71—42no—
147C22⋊C4×C71—42no—
148C22⋊C4×C71—42no—
149C22⋊C4×C71—42no—
150C22⋊C4×C71—42no—
151C22⋊C4×C71—41no—
152C22⋊C4×C71—42no—
153C22⋊C4×C71—41no—
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—21no—
167C23×C71—21no—
168C23×C71—21no—
169C23×C71—21no—
170C23×C71—21no—
171C23×C71—21no—
172C23×C71—22no—
173C23×C71—24no—
174C23×C71—44no—
175C23×C71—44no—
176C23×C71—44no—
177C23×C71—44no—
178C23×C71—42no—
179C23×C71—42no—
180C23×C71—42no—
181C23×C71—42no—
182C23×C71—42no—
183C23×C71—42no—
184C23×C71—42no—
185C23×C71—42no—
186C23×C71—41no—
187C23×C71—41no—
188C23×C71—41no—
189C23×C71—41no—
190C23×C71—41no—
191C23×C71—41no—
192C23×C71—42no—
193C23×C71—82no—
194C23×C71—82no—
195C23×C71—82no—
196C23×C71—82no—
197C23×C71#9912yes—
198C23×C71#10012yes—
199C23×C71C2×C22⋊C411yescharacteristic
200C23×C71C4×D412yes—
201C23×C71C2×M1611yescharacteristic
202C23×C71C2×M1611yescharacteristic
203C23×C71#13112yes—
204D4×C71—48no—
205D4×C71—48no—
206C2×C4×C71—24no—
207C2×C4×C71—24no—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—22no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—42no—
219C2×C4×C71—42no—
220C2×C4×C71—44no—
221C2×C4×C71—44no—
222C2×C4×C71—44no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71—42no—
226C2×C4×C71—41no—
227C2×C4×C71—41no—
228C2×C4×C71—42no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71—41no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—82no—
236C2×C4×C71—82no—
237C2×C4×C71#9912yes—
238C2×C4×C71#12011yescharacteristic
239C2×C4×C71#12112yes—
240C2×C4×C71#12311yescharacteristic
241C8×C71—44no—
242C8×C71—82no—
243C22×C71—22no—
244C22×C71—22no—
245C22×C71—21no—
246C22×C71—21no—
247C22×C71—21no—
248C22×C71—21no—
249C22×C71—22no—
250C22×C71—22no—
251C22×C71—22no—
252C22×C71—22no—
253C22×C71—22no—
254C22×C71—22no—
255C22×C71—22no—
256C22×C71—22no—
257C22×C71—48no—
258C22×C71—48no—
259C22×C71—41no—
260C22×C71—41no—
261C22×C71—41no—
262C22×C71—41no—
263C22×C71—41no—
264C22×C71—41no—
265C22×C71—41no—
266C22×C71—41no—
267C22×C71—42no—
268C22×C71—42no—
269C22×C71—42no—
270C22×C71—84no—
271C22×C71—84no—
272C22×C71—81no—
273C22×C71—81no—
274C22×C71—81no—
275C22×C71—81no—
276C22×C71#38611yescharacteristic
277C22×C71#40711yescharacteristic
278C22×C71#40811yescharacteristic
279C22×C71C2×#9911yescharacteristic
280C22×C71C2×#9911yescharacteristic
281C22×C71C2×#10011yescharacteristic
282C22×C71#41311yescharacteristic
283C22×C71#43512yes—
284C22×C71#43612yes—
285C4×C71—21no—
286C4×C71—22no—
287C4×C71—22no—
288C4×C71—21no—
289C4×C71—41no—
290C4×C71—42no—
291C4×C71—81no—
292C4×C71—81no—
293#9083C7111yesmaximal, characteristic, direct product factor, Sylow
294C2×C71—22no—
295C2×C71—22no—
296C2×C71—42no—
297C2×C71—42no—
298C2×C71—82no—
299C2×C71#146211yescharacteristic
300C2×C71#146511yescharacteristic
301C2×C71#146811yescharacteristic
302C2×C71#147111yescharacteristic
303C2×C71#187011yescharacteristic
304C2×C71#189811yescharacteristic
305C2×C71#190011yescharacteristic
306C2×#407C2×C7111yescharacteristic, complemented
307C2×#386C2×C7111yescharacteristic
308#1779C2×C7111yescharacteristic, complemented
309C2×#323C2×C7112yescomplemented
310#1280C2×C7112yescomplemented
311C71#908311yescharacteristic, direct product factor, Sylow
312C23×D4C4×C7111yescharacteristic
313C24×C4C22×C7111yescharacteristic
314C22×M16—21no—
315C22×C22⋊C4C4×C7111yescharacteristic
316C22×C22⋊C4C4×C7111yescharacteristic
317C22×C22⋊C4C22×C7111yescharacteristic
318#407—24no—
319C2×C22⋊C8—21no—
320C2×C22⋊C8—21no—
321C2×C22⋊C8—21no—
322C2×C22⋊C8C22×C7112yes—
323C2×C22⋊C8C22×C7112yes—
324#386—21no—
325#386—21no—
326#386—21no—
327#386—21no—
328C2×#95C4×C7111yescharacteristic
329C2×#95C22×C7111yescharacteristic
330#323—24no—
331C25C2×C4×C7111yescharacteristic
332C25C2×C4×C7111yescharacteristic
333C22×D4—21no—
334C22×D4—21no—
335C22×D4—22no—
336C22×D4—22no—
337C22×D4—42no—
338C22×D4—42no—
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—42no—
342C23×C4—42no—
343C23×C4C2×C4×C7111yescharacteristic
344C23×C4C2×C4×C7111yescharacteristic
345C23×C4C2×C4×C7111yescharacteristic
346C23×C4C2×C4×C7111yescharacteristic
347C23×C4D4×C7111yescharacteristic
348C23×C4D4×C7111yescharacteristic
349C23×C4D4×C7111yescharacteristic
350C23×C4D4×C7111yescharacteristic
351C23×C4C23×C7111yescharacteristic
352C2×M16—24no—
353C2×M16—42no—
354C2×M16—42no—
355C22×C8—22no—
356C22×C8—22no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—42no—
366C2×C22⋊C4—42no—
367C22⋊C8—24no—
368C22⋊C8—22no—
369C22⋊C8—22no—
370C22⋊C8—22no—
371C22⋊C8—42no—
372C22⋊C8—42no—
373C22⋊C8D4×C7114yescomplemented
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378C24—22no—
379C24—22no—
380C24—21no—
381C24—21no—
382C24—21no—
383C24—22no—
384C24—21no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24—41no—
393C24—41no—
394C24C22⋊C4×C7111yescharacteristic
395C24C22⋊C4×C7111yescharacteristic
396C24M16×C7112yescomplemented
397C24C22×C4×C7111yescharacteristic
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—48no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—48no—
404C2×D4—42no—
405C2×D4—42no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4C22⋊C4×C7111yescharacteristic
440C22×C4C22⋊C4×C7111yescharacteristic
441C22×C4M16×C7112yescomplemented
442C22×C4C2×D4×C7111yescharacteristic
443C22×C4C2×D4×C7111yescharacteristic
444C22×C4C2×D4×C7112yes—
445C22×C4#41×C7112yes—
446M16—48no—
447C2×C8—24no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—41no—
458C22⋊C4—42no—
459C22⋊C4—41no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—24no—
465C23—24no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—24no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23#99×C7112yes—
504C23#100×C7112yes—
505C23C2×C22⋊C4×C7111yescharacteristic
506C23C4×D4×C7112yes—
507C23C2×M16×C7111yescharacteristic
508C23C2×M16×C7111yescharacteristic
509C23#131×C7112yes—
510D4—48no—
511D4—48no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4#99×C7112yes—
544C2×C4#120×C7111yescharacteristic
545C2×C4#121×C7112yes—
546C2×C4#123×C7111yescharacteristic
547C8—44no—
548C8—82no—
549C22—22no—
550C22—22no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—48no—
564C22—48no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—84no—
577C22—84no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22#386×C7111yescharacteristic
583C22#407×C7111yescharacteristic
584C22#408×C7111yescharacteristic
585C22C2×#99×C7111yescharacteristic
586C22C2×#99×C7111yescharacteristic
587C22C2×#100×C7111yescharacteristic
588C22#413×C7111yescharacteristic
589C22#435×C7112yes—
590C22#436×C7112yes—
591C4—21no—
592C4—22no—
593C4—22no—
594C4—21no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C2—22no—
600C2—22no—
601C2—42no—
602C2—42no—
603C2—82no—
604C2C71×#146211yescharacteristic
605C2C71×#146511yescharacteristic
606C2C71×#146811yescharacteristic
607C2C71×#147111yescharacteristic
608C2C71×#187011yescharacteristic
609C2C71×#189811yescharacteristic
610C2C71×#190011yescharacteristic
subgroupG/H
center, socleC23×C71C2×C22⋊C4
commutatorC24C22×C4×C71
Frattini, cosocleC23×C4C23×C71
prime radicalC23×D4×C71C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23×C71 ◁ C24×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#407×C71C2semidirect
3. C71×#1779C2semidirect
4. C2×#323×C71C2semidirect
5. C71×#1280C2semidirect
67. C22⋊C8×C71D4semidirect
90. C24×C71M16semidirect
135. C22×C4×C71M16semidirect
293. #9083C71direct (N × H)
306. C2×#407C2×C71semidirect
308. #1779C2×C71semidirect
309. C2×#323C2×C71semidirect
310. #1280C2×C71semidirect
311. C71#9083direct (N × H)
373. C22⋊C8D4×C71semidirect
396. C24M16×C71semidirect
441. C22×C4M16×C71semidirect

Maximal subgroups

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

subgroupindex
C2×#407×C712
C2×#386×C712
C71×#17792
C2×#323×C712
C71×#12802
#908371