Finite Group Databasebeta

← C71×#10394 #31831266

Subgroups — C71×#10394 #31831266

Order 18176.

772 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#2617C211yesmaximal, characteristic
2C71×#1874C211yesmaximal, characteristic, complemented
3C71×#1852C211yesmaximal, characteristic, complemented
4C2×#327×C71C211yesmaximal, characteristic, complemented
5C2×#323×C71C211yesmaximal, characteristic, complemented
6C71×#1439C211yesmaximal, characteristic, complemented
7C71×#1294C211yesmaximal, characteristic, complemented
8C2×#121×C71C2211yescharacteristic
9C2×#120×C71—21no—
10C22×C4⋊C4×C71C2211yescharacteristic
11C22×C22⋊C4×C71—21no—
12C22×C22⋊C4×C71C2211yescharacteristic
13C2×#107×C71—21no—
14C2×#102×C71—21no—
15C2×C22⋊C8×C71C2211yescharacteristic
16C2×C22⋊C8×C71C2211yescharacteristic
17#397×C71—21no—
18#394×C71—21no—
19#394×C71—21no—
20#392×C71—21no—
21#386×C71—21no—
22#380×C71C2211yescharacteristic
23C2×#95×C71—21no—
24C2×#95×C71C2211yescharacteristic
25#340×C71—21no—
26#340×C71—21no—
27#327×C71—22no—
28#323×C71—22no—
29C25×C71D411yescharacteristic
30C22×D4×C71—21no—
31C22×D4×C71—21no—
32C22×D4×C71—21no—
33C22×D4×C71D411yescharacteristic
34C23×C4×C71—21no—
35C23×C4×C71—21no—
36C23×C4×C71D411yescharacteristic
37C23×C4×C71C2311yescharacteristic
38C22×C8×C71—21no—
39C22×C8×C71—21no—
40#121×C71—21no—
41#121×C71—21no—
42#121×C71—21no—
43#121×C71—21no—
44#120×C71—42no—
45#120×C71—42no—
46C2×C4⋊C4×C71—21no—
47C2×C4⋊C4×C71—21no—
48C2×C4⋊C4×C71—42no—
49C2×C4⋊C4×C71D411yescharacteristic
50C2×C4⋊C4×C71D411yescharacteristic
51C2×C4⋊C4×C71D411yescharacteristic
52C2×C22⋊C4×C71—21no—
53C2×C22⋊C4×C71—21no—
54C2×C22⋊C4×C71—21no—
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—21no—
62C2×C22⋊C4×C71—21no—
63C2×C22⋊C4×C71—21no—
64C2×C22⋊C4×C71—42no—
65C2×C22⋊C4×C71—42no—
66C2×C22⋊C4×C71—42no—
67#107×C71—42no—
68#102×C71—42no—
69C22⋊C8×C71—21no—
70C22⋊C8×C71—21no—
71C22⋊C8×C71—21no—
72C22⋊C8×C71D412yescomplemented
73#95×C71—21no—
74#95×C71—21no—
75#95×C71—21no—
76#95×C71—21no—
77#95×C71—21no—
78#95×C71—42no—
79C24×C71—21no—
80C24×C71—21no—
81C24×C71—21no—
82C24×C71—21no—
83C24×C71—21no—
84C24×C71—21no—
85C24×C71—21no—
86C24×C71—41no—
87C24×C71—41no—
88C24×C71—41no—
89C24×C71—41no—
90C24×C71D812yes—
91C24×C71C2×D411yescharacteristic
92C2×D4×C71—21no—
93C2×D4×C71—21no—
94C2×D4×C71—41no—
95C2×D4×C71—41no—
96C2×D4×C71—42no—
97C2×D4×C71—42no—
98C2×D4×C71—41no—
99C2×D4×C71—41no—
100C2×D4×C71—41no—
101C2×D4×C71—41no—
102C2×D4×C71—41no—
103C2×D4×C71—41no—
104C2×D4×C71—81no—
105C2×D4×C71—81no—
106C2×D4×C71—81no—
107C2×D4×C71—81no—
108C22×C4×C71—22no—
109C22×C4×C71—22no—
110C22×C4×C71—21no—
111C22×C4×C71—21no—
112C22×C4×C71—21no—
113C22×C4×C71—21no—
114C22×C4×C71—21no—
115C22×C4×C71—21no—
116C22×C4×C71—41no—
117C22×C4×C71—41no—
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—41no—
126C22×C4×C71—41no—
127C22×C4×C71—41no—
128C22×C4×C71—41no—
129C22×C4×C71—41no—
130C22×C4×C71—41no—
131C22×C4×C71—41no—
132C22×C4×C71C2×D411yescharacteristic
133C22×C4×C71C2×D411yescharacteristic
134C22×C4×C71C2×D411yescharacteristic
135C22×C4×C71#4111yescharacteristic
136C22×C4×C71#4111yescharacteristic
137C22×C4×C71#4111yescharacteristic
138C2×C8×C71—22no—
139C2×C8×C71—41no—
140C2×C8×C71—41no—
141C2×C8×C71—41no—
142C2×C8×C71—41no—
143C2×C8×C71—41no—
144C4⋊C4×C71—22no—
145C4⋊C4×C71—22no—
146C4⋊C4×C71—21no—
147C4⋊C4×C71—21no—
148C4⋊C4×C71—41no—
149C4⋊C4×C71—41no—
150C4⋊C4×C71—41no—
151C4⋊C4×C71—41no—
152C22⋊C4×C71—42no—
153C22⋊C4×C71—42no—
154C22⋊C4×C71—48no—
155C22⋊C4×C71—41no—
156C22⋊C4×C71—41no—
157C22⋊C4×C71—41no—
158C22⋊C4×C71—41no—
159C22⋊C4×C71—41no—
160C22⋊C4×C71—41no—
161C22⋊C4×C71—41no—
162C22⋊C4×C71—41no—
163C22⋊C4×C71—81no—
164C22⋊C4×C71—81no—
165C23×C71—21no—
166C23×C71—21no—
167C23×C71—21no—
168C23×C71—21no—
169C23×C71—22no—
170C23×C71—22no—
171C23×C71—22no—
172C23×C71—42no—
173C23×C71—42no—
174C23×C71—42no—
175C23×C71—42no—
176C23×C71—42no—
177C23×C71—42no—
178C23×C71—41no—
179C23×C71—41no—
180C23×C71—41no—
181C23×C71—41no—
182C23×C71—41no—
183C23×C71—41no—
184C23×C71—41no—
185C23×C71—41no—
186C23×C71—41no—
187C23×C71—41no—
188C23×C71—41no—
189C23×C71—41no—
190C23×C71—41no—
191C23×C71—41no—
192C23×C71—41no—
193C23×C71—81no—
194C23×C71—81no—
195C23×C71—81no—
196C23×C71—81no—
197C23×C71—81no—
198C23×C71—81no—
199C23×C71#12011yescharacteristic
200C23×C71#12111yescharacteristic
201C23×C71#12311yescharacteristic
202C23×C71C2×D811yescharacteristic
203C23×C71#13511yescharacteristic
204C23×C71AGL(1,ℤ/8)11yescharacteristic
205C23×C71#13711yescharacteristic
206D4×C71—42no—
207D4×C71—81no—
208D4×C71—84no—
209D4×C71—84no—
210D4×C71—81no—
211D4×C71—81no—
212D4×C71—81no—
213D4×C71—81no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—41no—
219C2×C4×C71—41no—
220C2×C4×C71—41no—
221C2×C4×C71—41no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71—42no—
226C2×C4×C71—42no—
227C2×C4×C71—42no—
228C2×C4×C71—41no—
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—41no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—42no—
239C2×C4×C71—41no—
240C2×C4×C71—41no—
241C2×C4×C71—81no—
242C2×C4×C71—81no—
243C2×C4×C71—81no—
244C2×C4×C71—81no—
245C2×C4×C71—81no—
246C2×C4×C71—81no—
247C2×C4×C71—81no—
248C2×C4×C71—81no—
249C2×C4×C71—81no—
250C2×C4×C71#12111yescharacteristic
251C2×C4×C71#12111yescharacteristic
252C2×C4×C71#12311yescharacteristic
253C2×C4×C71#12411yescharacteristic
254C8×C71—42no—
255C8×C71—81no—
256C22×C71—21no—
257C22×C71—21no—
258C22×C71—21no—
259C22×C71—21no—
260C22×C71—21no—
261C22×C71—21no—
262C22×C71—21no—
263C22×C71—21no—
264C22×C71—21no—
265C22×C71—21no—
266C22×C71—21no—
267C22×C71—21no—
268C22×C71—44no—
269C22×C71—42no—
270C22×C71—42no—
271C22×C71—42no—
272C22×C71—82no—
273C22×C71—82no—
274C22×C71—82no—
275C22×C71—81no—
276C22×C71—81no—
277C22×C71—81no—
278C22×C71—81no—
279C22×C71—81no—
280C22×C71—81no—
281C22×C71—81no—
282C22×C71—81no—
283C22×C71—81no—
284C22×C71—81no—
285C22×C71—81no—
286C22×C71#39411yescharacteristic
287C22×C71#44711yescharacteristic
288C22×C71#45211yescharacteristic
289C22×C71#45311yescharacteristic
290C22×C71#45511yescharacteristic
291C22×C71B(4,𝔽2)11yescharacteristic
292C22×C71#45811yescharacteristic
293C22×C71#46611yescharacteristic
294C22×C71#46811yescharacteristic
295C22×C71#48011yescharacteristic
296C22×C71#48211yescharacteristic
297C4×C71—21no—
298C4×C71—21no—
299C4×C71—21no—
300C4×C71—21no—
301C4×C71—82no—
302C4×C71—82no—
303C4×C71—81no—
304C4×C71—81no—
305C4×C71—81no—
306C4×C71—81no—
307#10394C7111yesmaximal, characteristic, direct product factor, Sylow
308C2×C71—21no—
309C2×C71—21no—
310C2×C71—21no—
311C2×C71—21no—
312C2×C71—82no—
313C2×C71—161no—
314C2×C71#159511yescharacteristic
315C2×C71#160611yescharacteristic
316C2×C71#160711yescharacteristic
317C2×C71#161811yescharacteristic
318C2×C71#203311yescharacteristic
319C2×C71#203711yescharacteristic
320C2×C71#204411yescharacteristic
321#2617C2×C7111yescharacteristic
322#1874C2×C7111yescharacteristic, complemented
323#1852C2×C7111yescharacteristic, complemented
324C2×#327C2×C7111yescharacteristic, complemented
325C2×#323C2×C7111yescharacteristic, complemented
326#1439C2×C7111yescharacteristic, complemented
327#1294C2×C7111yescharacteristic, complemented
328C71#1039411yescharacteristic, direct product factor, Sylow
329C2×#121C22×C7111yescharacteristic
330C2×#120—21no—
331C22×C4⋊C4C22×C7111yescharacteristic
332C22×C22⋊C4—21no—
333C22×C22⋊C4C22×C7111yescharacteristic
334C2×#107—21no—
335C2×#102—21no—
336C2×C22⋊C8C22×C7111yescharacteristic
337C2×C22⋊C8C22×C7111yescharacteristic
338#397—21no—
339#394—21no—
340#394—21no—
341#392—21no—
342#386—21no—
343#380C22×C7111yescharacteristic
344C2×#95—21no—
345C2×#95C22×C7111yescharacteristic
346#340—21no—
347#340—21no—
348#327—22no—
349#323—22no—
350C25D4×C7111yescharacteristic
351C22×D4—21no—
352C22×D4—21no—
353C22×D4—21no—
354C22×D4D4×C7111yescharacteristic
355C23×C4—21no—
356C23×C4—21no—
357C23×C4D4×C7111yescharacteristic
358C23×C4C23×C7111yescharacteristic
359C22×C8—21no—
360C22×C8—21no—
361#121—21no—
362#121—21no—
363#121—21no—
364#121—21no—
365#120—42no—
366#120—42no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—42no—
370C2×C4⋊C4D4×C7111yescharacteristic
371C2×C4⋊C4D4×C7111yescharacteristic
372C2×C4⋊C4D4×C7111yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—42no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—42no—
388#107—42no—
389#102—42no—
390C22⋊C8—21no—
391C22⋊C8—21no—
392C22⋊C8—21no—
393C22⋊C8D4×C7112yescomplemented
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—42no—
400C24—21no—
401C24—21no—
402C24—21no—
403C24—21no—
404C24—21no—
405C24—21no—
406C24—21no—
407C24—41no—
408C24—41no—
409C24—41no—
410C24—41no—
411C24D8×C7112yes—
412C24C2×D4×C7111yescharacteristic
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—41no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—81no—
426C2×D4—81no—
427C2×D4—81no—
428C2×D4—81no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4C2×D4×C7111yescharacteristic
454C22×C4C2×D4×C7111yescharacteristic
455C22×C4C2×D4×C7111yescharacteristic
456C22×C4#41×C7111yescharacteristic
457C22×C4#41×C7111yescharacteristic
458C22×C4#41×C7111yescharacteristic
459C2×C8—22no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—48no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—81no—
485C22⋊C4—81no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—81no—
515C23—81no—
516C23—81no—
517C23—81no—
518C23—81no—
519C23—81no—
520C23#120×C7111yescharacteristic
521C23#121×C7111yescharacteristic
522C23#123×C7111yescharacteristic
523C23C2×D8×C7111yescharacteristic
524C23#135×C7111yescharacteristic
525C23AGL(1,ℤ/8)×C7111yescharacteristic
526C23#137×C7111yescharacteristic
527D4—42no—
528D4—81no—
529D4—84no—
530D4—84no—
531D4—81no—
532D4—81no—
533D4—81no—
534D4—81no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4#121×C7111yescharacteristic
572C2×C4#121×C7111yescharacteristic
573C2×C4#123×C7111yescharacteristic
574C2×C4#124×C7111yescharacteristic
575C8—42no—
576C8—81no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22#394×C7111yescharacteristic
608C22#447×C7111yescharacteristic
609C22#452×C7111yescharacteristic
610C22#453×C7111yescharacteristic
611C22#455×C7111yescharacteristic
612C22B(4,𝔽2)×C7111yescharacteristic
613C22#458×C7111yescharacteristic
614C22#466×C7111yescharacteristic
615C22#468×C7111yescharacteristic
616C22#480×C7111yescharacteristic
617C22#482×C7111yescharacteristic
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—82no—
623C4—81no—
624C4—81no—
625C4—82no—
626C4—81no—
627C4—81no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—82no—
633C2—161no—
634C2C71×#159511yescharacteristic
635C2C71×#160611yescharacteristic
636C2C71×#160711yescharacteristic
637C2C71×#161811yescharacteristic
638C2C71×#203311yescharacteristic
639C2C71×#203711yescharacteristic
640C2C71×#204411yescharacteristic
subgroupG/H
center, socleC23×C71#120
commutator, Frattini, cosocleC23×C4C23×C71
prime radicalC71×#2617C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C71 ◁ C23×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#1874C2semidirect
3. C71×#1852C2semidirect
4. C2×#327×C71C2semidirect
5. C2×#323×C71C2semidirect
6. C71×#1439C2semidirect
7. C71×#1294C2semidirect
72. C22⋊C8×C71D4semidirect
307. #10394C71direct (N × H)
322. #1874C2×C71semidirect
323. #1852C2×C71semidirect
324. C2×#327C2×C71semidirect
325. C2×#323C2×C71semidirect
326. #1439C2×C71semidirect
327. #1294C2×C71semidirect
328. C71#10394direct (N × H)
393. C22⋊C8D4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#26172
C71×#18742
C71×#18522
C2×#327×C712
C2×#323×C712
C71×#14392
C71×#12942
#1039471