Finite Group Databasebeta

← C2×#397×D71 #32163656

Subgroups — C2×#397×D71 #32163656

Order 18176.

5230 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D71C211yesmaximal, characteristic, complemented
2C22×C22⋊C4×D71C212yesmaximal
3C22×C22⋊C4×D71C211yesmaximal, characteristic
4C2×#17985928C212yesmaximal, complemented
5C2×#17985927C212yesmaximal, complemented
6C2×#17985926C211yesmaximal, characteristic, complemented
7C2×#17985925C211yesmaximal, characteristic, complemented
8C2×#17985924C211yesmaximal, characteristic, complemented
9#397×D71C2116yesmaximal, direct product factor
10C2×#95×D71C212yesmaximal, complemented
11C2×#95×D71C211yesmaximal, characteristic, complemented
12C2×#397×C71C211yesmaximal, characteristic, complemented
13C25×D71C2211yescharacteristic
14C23×C4×D71C2212yes—
15C23×C4×D71C2212yes—
16C23×C4×D71C2211yescharacteristic
17C23×C4×D71C2211yescharacteristic
18C22×C4⋊Dic71C2211yescharacteristic, complemented
19C22×Dic71⋊C4C2212yescomplemented
20C2×C4⋊C4×D71—28no—
21C2×C4⋊C4×D71C2218yescomplemented
22C22×C22⋊Dic71C2212yes—
23C22×C22⋊Dic71C2211yescharacteristic
24C22×(C2×D71)⋊C4C2211yescharacteristic
25C22×(C2×D71)⋊C4C2212yes—
26C22×(C2×D71)⋊C4C2212yes—
27C22×(C2×D71)⋊C4C2211yescharacteristic
28C2×C22⋊C4×D71—216no—
29C2×C22⋊C4×D71—28no—
30C2×C22⋊C4×D71C22116yes—
31C2×C22⋊C4×D71C2218yes—
32#17985928C22116yescomplemented
33#17985927C22116yescomplemented
34#17985926C2218yescomplemented
35#17985925C2218yescomplemented
36#17985924C2218yescomplemented
37C2×C71⋊#95C2211yescharacteristic, complemented
38C2×C71⋊#95C2212yescomplemented
39C2×C71⋊#95C2212yescomplemented
40C2×C71⋊#95C2212yescomplemented
41C2×C71⋊#95C2212yescomplemented
42#95×D71C22116yescomplemented
43#95×D71C2218yescomplemented
44C22×C4⋊C4×C71C2211yescharacteristic, complemented
45C22×C22⋊C4×C71C2211yescharacteristic
46C22×C22⋊C4×C71C2212yes—
47#397×C71C2218yescomplemented
48C2×#95×C71C2211yescharacteristic, complemented
49C2×#95×C71C2212yescomplemented
50C24×D71—24no—
51C24×D71—24no—
52C24×D71—28no—
53C24×D71—28no—
54C24×D71D418yes—
55C24×D71C2314yes—
56C24×D71C2311yescharacteristic
57C24×D71C2311yescharacteristic
58C23×Dic71C2312yes—
59C23×Dic71C2311yescharacteristic
60C23×Dic71C2312yes—
61C23×Dic71C2311yescharacteristic
62C22×C4×D71—28no—
63C22×C4×D71—28no—
64C22×C4×D71—28no—
65C22×C4×D71—24no—
66C22×C4×D71—28no—
67C22×C4×D71—28no—
68C22×C4×D71—28no—
69C22×C4×D71—28no—
70C22×C4×D71—28no—
71C22×C4×D71D418yes—
72C22×C4×D71C2318yes—
73C22×C4×D71C2314yes—
74C22×C4×D71C2318yes—
75C22×C4×D71C2314yes—
76C2×C4⋊Dic71—24no—
77C2×C4⋊Dic71C2314yescomplemented
78C2×Dic71⋊C4—28no—
79C2×Dic71⋊C4C2318yescomplemented
80C4⋊C4×D71—232no—
81C2×C22⋊Dic71—24no—
82C2×C22⋊Dic71—28no—
83C2×C22⋊Dic71C2318yes—
84C2×C22⋊Dic71C2314yes—
85C2×(C2×D71)⋊C4—28no—
86C2×(C2×D71)⋊C4—28no—
87C2×(C2×D71)⋊C4—24no—
88C2×(C2×D71)⋊C4—24no—
89C2×(C2×D71)⋊C4C2314yes—
90C2×(C2×D71)⋊C4C2318yes—
91C2×(C2×D71)⋊C4C2318yes—
92C2×(C2×D71)⋊C4C2314yes—
93C22⋊C4×D71—264no—
94C22⋊C4×D71—232no—
95C71⋊#95C2314yescomplemented
96C71⋊#95C2318yescomplemented
97C71⋊#95C2318yescomplemented
98C71⋊#95C2318yescomplemented
99C71⋊#95C2318yescomplemented
100C25×C71C2311yescharacteristic
101C23×C4×C71C2311yescharacteristic
102C23×C4×C71C2312yes—
103C23×C4×C71C2312yes—
104C23×C4×C71C2311yescharacteristic
105C2×C4⋊C4×C71—24no—
106C2×C4⋊C4×C71C2314yescomplemented
107C2×C22⋊C4×C71—28no—
108C2×C22⋊C4×C71—24no—
109C2×C22⋊C4×C71C2314yes—
110C2×C22⋊C4×C71C2318yes—
111#95×C71C2314yescomplemented
112#95×C71C2318yescomplemented
113C23×D71—22no—
114C23×D71—22no—
115C23×D71—24no—
116C23×D71—24no—
117C23×D71—232no—
118C23×D71—264no—
119C23×D71—28no—
120C23×D71—28no—
121C23×D71—216no—
122C23×D71—216no—
123C23×D71—416no—
124C23×D71—416no—
125C23×D71—432no—
126C23×D71C2×D4116yes—
127C23×D71C2×D414yes—
128C23×D71C2×D414yes—
129C23×D71C2×D414yes—
130C23×D71#4114yes—
131C23×D71#4118yes—
132C23×D71#4118yes—
133C23×D71C2412yes—
134C23×D71C2412yes—
135C22×Dic71—24no—
136C22×Dic71—24no—
137C22×Dic71—24no—
138C22×Dic71—22no—
139C22×Dic71—24no—
140C22×Dic71—24no—
141C22×Dic71—24no—
142C22×Dic71—24no—
143C22×Dic71—24no—
144C22×Dic71C2×D414yes—
145C22×Dic71C2414yes—
146C22×Dic71C2412yes—
147C22×Dic71C2414yes—
148C22×Dic71C2412yes—
149C2×C4×D71—232no—
150C2×C4×D71—216no—
151C2×C4×D71—216no—
152C2×C4×D71—216no—
153C2×C4×D71—28no—
154C2×C4×D71—216no—
155C2×C4×D71—216no—
156C2×C4×D71—216no—
157C2×C4×D71—216no—
158C2×C4×D71—216no—
159C2×C4×D71—432no—
160C2×C4×D71—432no—
161C2×C4×D71—416no—
162C2×C4×D71C2×D4116yes—
163C4⋊Dic71—28no—
164Dic71⋊C4—216no—
165C22⋊Dic71—216no—
166C22⋊Dic71—28no—
167(C2×D71)⋊C4—216no—
168(C2×D71)⋊C4—216no—
169(C2×D71)⋊C4—28no—
170(C2×D71)⋊C4—28no—
171C24×C71—22no—
172C24×C71—22no—
173C24×C71—24no—
174C24×C71—24no—
175C24×C71C2×D414yes—
176C24×C71C2412yes—
177C24×C71C2411yescharacteristic
178C22×C4×C71—24no—
179C22×C4×C71—24no—
180C22×C4×C71—24no—
181C22×C4×C71—22no—
182C22×C4×C71—24no—
183C22×C4×C71—24no—
184C22×C4×C71—24no—
185C22×C4×C71—24no—
186C22×C4×C71—24no—
187C22×C4×C71C2×D414yes—
188C22×C4×C71C2414yes—
189C22×C4×C71C2412yes—
190C22×C4×C71C2414yes—
191C22×C4×C71C2412yes—
192C4⋊C4×C71—28no—
193C22⋊C4×C71—216no—
194C22⋊C4×C71—28no—
195C23×C71—28no—
196C23×C71—216no—
197C23×C71—22no—
198C23×C71—22no—
199C23×C71—24no—
200C23×C71—24no—
201C23×C71—44no—
202C23×C71—44no—
203C23×C71—48no—
204C23×C71C22×D414yes—
205C23×C71C22×D412yes—
206C23×C71C22×D412yes—
207C23×C71C2×#4112yes—
208C23×C71C2×#4114yes—
209C23×C71C2×#4114yes—
210C23×C71C2511yescharacteristic
211C22×D71—28no—
212C22×D71—216no—
213C22×D71—22no—
214C22×D71—22no—
215C22×D71—24no—
216C22×D71—24no—
217C22×D71—232no—
218C22×D71—264no—
219C22×D71—44no—
220C22×D71—44no—
221C22×D71—48no—
222C22×D71—416no—
223C22×D71—416no—
224C22×D71—432no—
225C22×D71—4128no—
226C22×D71#121116yes—
227C22×D71#123116yes—
228C22×D71#123132yes—
229C22×D71#124116yes—
230C22×D71#126132yes—
231C22×D71C22×D414yes—
232C22×D71C22×D414yes—
233C22×D71C22×D414yes—
234C22×D71C2×#4114yes—
235C22×D71C2×#4118yes—
236C22×D71C2×#4118yes—
237C2×C4×C71—28no—
238C2×C4×C71—24no—
239C2×C4×C71—24no—
240C2×C4×C71—24no—
241C2×C4×C71—22no—
242C2×C4×C71—24no—
243C2×C4×C71—24no—
244C2×C4×C71—24no—
245C2×C4×C71—24no—
246C2×C4×C71—24no—
247C2×C4×C71—48no—
248C2×C4×C71—48no—
249C2×C4×C71—44no—
250C2×C4×C71C22×D414yes—
251C2×Dic71—28no—
252C2×Dic71—24no—
253C2×Dic71—24no—
254C2×Dic71—24no—
255C2×Dic71—22no—
256C2×Dic71—24no—
257C2×Dic71—24no—
258C2×Dic71—24no—
259C2×Dic71—24no—
260C2×Dic71—24no—
261C2×Dic71—48no—
262C2×Dic71—48no—
263C2×Dic71—44no—
264C2×Dic71C22×D414yes—
265C4×D71—232no—
266C4×D71—432no—
267C4×D71—432no—
268C4×D71—416no—
269C22×C71—24no—
270C22×C71—28no—
271C22×C71—416no—
272C22×C71—42no—
273C22×C71—42no—
274C22×C71—44no—
275C22×C71C2×#12114yes—
276C22×C71C2×#12314yes—
277C22×C71C2×#12318yes—
278C22×C71C2×#12414yes—
279C22×C71C2×#12618yes—
280C22×C71C23×D411yescharacteristic
281C22×C71C23×D411yescharacteristic
282C22×C71C22×#4111yescharacteristic
283C22×C71C22×#4112yes—
284C22×C71C22×#4112yes—
285C2×D71—24no—
286C2×D71—28no—
287C2×D71—416no—
288C2×D71—42no—
289C2×D71—42no—
290C2×D71—44no—
291C2×D71—464no—
292C2×D71#397164yesdirect product factor
293C2×D71C2×#12118yes—
294C2×D71C2×#12318yes—
295C2×D71C2×#123116yes—
296C2×D71C2×#12418yes—
297C2×D71C2×#126116yes—
298C4×C71—24no—
299C4×C71—44no—
300C4×C71—44no—
301C4×C71—42no—
302Dic71—24no—
303Dic71—44no—
304Dic71—44no—
305Dic71—42no—
306C22×#397—711nomaximal, Sylow
307C2×C71—44no—
308C2×C71C2×#39718yescomplemented
309C2×C71C22×#12111yescharacteristic
310C2×C71C22×#12311yescharacteristic
311C2×C71C22×#12312yes—
312C2×C71C22×#12411yescharacteristic
313C2×C71C22×#12612yes—
314D71—44no—
315D71C2×#397116yesdirect product factor
316C23×C4⋊C4—711no—
317C23×C22⋊C4—711no—
318C23×C22⋊C4—712no—
319C2×#397—7116no—
320C2×#397—711no—
321C2×#397—712no—
322C2×#397—711no—
323C2×#397—711no—
324C2×#397—712no—
325C2×#397D7111yescharacteristic, direct product factor
326C22×#95—711no—
327C22×#95—712no—
328C71C22×#39711yescharacteristic, complemented, Sylow
329C26—711no—
330C24×C4—711no—
331C24×C4—712no—
332C24×C4—711no—
333C24×C4—712no—
334C22×C4⋊C4—712no—
335C22×C4⋊C4—711no—
336C22×C4⋊C4—718no—
337C22×C4⋊C4—1428no—
338C22×C4⋊C4C2×D7111yescharacteristic, complemented
339C22×C22⋊C4—711no—
340C22×C22⋊C4—712no—
341C22×C22⋊C4—711no—
342C22×C22⋊C4—712no—
343C22×C22⋊C4—712no—
344C22×C22⋊C4—711no—
345C22×C22⋊C4—718no—
346C22×C22⋊C4—7116no—
347C22×C22⋊C4—1428no—
348C22×C22⋊C4—14216no—
349C22×C22⋊C4C2×D7112yes—
350C22×C22⋊C4C2×D7111yescharacteristic
351#397—718no—
352#397—718no—
353#397—7116no—
354#397—718no—
355#397—7116no—
356#397C2×D7118yesdirect product factor
357C2×#95—712no—
358C2×#95—712no—
359C2×#95—712no—
360C2×#95—712no—
361C2×#95—711no—
362C2×#95—718no—
363C2×#95—7116no—
364C2×#95C2×D7112yescomplemented
365C2×#95C2×D7111yescharacteristic, complemented
366C25—711no—
367C25—718no—
368C25—714no—
369C25—711no—
370C25—1424no—
371C25—1424no—
372C25—1428no—
373C25—1428no—
374C25C22×D7111yescharacteristic
375C23×C4—711no—
376C23×C4—718no—
377C23×C4—714no—
378C23×C4—712no—
379C23×C4—712no—
380C23×C4—711no—
381C23×C4—718no—
382C23×C4—714no—
383C23×C4—718no—
384C23×C4—1424no—
385C23×C4—1428no—
386C23×C4—1428no—
387C23×C4—1428no—
388C23×C4—1428no—
389C23×C4—1428no—
390C23×C4—1428no—
391C23×C4—1428no—
392C23×C4—1428no—
393C23×C4C22×D7112yes—
394C23×C4C22×D7112yes—
395C23×C4C22×D7111yescharacteristic
396C23×C4C22×D7111yescharacteristic
397C2×C4⋊C4—24no—
398C2×C4⋊C4—718no—
399C2×C4⋊C4—714no—
400C2×C4⋊C4—14232no—
401C2×C4⋊C4—1428no—
402C2×C4⋊C4—1424no—
403C2×C4⋊C4C22×D7114yescomplemented
404C2×C22⋊C4—28no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—714no—
407C2×C22⋊C4—718no—
408C2×C22⋊C4—714no—
409C2×C22⋊C4—718no—
410C2×C22⋊C4—718no—
411C2×C22⋊C4—714no—
412C2×C22⋊C4—14232no—
413C2×C22⋊C4—14264no—
414C2×C22⋊C4—1424no—
415C2×C22⋊C4—1428no—
416C2×C22⋊C4—1428no—
417C2×C22⋊C4—1424no—
418C2×C22⋊C4—1428no—
419C2×C22⋊C4—1424no—
420C2×C22⋊C4C22×D7118yes—
421C2×C22⋊C4C22×D7114yes—
422#95—718no—
423#95—718no—
424#95—714no—
425#95—718no—
426#95—718no—
427#95C22×D7118yescomplemented
428#95C22×D7114yescomplemented
429C24—22no—
430C24—22no—
431C24—24no—
432C24—24no—
433C24—28416no—
434C24—28416no—
435C24—28432no—
436C24—714no—
437C24—714no—
438C24—714no—
439C24—718no—
440C24—718no—
441C24—712no—
442C24—7116no—
443C24—714no—
444C24—712no—
445C24—1428no—
446C24—1428no—
447C24—14216no—
448C24—14216no—
449C24—14232no—
450C24—14264no—
451C24—1422no—
452C24—1424no—
453C24—1424no—
454C24—1422no—
455C24D4×D7114yes—
456C24C23×D7112yes—
457C24C23×D7111yescharacteristic
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—28432no—
468C22×C4—28432no—
469C22×C4—28416no—
470C22×C4—714no—
471C22×C4—712no—
472C22×C4—7116no—
473C22×C4—714no—
474C22×C4—712no—
475C22×C4—714no—
476C22×C4—1428no—
477C22×C4—14216no—
478C22×C4—14216no—
479C22×C4—14216no—
480C22×C4—14216no—
481C22×C4—14216no—
482C22×C4—14216no—
483C22×C4—14216no—
484C22×C4—14216no—
485C22×C4—14232no—
486C22×C4—1424no—
487C22×C4—1424no—
488C22×C4—1422no—
489C22×C4—1424no—
490C22×C4—1424no—
491C22×C4—1424no—
492C22×C4—1424no—
493C22×C4—1424no—
494C22×C4—1424no—
495C22×C4D4×D7114yes—
496C22×C4C23×D7114yes—
497C22×C4C23×D7112yes—
498C22×C4C23×D7114yes—
499C22×C4C23×D7112yes—
500C4⋊C4—28no—
501C4⋊C4—14216no—
502C4⋊C4—1428no—
503C22⋊C4—216no—
504C22⋊C4—28no—
505C22⋊C4—1428no—
506C22⋊C4—14216no—
507C22⋊C4—14216no—
508C22⋊C4—1428no—
509C22⋊C4—14216no—
510C22⋊C4—1428no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—216no—
517C23—44no—
518C23—44no—
519C23—48no—
520C23—284128no—
521C23—28416no—
522C23—28416no—
523C23—28432no—
524C23—2844no—
525C23—2844no—
526C23—2848no—
527C23—714no—
528C23—714no—
529C23—714no—
530C23—718no—
531C23—718no—
532C23—7116no—
533C23—7132no—
534C23—7116no—
535C23—7116no—
536C23—7132no—
537C23—714no—
538C23—14232no—
539C23—14264no—
540C23—1422no—
541C23—1424no—
542C23—1424no—
543C23—1422no—
544C23—1428no—
545C23—14216no—
546C23C2×D4×D7114yes—
547C23C2×D4×D7112yes—
548C23C2×D4×D7112yes—
549C23#41×D7112yes—
550C23#41×D7114yes—
551C23#41×D7114yes—
552C23C24×D7111yescharacteristic
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—44no—
566C2×C4—28432no—
567C2×C4—28432no—
568C2×C4—28416no—
569C2×C4—2848no—
570C2×C4—2848no—
571C2×C4—2844no—
572C2×C4—714no—
573C2×C4—14232no—
574C2×C4—1424no—
575C2×C4—1424no—
576C2×C4—1422no—
577C2×C4—1424no—
578C2×C4—1424no—
579C2×C4—1424no—
580C2×C4—1424no—
581C2×C4—1424no—
582C2×C4—1424no—
583C2×C4—1428no—
584C2×C4C2×D4×D7114yes—
585C22—24no—
586C22—28no—
587C22—416no—
588C22—42no—
589C22—42no—
590C22—44no—
591C22—28464no—
592C22—28416no—
593C22—2842no—
594C22—2842no—
595C22—2844no—
596C22—7164no—
597C22—718no—
598C22—7116no—
599C22—718no—
600C22—718no—
601C22—7116no—
602C22—1424no—
603C22—1428no—
604C22#121×D7114yes—
605C22#123×D7114yes—
606C22#123×D7118yes—
607C22#124×D7114yes—
608C22#126×D7118yes—
609C22C22×D4×D7111yescharacteristic
610C22C22×D4×D7111yescharacteristic
611C22C2×#41×D7111yescharacteristic
612C22C2×#41×D7112yes—
613C22C2×#41×D7112yes—
614C4—24no—
615C4—44no—
616C4—44no—
617C4—42no—
618C4—2844no—
619C4—2844no—
620C4—2842no—
621C4—1424no—
622C2—44no—
623C2—2844no—
624C2—7116no—
625C2#397×D7118yesdirect product factor
626C2C2×#121×D7111yescharacteristic
627C2C2×#123×D7111yescharacteristic
628C2C2×#123×D7112yes—
629C2C2×#124×D7111yescharacteristic
630C2C2×#126×D7112yes—
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#397×C71C2
socleC24×C71C24
prime radicalC25×D71C22
hypercenterC2×#397D71
hypocenterC71C22×#397
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊C4×D71C2semidirect
4. C2×#17985928C2semidirect
5. C2×#17985927C2semidirect
6. C2×#17985926C2semidirect
7. C2×#17985925C2semidirect
8. C2×#17985924C2semidirect
9. #397×D71C2direct (N × H)
10. C2×#95×D71C2semidirect
11. C2×#95×D71C2semidirect
12. C2×#397×C71C2semidirect
18. C22×C4⋊Dic71C22semidirect
19. C22×Dic71⋊C4C22semidirect
21. C2×C4⋊C4×D71C22semidirect
32. #17985928C22semidirect
33. #17985927C22semidirect
34. #17985926C22semidirect
35. #17985925C22semidirect
36. #17985924C22semidirect
37. C2×C71⋊#95C22semidirect
38. C2×C71⋊#95C22semidirect
39. C2×C71⋊#95C22semidirect
40. C2×C71⋊#95C22semidirect
41. C2×C71⋊#95C22semidirect
42. #95×D71C22semidirect
43. #95×D71C22semidirect
44. C22×C4⋊C4×C71C22semidirect
47. #397×C71C22semidirect
48. C2×#95×C71C22semidirect
49. C2×#95×C71C22semidirect
77. C2×C4⋊Dic71C23semidirect
79. C2×Dic71⋊C4C23semidirect
95. C71⋊#95C23semidirect
96. C71⋊#95C23semidirect
97. C71⋊#95C23semidirect
98. C71⋊#95C23semidirect
99. C71⋊#95C23semidirect
106. C2×C4⋊C4×C71C23semidirect
111. #95×C71C23semidirect
112. #95×C71C23semidirect
292. C2×D71#397direct (N × H)
308. C2×C71C2×#397semidirect
315. D71C2×#397direct (N × H)
325. C2×#397D71direct (N × H)
328. C71C22×#397semidirect
338. C22×C4⋊C4C2×D71semidirect
356. #397C2×D71direct (N × H)
364. C2×#95C2×D71semidirect
365. C2×#95C2×D71semidirect
403. C2×C4⋊C4C22×D71semidirect
427. #95C22×D71semidirect
428. #95C22×D71semidirect
625. C2#397×D71direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D712
C22×C22⋊C4×D712
C2×#179859282
C2×#179859272
C2×#179859262
C2×#179859252
C2×#179859242
#397×D712
C2×#95×D712
C2×#397×C712
C22×#39771