Finite Group Databasebeta

← #562×#6157 #28328987

Subgroups — #562×#6157 #28328987

Order 13568.

2248 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291261C211yesmaximal, characteristic, complemented
2#20291258C212yesmaximal, complemented
3#20291257C214yesmaximal, complemented
4#20291256C214yesmaximal, complemented
5#20291255C214yesmaximal, complemented
6#125×#6157C211yesmaximal, characteristic, complemented
7#124×#6157C212yesmaximal, complemented
8C22⋊Q8×#6157C214yesmaximal, complemented
9#123×#6157C214yesmaximal, complemented
10#121×#6157C214yesmaximal, complemented
11#562×D53C211yesmaximal, characteristic
12C2×Q8×#6157C2212yes—
13C2×D4×#6157C2214yes—
14C2×D4×#6157C2212yes—
15C22×C4×#6157C2214yes—
16(C2×#6157)⋊Q8C2214yes—
17(C2×#6157)⋊Q8C2212yes—
18#16811575C2214yes—
19#16811575C2212yes—
20#16811574C2214yes—
21#16811574C2214yes—
22#16811574C2212yes—
23#16811573C2214yes—
24#16811573C2214yes—
25#16811573C2214yes—
26#16811573C2214yes—
27#16811573C2214yes—
28#16811573C2214yes—
29#16811572C2214yes—
30#16811572C2214yes—
31#16811572C2214yes—
32#16811572C2214yes—
33#16811572C2214yes—
34#16811571C2214yes—
35#16811571C2214yes—
36#16811571C2214yes—
37#16811570C2214yes—
38#16811570C2214yes—
39#16811570C2214yes—
40#16811567C2212yes—
41#16811567C2211yescharacteristic
42#16811566C2214yes—
43#16811565C2214yes—
44#16811565C2214yes—
45C4⋊C4×#6157C2214yes—
46C4⋊C4×#6157C2212yes—
47C4⋊C4×#6157C2214yes—
48C22⋊C4×#6157C2214yes—
49C22⋊C4×#6157C2214yes—
50C22⋊C4×#6157C2214yes—
51C42×#6157C2211yescharacteristic
52#16811371C411yescharacteristic, complemented
53#16811370C414yescomplemented
54#16811369C414yescomplemented
55#16811368C414yescomplemented
56#16811367C412yescomplemented
57#125×D53C2211yescharacteristic
58#124×D53C2212yes—
59#123×D53C2214yes—
60C22⋊Q8×D53C2214yes—
61#121×D53C2214yes—
62C53×#562C411yescharacteristic, complemented
63C23×#6157C2314yes—
64C23×#6157C2314yes—
65Q8×#6157—44no—
66D4×#6157—48no—
67D4×#6157—44no—
68C2×C4⋊#6157C2314yes—
69C2×C4⋊#6157C2314yes—
70C2×C4⋊#6157C2312yes—
71C2×C4⋊#6157C2314yes—
72C2×C4⋊#6157C2314yes—
73C2×C4⋊#6157C2312yes—
74C2×C4⋊#6157C2312yes—
75C2×C22⋊#6157C2314yes—
76C2×C22⋊#6157C2314yes—
77C2×C22⋊#6157C2314yes—
78C2×C22⋊#6157C2314yes—
79C2×C22⋊#6157C2314yes—
80C2×C22⋊#6157C2314yes—
81C2×C4×#6157—48no—
82C2×C4×#6157C2311yescharacteristic
83C2×C4×#6157C2312yes—
84C2×C4×#6157C2314yes—
85C2×C4×#6157C2314yes—
86C2×C4×#6157C2312yes—
87C2×C4×#6157C2311yescharacteristic
88C2×C4×#6157C2314yes—
89C2×C4×#6157C2314yes—
90(C2×#6157)⋊C4C2314yes—
91(C2×#6157)⋊C4C2314yes—
92(C2×#6157)⋊C4C2314yes—
93(C2×#6157)⋊C4C2314yes—
94(C2×#6157)⋊C4C2314yes—
95(C2×#6157)⋊C4C2314yes—
96(C2×#6157)⋊C4C2314yes—
97(C2×#6157)⋊C4C2314yes—
98(C2×#6157)⋊C4C2314yes—
99(C2×#6157)⋊C4C2314yes—
100(C2×#6157)⋊C4C2314yes—
101C2×Q8×D53C2312yes—
102C2×D4×D53C2314yes—
103C2×D4×D53C2312yes—
104C22×C4×D53C2314yes—
105#13133209C2×C411yescharacteristic, complemented
106#13133208C2×C412yescomplemented
107#13133207C2×C414yescomplemented
108#13133206C2×C412yescomplemented
109#13133205C2×C412yescomplemented
110#13133204C2×C412yescomplemented
111#13133203C2×C414yescomplemented
112#13133203C2×C414yescomplemented
113#13133202C2×C414yescomplemented
114#13133202C2×C414yescomplemented
115#13133201C2×C414yescomplemented
116#13133200C2×C414yescomplemented
117#13133199C2×C414yescomplemented
118#13133199C2×C414yescomplemented
119#13133198C2×C414yescomplemented
120#13133197C2×C414yescomplemented
121#13133196C2×C414yescomplemented
122#13133195C2×C414yescomplemented
123#13133195C2×C414yescomplemented
124#13133194C2×C414yescomplemented
125#13133194C2×C414yescomplemented
126#13133193C2×C414yescomplemented
127#13133192C2×C414yescomplemented
128#13133191C2×C414yescomplemented
129#13133190C2×C414yescomplemented
130#13133190C2×C414yescomplemented
131#13133189C2×C414yescomplemented
132#13133189C2×C414yescomplemented
133#13133188C2×C414yescomplemented
134C4⋊C4×D53C2314yes—
135C4⋊C4×D53C2314yes—
136C4⋊C4×D53C2312yes—
137C22⋊C4×D53C2314yes—
138C22⋊C4×D53C2314yes—
139C22⋊C4×D53C2314yes—
140C42×D53C2311yescharacteristic
141#125×C53C2×C411yescharacteristic, complemented
142#124×C53C2×C412yescomplemented
143#123×C53C2×C414yescomplemented
144C22⋊Q8×C53C2×C414yescomplemented
145#121×C53C2×C414yescomplemented
146C23×D53C2414yes—
147C22×#6157—28no—
148C22×#6157—28no—
149C22×#6157—28no—
150C22×#6157—44no—
151C22×#6157—44no—
152C22×#6157—44no—
153C22×#6157C2411yescharacteristic
154C22×#6157C2414yes—
155C22×#6157C2414yes—
156C22×#6157C2414yes—
157C22×#6157C2412yes—
158C22×#6157C2411yescharacteristic
159C2×Q8×C53C22×C412yes—
160C2×D4×C53C22×C414yes—
161C2×D4×C53C22×C412yes—
162C22×C4×C53C22×C414yes—
163C2×C22⋊D53C22×C414yes—
164C2×C22⋊D53C22×C414yes—
165C2×C22⋊D53C22×C414yes—
166C22×Dic53C22×C414yes—
167Q8×D53—42no—
168Q8×D53—42no—
169D4×D53—42no—
170D4×D53—44no—
171D4×D53—44no—
172D4×D53—42no—
173C2×D212C22×C412yes—
174C2×D212C22×C414yes—
175C2×C4×D53—44no—
176C2×C4×D53—44no—
177C2×C4×D53C22×C414yes—
178C2×C4×D53C22×C414yes—
179C2×C4×D53C2411yescharacteristic
180C2×C4×D53C2412yes—
181C2×C4×D53C2414yes—
182C2×C4×D53C2414yes—
183C2×Dic106C22×C412yes—
184C2×Dic106C22×C414yes—
185C22⋊#6157—44no—
186C22⋊#6157—44no—
187C22⋊#6157—44no—
188C22⋊#6157—44no—
189C22⋊#6157—44no—
190C22⋊#6157—44no—
191C4⋊#6157—42no—
192C4⋊#6157—44no—
193C4⋊#6157—44no—
194C4⋊#6157—44no—
195C4⋊#6157—42no—
196C4⋊#6157—44no—
197C4⋊#6157—42no—
198C4×#6157—28no—
199C4×#6157—28no—
200C4×#6157—24no—
201C4×#6157—22no—
202C4×#6157—44no—
203C4×#6157—44no—
204C4×#6157—41no—
205C4×#6157—42no—
206C4⋊C4×C53C22×C414yes—
207C4⋊C4×C53C22×C414yes—
208C4⋊C4×C53C22×C412yes—
209C22⋊C4×C53C22×C414yes—
210C22⋊C4×C53C22×C414yes—
211C22⋊C4×C53C22×C414yes—
212C42×C53C22×C411yescharacteristic
213C22⋊Dic53C22×C414yes—
214C22⋊Dic53C22×C414yes—
215C22⋊Dic53C22×C414yes—
216#11719921C22×C414yes—
217#11719921C22×C414yes—
218#11719921C22×C414yes—
219#11719921C22×C414yes—
220#11719921C22×C414yes—
221#11719921C22×C414yes—
222C4⋊Dic53C22×C414yes—
223C4⋊Dic53C22×C414yes—
224C4⋊Dic53C22×C412yes—
225#11719919C22×C414yes—
226#11719919C22×C414yes—
227#11719919C22×C414yes—
228#11719919C22×C414yes—
229#11719919C22×C414yes—
230C4×Dic53C22×C411yescharacteristic
231C4×Dic53C22×C412yes—
232C23×C53C23×C414yes—
233C22×D53—24no—
234C22×D53—24no—
235C22×D53—24no—
236C22×D53—24no—
237C22×D53—24no—
238C22×D53—24no—
239C22×D53C23×C414yes—
240C22×D53C2511yescharacteristic
241C2×#6157—24no—
242C2×#6157—24no—
243C2×#6157—22no—
244C2×#6157—24no—
245C2×#6157—24no—
246C2×#6157—21no—
247C2×#6157—24no—
248C2×#6157—22no—
249C2×#6157—22no—
250C2×#6157—22no—
251C2×#6157—24no—
252C2×#6157—24no—
253C2×#6157—24no—
254C2×#6157—24no—
255C2×#6157—416no—
256C2×#6157—416no—
257C2×#6157H(5,𝔽2)14yes—
258C2×#61572−1+412yes—
259Q8×C53—42no—
260D4×C53—42no—
261D4×C53—44no—
262C2×C4×C53—44no—
263C2×C4×C53C23×C414yes—
264C2×C4×C53C23×C412yes—
265C2×C4×C53C23×C414yes—
266C2×C4×C53C23×C411yescharacteristic
267C22⋊D53—44no—
268C22⋊D53—44no—
269C22⋊D53—44no—
270C2×Dic53—44no—
271C2×Dic53C23×C414yes—
272C2×Dic53C23×C412yes—
273C2×Dic53C23×C414yes—
274C2×Dic53C23×C411yescharacteristic
275D212—42no—
276D212—44no—
277C4×D53—24no—
278C4×D53—24no—
279C4×D53—22no—
280C4×D53—21no—
281C4×D53—21no—
282C4×D53—22no—
283C4×D53—24no—
284C4×D53—24no—
285C4×D53—44no—
286C4×D53—44no—
287Dic106—42no—
288Dic106—44no—
289C4×#562—531nomaximal, Sylow
290C22×C53—24no—
291C22×C53—24no—
292C22×C53—24no—
293C22×C53C24×C411yescharacteristic
294C2×D53—24no—
295C2×D53—24no—
296C2×D53—24no—
297C2×D53—44no—
298C2×D53—44no—
299C2×D53—44no—
300C2×D53—44no—
301C2×D53C2×H(5,𝔽2)12yes—
302C2×D53C2×2−1+411yescharacteristic
303C2×D53#58511yescharacteristic
304C2×D53#58512yes—
305#6157—44no—
306#6157—42no—
307#6157—44no—
308#6157—44no—
309#6157—41no—
310#6157#56214yesdirect product factor
311C4×C53—24no—
312C4×C53—22no—
313C4×C53—24no—
314C4×C53—21no—
315Dic53—24no—
316Dic53—22no—
317Dic53—24no—
318Dic53—21no—
319C2×#562—531no—
320#2382—531no—
321#2379—532no—
322#2378—534no—
323#2377—534no—
324#2376—534no—
325C4×#125—531no—
326C4×#124—532no—
327C4×C22⋊Q8—534no—
328C4×#123—534no—
329C4×#121—534no—
330C2×C53—44no—
331C2×C53C4×H(5,𝔽2)12yes—
332C2×C53C4×2−1+411yescharacteristic
333D53—44no—
334D53C2×#56211yescharacteristic
335D53#352912yes—
336D53#353111yescharacteristic
337#562—531no—
338#562—532no—
339#562—534no—
340#562—534no—
341#562—534no—
342#562#615711yescharacteristic, direct product factor
343C2×#125—531no—
344C2×#124—532no—
345C2×#123—534no—
346C2×C22⋊Q8—534no—
347C2×#121—534no—
348C2×C4×Q8—532no—
349C2×C4×D4—534no—
350C2×C4×D4—532no—
351C22×C42—534no—
352#391—532no—
353#391—534no—
354#390—534no—
355#390—532no—
356#389—532no—
357#389—534no—
358#389—534no—
359#388—534no—
360#388—534no—
361#388—534no—
362#388—534no—
363#388—534no—
364#388—534no—
365#387—534no—
366#387—534no—
367#387—534no—
368#387—534no—
369#387—534no—
370#386—534no—
371#386—534no—
372#386—534no—
373#385—534no—
374#385—534no—
375#385—534no—
376#382—531no—
377#382—532no—
378#381—534no—
379#380—534no—
380#380—534no—
381C4×C4⋊C4—534no—
382C4×C4⋊C4—532no—
383C4×C4⋊C4—534no—
384C4×C22⋊C4—534no—
385C4×C22⋊C4—534no—
386C4×C22⋊C4—534no—
387C43—531no—
388C53C4×#56211yescharacteristic, complemented, Sylow
389C22×Q8—532no—
390C22×D4—534no—
391C22×D4—532no—
392C23×C4—534no—
393C23×C4—534no—
394C23×C4—534no—
395#125—534no—
396#125—532no—
397#125—531no—
398#125C2×#615711yescharacteristic, complemented
399#124—532no—
400#124—532no—
401#124—534no—
402#124—534no—
403#124—532no—
404#124C2×#615712yescomplemented
405#123—534no—
406#123—534no—
407#123—534no—
408#123—534no—
409#123—534no—
410#123—534no—
411#123—534no—
412#123C2×#615714yescomplemented
413C22⋊Q8—534no—
414C22⋊Q8—534no—
415C22⋊Q8—534no—
416C22⋊Q8—534no—
417C22⋊Q8—534no—
418C22⋊Q8—534no—
419C22⋊Q8—534no—
420C22⋊Q8C2×#615714yescomplemented
421#121—534no—
422#121—534no—
423#121—534no—
424#121—534no—
425#121—534no—
426#121—534no—
427#121—534no—
428#121C2×#615714yescomplemented
429C4×Q8—2124no—
430C4×D4—2128no—
431C4×D4—2124no—
432C2×C4⋊C4—534no—
433C2×C4⋊C4—534no—
434C2×C4⋊C4—532no—
435C2×C4⋊C4—534no—
436C2×C4⋊C4—532no—
437C2×C4⋊C4—534no—
438C2×C4⋊C4—532no—
439C2×C4⋊C4—534no—
440C2×C4⋊C4—534no—
441C2×C4⋊C4—532no—
442C2×C22⋊C4—534no—
443C2×C22⋊C4—534no—
444C2×C22⋊C4—534no—
445C2×C22⋊C4—534no—
446C2×C22⋊C4—534no—
447C2×C22⋊C4—534no—
448C2×C22⋊C4—534no—
449C2×C22⋊C4—534no—
450C2×C22⋊C4—534no—
451C2×C42—531no—
452C2×C42—532no—
453C2×C42—534no—
454C2×C42—534no—
455C2×C42—531no—
456C2×C42—532no—
457C2×C42—531no—
458C2×C42—534no—
459C2×C42—534no—
460C2×C42—2128no—
461#95—534no—
462#95—534no—
463#95—534no—
464#95—534no—
465#95—534no—
466#95—534no—
467#95—534no—
468#95—534no—
469#95—534no—
470#95—534no—
471#95—534no—
472C24—534no—
473C2×Q8—534no—
474C2×Q8—532no—
475C2×Q8—2122no—
476C2×Q8—2122no—
477C2×Q8C22×#615712yes—
478C2×D4—534no—
479C2×D4—534no—
480C2×D4—534no—
481C2×D4—534no—
482C2×D4—532no—
483C2×D4—2124no—
484C2×D4—2122no—
485C2×D4—2122no—
486C2×D4—2124no—
487C2×D4C22×#615714yes—
488C2×D4C22×#615712yes—
489C22×C4—531no—
490C22×C4—534no—
491C22×C4—531no—
492C22×C4—532no—
493C22×C4—534no—
494C22×C4—534no—
495C22×C4—534no—
496C22×C4—534no—
497C22×C4—531no—
498C22×C4—534no—
499C22×C4—534no—
500C22×C4—534no—
501C22×C4—532no—
502C22×C4—1068no—
503C22×C4—1068no—
504C22×C4—1068no—
505C22×C4—2124no—
506C22×C4—2124no—
507C22×C4—2124no—
508C22×C4—2124no—
509C22×C4—2124no—
510C22×C4C22×#615714yes—
511C4⋊C4—534no—
512C4⋊C4—534no—
513C4⋊C4—534no—
514C4⋊C4—534no—
515C4⋊C4—534no—
516C4⋊C4—534no—
517C4⋊C4—534no—
518C4⋊C4—532no—
519C4⋊C4—2122no—
520C4⋊C4—2124no—
521C4⋊C4—2124no—
522C4⋊C4—2124no—
523C4⋊C4—2122no—
524C4⋊C4—2124no—
525C4⋊C4—2122no—
526C4⋊C4C22×#615714yes—
527C4⋊C4C22×#615714yes—
528C4⋊C4C22×#615712yes—
529C22⋊C4—534no—
530C22⋊C4—534no—
531C22⋊C4—534no—
532C22⋊C4—534no—
533C22⋊C4—534no—
534C22⋊C4—534no—
535C22⋊C4—534no—
536C22⋊C4—534no—
537C22⋊C4—534no—
538C22⋊C4—2124no—
539C22⋊C4—2124no—
540C22⋊C4—2124no—
541C22⋊C4—2124no—
542C22⋊C4—2124no—
543C22⋊C4—2124no—
544C22⋊C4C22×#615714yes—
545C22⋊C4C22×#615714yes—
546C22⋊C4C22×#615714yes—
547C42—532no—
548C42—531no—
549C42—1062no—
550C42—1068no—
551C42—1068no—
552C42—1064no—
553C42—2121no—
554C42—2122no—
555C42—2124no—
556C42—2124no—
557C42C22×#615711yescharacteristic
558C23—531no—
559C23—534no—
560C23—1064no—
561C23—1064no—
562C23—1064no—
563C23—1064no—
564C23—1064no—
565C23—1064no—
566C23C23×#615714yes—
567Q8—42no—
568Q8—2124no—
569Q8—2122no—
570D4—44no—
571D4—42no—
572D4—2124no—
573D4—2122no—
574D4—2124no—
575D4—2124no—
576D4—2124no—
577C2×C4—44no—
578C2×C4—534no—
579C2×C4—532no—
580C2×C4—531no—
581C2×C4—532no—
582C2×C4—534no—
583C2×C4—534no—
584C2×C4—1064no—
585C2×C4—1064no—
586C2×C4—1062no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1062no—
590C2×C4—1064no—
591C2×C4—1064no—
592C2×C4—1064no—
593C2×C4—1064no—
594C2×C4—1062no—
595C2×C4—1061no—
596C2×C4—1064no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1064no—
601C2×C4—1064no—
602C2×C4—1064no—
603C2×C4—1064no—
604C2×C4—1064no—
605C2×C4—1064no—
606C2×C4—2124no—
607C2×C4—2124no—
608C2×C4—21216no—
609C2×C4—2124no—
610C2×C4—21216no—
611C2×C4C23×#615711yescharacteristic
612C2×C4C23×#615712yes—
613C2×C4C23×#615714yes—
614C2×C4C23×#615714yes—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—532no—
619C22—532no—
620C22—531no—
621C22—531no—
622C22—1064no—
623C22—1064no—
624C22—1064no—
625C22—2124no—
626C22—2124no—
627C22—2124no—
628C22—2124no—
629C22C24×#615711yescharacteristic
630C4—24no—
631C4—24no—
632C4—22no—
633C4—21no—
634C4—534no—
635C4—1064no—
636C4—1064no—
637C4—1062no—
638C4—1061no—
639C4—2124no—
640C4—2124no—
641C4—2124no—
642C4—2122no—
643C4—2121no—
644C2—44no—
645C2—531no—
646C2—532no—
647C2—531no—
648C2—2124no—
649C2H(5,𝔽2)×#615712yes—
650C22−1+4×#615711yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6157
commutator, socleC22×C53C24×C4
FittingC53×#562C4
prime radical#562×D53C2
cosocleC22×D53C25
hypercenter#562#6157
hypocenterC53C4×#562
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #20291261C2semidirect
2. #20291258C2semidirect
3. #20291257C2semidirect
4. #20291256C2semidirect
5. #20291255C2semidirect
6. #125×#6157C2semidirect
7. #124×#6157C2semidirect
8. C22⋊Q8×#6157C2semidirect
9. #123×#6157C2semidirect
10. #121×#6157C2semidirect
52. #16811371C4semidirect
53. #16811370C4semidirect
54. #16811369C4semidirect
55. #16811368C4semidirect
56. #16811367C4semidirect
62. C53×#562C4semidirect
105. #13133209C2×C4semidirect
106. #13133208C2×C4semidirect
107. #13133207C2×C4semidirect
108. #13133206C2×C4semidirect
109. #13133205C2×C4semidirect
110. #13133204C2×C4semidirect
111. #13133203C2×C4semidirect
112. #13133203C2×C4semidirect
113. #13133202C2×C4semidirect
114. #13133202C2×C4semidirect
115. #13133201C2×C4semidirect
116. #13133200C2×C4semidirect
117. #13133199C2×C4semidirect
118. #13133199C2×C4semidirect
119. #13133198C2×C4semidirect
120. #13133197C2×C4semidirect
121. #13133196C2×C4semidirect
122. #13133195C2×C4semidirect
123. #13133195C2×C4semidirect
124. #13133194C2×C4semidirect
125. #13133194C2×C4semidirect
126. #13133193C2×C4semidirect
127. #13133192C2×C4semidirect
128. #13133191C2×C4semidirect
129. #13133190C2×C4semidirect
130. #13133190C2×C4semidirect
131. #13133189C2×C4semidirect
132. #13133189C2×C4semidirect
133. #13133188C2×C4semidirect
141. #125×C53C2×C4semidirect
142. #124×C53C2×C4semidirect
143. #123×C53C2×C4semidirect
144. C22⋊Q8×C53C2×C4semidirect
145. #121×C53C2×C4semidirect
310. #6157#562direct (N × H)
342. #562#6157direct (N × H)
388. C53C4×#562semidirect
398. #125C2×#6157semidirect
404. #124C2×#6157semidirect
412. #123C2×#6157semidirect
420. C22⋊Q8C2×#6157semidirect
428. #121C2×#6157semidirect

Maximal subgroups

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

subgroupindex
#202912612
#202912582
#202912572
#202912562
#202912552
#125×#61572
#124×#61572
C22⋊Q8×#61572
#123×#61572
#121×#61572
#562×D532
C4×#56253