Finite Group Databasebeta

← #41×#11719913 #28265513

Subgroups — #41×#11719913 #28265513

Order 13568.

2116 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×Dic106C211yesmaximal, characteristic, complemented
2#41×D212C211yesmaximal, characteristic
3#20287697C211yesmaximal, characteristic, complemented
4Q8×#11719913C211yesmaximal, characteristic, complemented
5#20287387C211yesmaximal, characteristic, complemented
6#20287358C211yesmaximal, characteristic, complemented
7#20287073C213yesmaximal, complemented
8#20286581C213yesmaximal, complemented
9#20286540C213yesmaximal, complemented
10D4×#11719913C213yesmaximal, complemented
11#20285274C213yesmaximal, complemented
12#20284747C213yesmaximal, complemented
13#20284724C213yesmaximal, complemented
14C2×C4×#11719913C213yesmaximal, complemented
15C8×#41×C53C211yesmaximal, characteristic, complemented
16C2×#41×D53—21no—
17C22×#11719913C2213yes—
18Q8×Dic106C2211yescharacteristic, complemented
19#16811352C2211yescharacteristic, complemented
20#16811349—21no—
21#16811332—23no—
22#16811330C2213yescomplemented
23#16811295C2211yescharacteristic
24Q8×D212C2211yescharacteristic
25D4×Dic106C2213yescomplemented
26#16811285—21no—
27#16811277C2213yes—
28#16811268—23no—
29D4×D212C2213yes—
30#16811180C2213yescomplemented
31#16811176—23no—
32#16811173C2213yes—
33#16811135—23no—
34#41×Dic53—21no—
35C2×C4×Dic106C2213yescomplemented
36C2×C4×D212C2213yes—
37#16811021C2213yescomplemented
38#16810996C2213yescomplemented
39#16810974C2213yescomplemented
40#16810970C2213yescomplemented
41#16810954C2216yescomplemented
42#16810942C2216yescomplemented
43#16810914C2213yes—
44#16810914C2213yescomplemented
45#16810909C2213yescomplemented
46#16810909C2213yescomplemented
47#16810852C2216yescomplemented
48#16810852C2216yescomplemented
49#16810829C2213yes—
50#16810829C2213yescomplemented
51#16810824C2213yescomplemented
52#16810824C2213yescomplemented
53#16810776C2216yes—
54#16810764C2216yescomplemented
55C4×#11719913C2213yes—
56C4×#11719913C2213yes—
57C4×#11719913C2211yescharacteristic
58C4×#11719913C2213yes—
59C4×#11719913C2216yescomplemented
60C4×#11719913C2213yescomplemented
61C4×#11719913C2213yescomplemented
62C4×#11719913C2213yescomplemented
63C4×#11719913C2213yescomplemented
64C2×#13133149C2213yescomplemented
65C2×#13133142C2213yescomplemented
66C2×#13133139C2213yes—
67C4×#41×C53C2211yescharacteristic
68C8×Q8×C53C2211yescharacteristic, complemented
69C8×Q8×C53C2211yescharacteristic, complemented
70C8×D4×C53C2213yescomplemented
71C8×D4×C53C2213yescomplemented
72C53×#431C2213yescomplemented
73C2×C4×C8×C53C2213yescomplemented
74C2×#11719949—21no—
75C2×#11719945—23no—
76C2×#11719947—23no—
77#41×D53—48no—
78C22×Dic106C2313yes—
79C2×Q8×D53—21no—
80C22×D212C2313yes—
81C2×D4×D53—23no—
82C22×C4×D53—23no—
83C2×#11719913—212no—
84C2×#11719913C2313yes—
85C2×#11719913C2313yes—
86C2×#11719913C2313yes—
87C2×#11719913C2316yes—
88C2×#11719913C2311yescharacteristic
89#13133217C2313yes—
90#13133217C2313yes—
91#13133217C2313yes—
92#13133216—23no—
93#13133216—23no—
94#13133214C2313yes—
95#13133209—23no—
96#13133208—23no—
97#13133208—23no—
98#13133205—23no—
99#13133201—26no—
100#13133197—23no—
101#13133197—23no—
102#13133196—23no—
103#13133196C2316yes—
104#13133194—26no—
105#13133194—26no—
106#13133192—23no—
107#13133192—23no—
108#13133191—23no—
109#13133191C2316yes—
110#13133187—26no—
111Q8×Dic53—21no—
112Q8×Dic53—21no—
113C4×Dic106—23no—
114C4×Dic106—23no—
115C4×Dic106—23no—
116C4×Dic106C2313yes—
117C4×Dic106C2313yes—
118C4×Dic106C2311yescharacteristic
119C4×Dic106C2313yes—
120C4×D212—23no—
121C4×D212—23no—
122C4×D212—23no—
123C4×D212C2313yes—
124C4×D212C2313yes—
125C4×D212C2311yescharacteristic
126C4×D212C2313yes—
127D4×Dic53—23no—
128D4×Dic53—23no—
129#13133176—23no—
130C2×C4⋊Dic53C2313yes—
131C2×C4⋊Dic53C2313yes—
132C2×#11719919—23no—
133C2×#11719921—23no—
134C2×C4×Dic53—23no—
135#13133149C2313yes—
136#13133149C2316yes—
137#13133149C2313yes—
138#13133149C2311yescharacteristic
139#13133149C2313yes—
140#13133142C2313yes—
141#13133142C2316yes—
142#13133142C2313yes—
143#13133142C2313yes—
144#13133142C2311yescharacteristic
145#13133139C2313yes—
146#13133139C2316yes—
147#13133139C2313yes—
148#13133139C2313yes—
149#13133139C2311yescharacteristic
150C2×#41×C53D411yescharacteristic
151C22×C8×C53C2313yes—
152C22×C8×C53C2313yes—
153C4×Q8×C53D411yescharacteristic
154C4×Q8×C53C2311yescharacteristic
155C4×D4×C53D413yes—
156C4×D4×C53C2313yes—
157#117×C53D413yes—
158C2×C42×C53C2313yes—
159C4⋊C8×C53C2313yes—
160C4⋊C8×C53C2313yes—
161C22⋊C8×C53C2316yes—
162C4×C8×C53C2313yes—
163C4×C8×C53C2311yescharacteristic
164C4×C8×C53C2313yes—
165C4×C8×C53C2313yes—
166C23×D53—23no—
167#41×C53SD1618yescomplemented
168C2×Q8×C53C2×D411yescharacteristic
169C2×D4×C53C2×D413yes—
170C22×C4×C53C2×D413yes—
171C22×C4×C53C2×D413yes—
172C22×C4×C53C2413yes—
173C2×C22⋊D53—26no—
174C22×Dic53—23no—
175C22×Dic53—23no—
176C22×Dic53—23no—
177#11719949—44no—
178Q8×D53—44no—
179#11719947—412no—
180D4×D53—412no—
181#11719945—412no—
182C2×D212—26no—
183C2×D212—23no—
184C2×D212—23no—
185C2×D212—23no—
186C2×D212—43no—
187C2×D212C2413yes—
188C2×D212C2411yescharacteristic
189C2×C4×D53—21no—
190C2×C4×D53—23no—
191C2×C4×D53—23no—
192C2×C4×D53—23no—
193C2×C4×D53—412no—
194C2×Dic106—26no—
195C2×Dic106—23no—
196C2×Dic106—23no—
197C2×Dic106—23no—
198C2×Dic106—43no—
199C2×Dic106C2413yes—
200C2×Dic106C2411yescharacteristic
201C2×C8×C53—26no—
202C2×C8×C53—43no—
203C2×C8×C53C2411yescharacteristic
204C2×C8×C53C2413yes—
205C2×C8×C53C2413yes—
206C2×C8×C53C2411yescharacteristic
207C4⋊C4×C53C2×D413yes—
208C4⋊C4×C53C2×D413yes—
209C22⋊C4×C53C2×D416yes—
210C42×C53C2×D413yes—
211C42×C53C2×D413yes—
212C42×C53C2411yescharacteristic
213C42×C53C2413yes—
214C22⋊Dic53—26no—
215#11719921—21no—
216#11719921—23no—
217#11719921—23no—
218#11719921—23no—
219#11719921—26no—
220C4⋊Dic53—23no—
221C4⋊Dic53—23no—
222C4⋊Dic53—23no—
223C4⋊Dic53—23no—
224C4⋊Dic53—23no—
225C4⋊Dic53—23no—
226C4⋊Dic53C2411yescharacteristic
227C4⋊Dic53C2413yes—
228C4⋊Dic53C2411yescharacteristic
229C4⋊Dic53C2413yes—
230#11719919—21no—
231#11719919—23no—
232#11719919—23no—
233#11719919—23no—
234#11719919—26no—
235C4×Dic53—21no—
236C4×Dic53—23no—
237C4×Dic53—23no—
238C4×Dic53—23no—
239#11719913—26no—
240#11719913—26no—
241#11719913—26no—
242#11719913—46no—
243#11719913#4114yesdirect product factor
244C23×C53C22×D413yes—
245C22×D53—21no—
246C22×D53—23no—
247C22×D53—46no—
248C22×D53—46no—
249C22×D53—83no—
250Q8×C53C2×SD1614yescomplemented
251D4×C53C2×SD16112yescomplemented
252C2×C4×C53—23no—
253C2×C4×C53—23no—
254C2×C4×C53—43no—
255C2×C4×C53C2×SD16112yescomplemented
256C2×C4×C53C22×D413yes—
257C2×C4×C53C22×D413yes—
258C2×C4×C53C22×D411yescharacteristic
259C2×C4×C53C22×D413yes—
260C2×C4×C53C22×D411yescharacteristic
261C2×C4×C53C2511yescharacteristic
262C22⋊D53—412no—
263C2×Dic53—21no—
264C2×Dic53—23no—
265C2×Dic53—23no—
266C2×Dic53—23no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—46no—
270C2×Dic53—46no—
271C2×Dic53—83no—
272D212—23no—
273D212—21no—
274D212—23no—
275D212—46no—
276D212—46no—
277D212—83no—
278D212C2×#4112yes—
279C4×D53—42no—
280C4×D53—46no—
281C4×D53—46no—
282C4×D53—46no—
283Dic106—21no—
284Dic106—23no—
285Dic106—23no—
286Dic106—46no—
287Dic106—46no—
288Dic106—83no—
289Dic106C2×#4112yescomplemented
290C8×C53—23no—
291C8×C53—23no—
292C8×C53—21no—
293C8×C53C2×#4112yescomplemented
294SD16×#41—531nomaximal, Sylow
295C22×C53—23no—
296C22×C53—23no—
297C22×C53C22×SD1616yes—
298C22×C53C23×D411yescharacteristic
299C2×D53—22no—
300C2×D53—41no—
301C2×D53—41no—
302C2×D53—43no—
303C2×D53—43no—
304C2×D53—43no—
305C2×D53—812no—
306C4×C53—23no—
307C4×C53—21no—
308C4×C53—23no—
309C4×C53C22×SD1616yes—
310C4×C53C22×SD1612yes—
311C4×C53C22×#4111yescharacteristic
312C4×C53#58511yescharacteristic
313Dic53—22no—
314Dic53—43no—
315Dic53—41no—
316Dic53—43no—
317Dic53—43no—
318Dic53—41no—
319Q8×#41—531no—
320D4×#41—531no—
321#3392—531no—
322Q8×SD16—531no—
323#3362—531no—
324#3359—531no—
325#3335—533no—
326#3298—533no—
327#3295—533no—
328D4×SD16—533no—
329#3167—533no—
330#3103—533no—
331#3101—533no—
332C8×#41D5311yescharacteristic, complemented
333C2×C4×SD16—533no—
334C2×C53—26no—
335C2×C53D4×#4111yescharacteristic
336C2×C53C23×SD1611yescharacteristic
337C2×C53#358411yescharacteristic
338D53—42no—
339D53—83no—
340C22×#41—1061no—
341C22×SD16—533no—
342Q82—531no—
343#557—531no—
344#557—1061no—
345#554—533no—
346#554—1063no—
347#550—531no—
348D4×Q8—531no—
349D4×Q8—533no—
350#548—1061no—
351#547—533no—
352#546—1063no—
353D42—533no—
354#533—533no—
355#533—1063no—
356#532—533no—
357#525—1063no—
358C4×#41—1061no—
359C4×#41C2×D5311yescharacteristic
360C2×C4×Q8—533no—
361C2×C4×D4—533no—
362#498—533no—
363#492—533no—
364#487—533no—
365#486—533no—
366#483—536no—
367#481—536no—
368#476—533no—
369#476—533no—
370#475—533no—
371#475—533no—
372#465—536no—
373#465—536no—
374#461—533no—
375#461—533no—
376#460—533no—
377#460—533no—
378#450—536no—
379#448—536no—
380C8×Q8C2×D5311yescharacteristic, complemented
381C8×Q8C2×D5311yescharacteristic, complemented
382C4×SD16—533no—
383C4×SD16—533no—
384C4×SD16—536no—
385C4×SD16—533no—
386C4×SD16—533no—
387C4×SD16—533no—
388C4×SD16—533no—
389C4×SD16—533no—
390C4×SD16—531no—
391C8×D4C2×D5313yescomplemented
392C8×D4C2×D5313yescomplemented
393#431C2×D5313yescomplemented
394C2×#106—533no—
395C2×#103—533no—
396C2×#102—533no—
397C2×C4×C8C2×D5313yescomplemented
398C53SD16×#4111yescharacteristic, complemented, Sylow
399C2×#41—1061no—
400C2×#41—1063no—
401C2×#41—1063no—
402C2×#41—2128no—
403C2×#41D21211yescharacteristic
404C22×Q8—533no—
405C22×Q8—1061no—
406C22×D4—533no—
407C22×D4—1063no—
408C23×C4—1063no—
409C2×SD16—531no—
410C2×SD16—533no—
411C2×SD16—533no—
412C2×SD16—533no—
413C2×SD16—536no—
414C2×SD16—10612no—
415C22×C8C22×D5313yes—
416C22×C8C22×D5313yes—
417C4⋊Q8—533no—
418C4⋊Q8—533no—
419C4⋊Q8—533no—
420C4⋊Q8—1063no—
421C4⋊Q8—1063no—
422#127—533no—
423#125—1063no—
424#125—1063no—
425#125—1063no—
426#124—1063no—
427#123—1066no—
428C22⋊Q8—536no—
429C22⋊Q8—1063no—
430C22⋊Q8—1063no—
431C22⋊Q8—1063no—
432C22⋊Q8—1066no—
433C22⋊Q8—1066no—
434#121—536no—
435#121—1063no—
436#121—1063no—
437#121—1063no—
438#120—1066no—
439C4×Q8—531no—
440C4×Q8—533no—
441C4×Q8—533no—
442C4×Q8—533no—
443C4×Q8—1061no—
444C4×Q8—1061no—
445C4×Q8—1063no—
446C4×Q8—1063no—
447C4×Q8—1063no—
448C4×Q8D21211yescharacteristic
449C4×Q8C22×D5311yescharacteristic
450C4×D4—531no—
451C4×D4—533no—
452C4×D4—533no—
453C4×D4—533no—
454C4×D4—1063no—
455C4×D4—1063no—
456C4×D4—1063no—
457C4×D4—1063no—
458C4×D4—1063no—
459C4×D4D21213yes—
460C4×D4C22×D5313yes—
461#117—1063no—
462#117D21213yes—
463C2×C4⋊C4—533no—
464C2×C4⋊C4—533no—
465C2×C4⋊C4—1063no—
466C2×C22⋊C4—1063no—
467C2×C42—1063no—
468C2×C42C22×D5313yes—
469#106—531no—
470#106—533no—
471#106—533no—
472#106—533no—
473#106—536no—
474C4⋊C8C22×D5313yes—
475C4⋊C8C22×D5313yes—
476#103—531no—
477#103—533no—
478#103—533no—
479#103—533no—
480#103—536no—
481#102—531no—
482#102—533no—
483#102—533no—
484#102—533no—
485#102—536no—
486C22⋊C8C22×D5316yes—
487C4×C8C22×D5311yescharacteristic
488C4×C8C22×D5313yes—
489C4×C8C22×D5313yes—
490C4×C8C22×D5313yes—
491C24—1063no—
492#41—2124no—
493#41—21212no—
494#41—21212no—
495#41#1171991318yesdirect product factor
496C2×Q8—531no—
497C2×Q8—533no—
498C2×Q8—1066no—
499C2×Q8—1063no—
500C2×Q8—1063no—
501C2×Q8—1063no—
502C2×Q8—2123no—
503C2×Q8—2124no—
504C2×Q8C2×D21211yescharacteristic
505C2×D4—531no—
506C2×D4—533no—
507C2×D4—1066no—
508C2×D4—1066no—
509C2×D4—1063no—
510C2×D4—1063no—
511C2×D4—1063no—
512C2×D4—2123no—
513C2×D4—21212no—
514C2×D4C2×D21213yes—
515C22×C4—1063no—
516C22×C4—1063no—
517C22×C4—1063no—
518C22×C4—1063no—
519C22×C4—1063no—
520C22×C4—1061no—
521C22×C4—1063no—
522C22×C4—21212no—
523C22×C4C2×D21213yes—
524C22×C4C2×D21213yes—
525C22×C4C23×D5313yes—
526SD16—534no—
527SD16—1066no—
528SD16—1066no—
529SD16—1066no—
530SD16—2126no—
531C2×C8—26no—
532C2×C8—43no—
533C2×C8C23×D5311yescharacteristic
534C2×C8C23×D5313yes—
535C2×C8C23×D5313yes—
536C2×C8C23×D5311yescharacteristic
537C4⋊C4—531no—
538C4⋊C4—533no—
539C4⋊C4—531no—
540C4⋊C4—533no—
541C4⋊C4—1063no—
542C4⋊C4—1063no—
543C4⋊C4—1061no—
544C4⋊C4—1063no—
545C4⋊C4—1063no—
546C4⋊C4—1063no—
547C4⋊C4—1063no—
548C4⋊C4—1063no—
549C4⋊C4—1063no—
550C4⋊C4—1066no—
551C4⋊C4—1063no—
552C4⋊C4C2×D21213yes—
553C4⋊C4C2×D21213yes—
554C22⋊C4—1066no—
555C22⋊C4—1061no—
556C22⋊C4—1063no—
557C22⋊C4—1063no—
558C22⋊C4—1063no—
559C22⋊C4—1066no—
560C22⋊C4C2×D21216yes—
561C42—1063no—
562C42—1063no—
563C42—1061no—
564C42—1063no—
565C42C2×D21213yes—
566C42C2×D21213yes—
567C42C23×D5313yes—
568C42C23×D5311yescharacteristic
569C23—1061no—
570C23—1063no—
571C23—4243no—
572C23—2126no—
573C23—2126no—
574C23C22×D21213yes—
575Q8—532no—
576Q8—1061no—
577Q8—1063no—
578Q8—1063no—
579Q8—4243no—
580Q8—2126no—
581Q8—2126no—
582Q8C2×#1171991314yescomplemented
583D4—532no—
584D4—1063no—
585D4—1061no—
586D4—1063no—
587D4—4243no—
588D4—2126no—
589D4—21212no—
590D4—2126no—
591D4C2×#11719913112yescomplemented
592C2×C4—23no—
593C2×C4—23no—
594C2×C4—43no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1063no—
598C2×C4—1063no—
599C2×C4—1061no—
600C2×C4—1063no—
601C2×C4—4243no—
602C2×C4—2126no—
603C2×C4—2126no—
604C2×C4—2122no—
605C2×C4—2126no—
606C2×C4—2126no—
607C2×C4—2126no—
608C2×C4C2×#11719913112yescomplemented
609C2×C4C22×D21211yescharacteristic
610C2×C4C22×D21213yes—
611C2×C4C22×D21213yes—
612C2×C4C22×D21213yes—
613C2×C4C22×D21211yescharacteristic
614C2×C4C24×D5311yescharacteristic
615C8—23no—
616C8—21no—
617C8—23no—
618C8#41×D5312yescomplemented
619C22—23no—
620C22—23no—
621C22—1062no—
622C22—42412no—
623C22—2123no—
624C22—2123no—
625C22—2123no—
626C22—2121no—
627C22—2121no—
628C22C22×#1171991316yes—
629C22C23×D21211yescharacteristic
630C4—23no—
631C4—21no—
632C4—23no—
633C4—1062no—
634C4—2121no—
635C4—2123no—
636C4—2123no—
637C4—2121no—
638C4—2123no—
639C4C22×#1171991312yes—
640C4C22×#1171991316yes—
641C4C2×#41×D5311yescharacteristic
642C4#1681145911yescharacteristic
643C2—26no—
644C2—4243no—
645C2—2122no—
646C2#41×D21211yescharacteristic
647C2C23×#1171991311yescharacteristic
648C2#2028948911yescharacteristic
subgroupG/H
centerC2×C4C22×D212
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC8×#41×C53C2
socleC22×C53C23×D4
prime radical#41×D212C2
hypercenterC8×#41D53
hypocenterC53SD16×#41
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
1. #41×Dic106C2semidirect
3. #20287697C2semidirect
4. Q8×#11719913C2semidirect
5. #20287387C2semidirect
6. #20287358C2semidirect
7. #20287073C2semidirect
8. #20286581C2semidirect
9. #20286540C2semidirect
10. D4×#11719913C2semidirect
11. #20285274C2semidirect
12. #20284747C2semidirect
13. #20284724C2semidirect
14. C2×C4×#11719913C2semidirect
15. C8×#41×C53C2semidirect
18. Q8×Dic106C22semidirect
19. #16811352C22semidirect
22. #16811330C22semidirect
25. D4×Dic106C22semidirect
30. #16811180C22semidirect
35. C2×C4×Dic106C22semidirect
37. #16811021C22semidirect
38. #16810996C22semidirect
39. #16810974C22semidirect
40. #16810970C22semidirect
41. #16810954C22semidirect
42. #16810942C22semidirect
44. #16810914C22semidirect
45. #16810909C22semidirect
46. #16810909C22semidirect
47. #16810852C22semidirect
48. #16810852C22semidirect
50. #16810829C22semidirect
51. #16810824C22semidirect
52. #16810824C22semidirect
54. #16810764C22semidirect
59. C4×#11719913C22semidirect
60. C4×#11719913C22semidirect
61. C4×#11719913C22semidirect
62. C4×#11719913C22semidirect
63. C4×#11719913C22semidirect
64. C2×#13133149C22semidirect
65. C2×#13133142C22semidirect
68. C8×Q8×C53C22semidirect
69. C8×Q8×C53C22semidirect
70. C8×D4×C53C22semidirect
71. C8×D4×C53C22semidirect
72. C53×#431C22semidirect
73. C2×C4×C8×C53C22semidirect
167. #41×C53SD16semidirect
243. #11719913#41direct (N × H)
250. Q8×C53C2×SD16semidirect
251. D4×C53C2×SD16semidirect
255. C2×C4×C53C2×SD16semidirect
289. Dic106C2×#41semidirect
293. C8×C53C2×#41semidirect
332. C8×#41D53semidirect
380. C8×Q8C2×D53semidirect
381. C8×Q8C2×D53semidirect
391. C8×D4C2×D53semidirect
392. C8×D4C2×D53semidirect
393. #431C2×D53semidirect
397. C2×C4×C8C2×D53semidirect
398. C53SD16×#41semidirect
495. #41#11719913direct (N × H)
582. Q8C2×#11719913semidirect
591. D4C2×#11719913semidirect
608. C2×C4C2×#11719913semidirect
618. C8#41×D53semidirect

Maximal subgroups

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

subgroupindex
#41×Dic1062
#41×D2122
#202876972
Q8×#117199132
#202873872
#202873582
#202870732
#202865812
#202865402
D4×#117199132
#202852742
#202847472
#202847242
C2×C4×#117199132
C8×#41×C532
SD16×#4153