Finite Group Databasebeta

← C47×#19867 #26059565

Subgroups — C47×#19867 #26059565

Order 12032.

692 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#3397C211yesmaximal, characteristic, complemented
2C47×#3391C211yesmaximal, characteristic, complemented
3C47×#3304C211yesmaximal, characteristic, complemented
4C47×#3287C211yesmaximal, characteristic, complemented
5C47×#1721C211yesmaximal, characteristic, complemented
6C47×#1688C211yesmaximal, characteristic, complemented
7C47×#1681C211yesmaximal, characteristic, complemented
8C47×#1667C211yesmaximal, characteristic, complemented
9C47×#1666C211yesmaximal, characteristic, complemented
10C47×#1642C211yesmaximal, characteristic, complemented
11C47×#1628C211yesmaximal, characteristic, complemented
12C47×#1621C211yesmaximal, characteristic, complemented
13C47×#1621C211yesmaximal, characteristic, complemented
14C47×#1584C211yesmaximal, characteristic, complemented
15C47×#1580C211yesmaximal, characteristic, complemented
16C2×#135×C47—21no—
17C47×#556C2211yescharacteristic
18C47×#550C2211yescharacteristic
19C47×#550C2211yescharacteristic
20C47×#550C2211yescharacteristic
21C47×#496—21no—
22C47×#493—21no—
23C47×#493C2211yescharacteristic
24C47×#493C2211yescharacteristic
25C47×#493C2211yescharacteristic
26C47×#492C2211yescharacteristic
27C47×#486C2211yescharacteristic
28C47×#486C2211yescharacteristic
29C47×#482—21no—
30C47×#475C2211yescharacteristic
31C47×#466—21no—
32C47×#465—21no—
33C47×#463—21no—
34C47×#461C2211yescharacteristic
35C47×#460—21no—
36C47×#460C2211yescharacteristic
37C47×#460C2211yescharacteristic
38C47×#460C2211yescharacteristic
39C47×#459—21no—
40C47×#459—21no—
41C47×#459C2211yescharacteristic
42C47×#459C2211yescharacteristic
43C47×#459C2211yescharacteristic
44C47×#452—21no—
45C47×#449—21no—
46C47×#449—21no—
47C47×#446C2211yescharacteristic
48C8×Q8×C47C2211yescharacteristic
49C47×#440—21no—
50C4×Q16×C47C2211yescharacteristic
51C4×D8×C47—21no—
52C4×D8×C47C2211yescharacteristic
53C8×D4×C47C2211yescharacteristic
54C8×D4×C47C2211yescharacteristic
55C47×#416—21no—
56C47×#335C2211yescharacteristic
57C47×#334C2211yescharacteristic
58C47×#332C2211yescharacteristic
59C47×#331C2211yescharacteristic
60C47×#331C2211yescharacteristic
61C47×#331C2211yescharacteristic
62C47×Q8⋊C8C2211yescharacteristic
63C47×Q8⋊C8C2211yescharacteristic
64C47×D4⋊C8C2211yescharacteristic
65C47×D4⋊C8C2211yescharacteristic
66C47×#322C2211yescharacteristic
67C2×#41×C47—21no—
68C2×#41×C47—21no—
69C2×#41×C47—21no—
70#135×C47—44no—
71C2×Q16×C47D411yescharacteristic
72C2×SD16×C47—21no—
73C2×SD16×C47—21no—
74C2×SD16×C47—21no—
75C2×SD16×C47—21no—
76C2×SD16×C47—21no—
77C2×SD16×C47—21no—
78C2×D8×C47—21no—
79C2×D8×C47—21no—
80C2×D8×C47—21no—
81C2×D8×C47—21no—
82C2×D8×C47—21no—
83C2×D8×C47—21no—
84C2×D8×C47—21no—
85C2×D8×C47—41no—
86C2×D8×C47D411yescharacteristic
87C22×C8×C47—21no—
88C22×C8×C47—21no—
89C4⋊Q8×C47C2311yescharacteristic
90#127×C47—21no—
91#127×C47—21no—
92#127×C47—21no—
93#127×C47C2311yescharacteristic
94#127×C47C2311yescharacteristic
95#127×C47C2311yescharacteristic
96#126×C47—41no—
97#124×C47—21no—
98C22⋊Q8×C47—21no—
99#121×C47—21no—
100#121×C47—21no—
101#121×C47—21no—
102#121×C47—41no—
103#121×C47—41no—
104#121×C47—41no—
105C4×Q8×C47—21no—
106C4×Q8×C47C2311yescharacteristic
107C4×Q8×C47C2311yescharacteristic
108C4×D4×C47—21no—
109C4×D4×C47—21no—
110C4×D4×C47—21no—
111C4×D4×C47C2311yescharacteristic
112C4×D4×C47C2311yescharacteristic
113#117×C47—21no—
114#107×C47D411yescharacteristic
115#107×C47D411yescharacteristic
116#106×C47—21no—
117C4⋊C8×C47—21no—
118C4⋊C8×C47—21no—
119C4⋊C8×C47C2311yescharacteristic
120C4⋊C8×C47C2311yescharacteristic
121C4⋊C8×C47C2311yescharacteristic
122C4⋊C8×C47C2311yescharacteristic
123#103×C47—21no—
124#103×C47—21no—
125#103×C47D411yescharacteristic
126#103×C47D411yescharacteristic
127#102×C47—21no—
128#102×C47—21no—
129#102×C47—21no—
130#102×C47—21no—
131#102×C47—21no—
132#102×C47—21no—
133#102×C47—21no—
134#102×C47—21no—
135#102×C47—41no—
136#102×C47D411yescharacteristic
137#102×C47D411yescharacteristic
138C22⋊C8×C47—21no—
139C22⋊C8×C47—21no—
140#97×C47—21no—
141C4×C8×C47—21no—
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C4×C8×C47C2311yescharacteristic
145#41×C47—42no—
146#41×C47—42no—
147#41×C47—42no—
148#41×C47—81no—
149#41×C47—81no—
150#41×C47—81no—
151C2×Q8×C47—21no—
152C2×Q8×C47C2×D411yescharacteristic
153C2×Q8×C47C2×D411yescharacteristic
154C2×D4×C47—21no—
155C2×D4×C47—21no—
156C2×D4×C47—21no—
157C2×D4×C47—21no—
158C2×D4×C47—21no—
159C2×D4×C47—21no—
160C2×D4×C47—21no—
161C2×D4×C47—21no—
162C2×D4×C47—21no—
163C2×D4×C47—41no—
164C2×D4×C47—41no—
165C2×D4×C47—41no—
166C2×D4×C47—41no—
167C2×D4×C47—41no—
168C2×D4×C47—41no—
169C2×D4×C47C2×D411yescharacteristic
170C2×D4×C47C2×D411yescharacteristic
171C22×C4×C47—21no—
172C22×C4×C47—21no—
173C22×C4×C47—41no—
174C22×C4×C47—41no—
175C22×C4×C47—41no—
176Q16×C47—22no—
177SD16×C47—42no—
178SD16×C47—42no—
179SD16×C47—42no—
180SD16×C47—81no—
181SD16×C47—81no—
182SD16×C47—81no—
183D8×C47—22no—
184D8×C47—42no—
185D8×C47—42no—
186D8×C47—81no—
187D8×C47—81no—
188D8×C47—81no—
189D8×C47—81no—
190D8×C47—81no—
191D8×C47—82no—
192C2×C8×C47—21no—
193C2×C8×C47—21no—
194C2×C8×C47—21no—
195C2×C8×C47—21no—
196C2×C8×C47—21no—
197C2×C8×C47—42no—
198C2×C8×C47—81no—
199C2×C8×C47C2×D411yescharacteristic
200C2×C8×C47C2×D411yescharacteristic
201C2×C8×C47C2×D411yescharacteristic
202C2×C8×C47C2×D411yescharacteristic
203C4⋊C4×C47—21no—
204C4⋊C4×C47—21no—
205C4⋊C4×C47—21no—
206C4⋊C4×C47—21no—
207C4⋊C4×C47—41no—
208C4⋊C4×C47C2×D411yescharacteristic
209C4⋊C4×C47C2×D411yescharacteristic
210C4⋊C4×C47C2×D411yescharacteristic
211C4⋊C4×C47C2×D411yescharacteristic
212C22⋊C4×C47—21no—
213C22⋊C4×C47—21no—
214C22⋊C4×C47—41no—
215C22⋊C4×C47—41no—
216C22⋊C4×C47—41no—
217C22⋊C4×C47—41no—
218C22⋊C4×C47—41no—
219C42×C47—21no—
220C42×C47—21no—
221C42×C47—41no—
222C42×C47C2411yescharacteristic
223C23×C47—21no—
224C23×C47—21no—
225C23×C47—41no—
226C23×C47—41no—
227C23×C47—41no—
228Q8×C47—21no—
229Q8×C47—21no—
230Q8×C47—21no—
231Q8×C47—41no—
232Q8×C47—41no—
233Q8×C47AGL(1,ℤ/8)12yes—
234D4×C47—21no—
235D4×C47—21no—
236D4×C47—21no—
237D4×C47—21no—
238D4×C47—41no—
239D4×C47—41no—
240D4×C47—41no—
241D4×C47—41no—
242D4×C47—41no—
243D4×C47—41no—
244D4×C47—41no—
245D4×C47—41no—
246D4×C47—41no—
247D4×C47—41no—
248D4×C47—41no—
249D4×C47—41no—
250D4×C47—82no—
251D4×C47—82no—
252D4×C47—81no—
253D4×C47—81no—
254D4×C47—81no—
255D4×C47—82no—
256D4×C47—161no—
257D4×C47—161no—
258D4×C47—161no—
259C2×C4×C47—21no—
260C2×C4×C47—21no—
261C2×C4×C47—21no—
262C2×C4×C47—21no—
263C2×C4×C47—21no—
264C2×C4×C47—21no—
265C2×C4×C47—41no—
266C2×C4×C47—41no—
267C2×C4×C47—41no—
268C2×C4×C47—41no—
269C2×C4×C47—41no—
270C2×C4×C47—82no—
271C2×C4×C47—82no—
272C2×C4×C47—82no—
273C2×C4×C47C22×D411yescharacteristic
274C2×C4×C47C22×D411yescharacteristic
275C2×C4×C47H(5,𝔽2)11yescharacteristic
276C8×C47—21no—
277C8×C47—21no—
278C8×C47—21no—
279C8×C47—41no—
280C8×C47—41no—
281C8×C47—41no—
282C8×C47—41no—
283C8×C47—41no—
284C8×C47#13512yescomplemented
285#19867C4711yesmaximal, characteristic, direct product factor, Sylow
286C22×C47—41no—
287C22×C47—41no—
288C22×C47—41no—
289C22×C47—41no—
290C22×C47—41no—
291C22×C47—41no—
292C22×C47—81no—
293C22×C47—81no—
294C22×C47—81no—
295C22×C47—81no—
296C22×C47—81no—
297C22×C47—81no—
298C22×C47—81no—
299C22×C47—81no—
300C22×C47—81no—
301C22×C47D4211yescharacteristic
302C4×C47—22no—
303C4×C47—21no—
304C4×C47—41no—
305C4×C47—41no—
306C4×C47—41no—
307C4×C47—41no—
308C4×C47—41no—
309C4×C47—81no—
310C4×C47C2×#13511yescharacteristic
311C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
312C4×C47#57611yescharacteristic
313C4×C47#57611yescharacteristic
314#3397C2×C4711yescharacteristic, complemented
315#3391C2×C4711yescharacteristic, complemented
316#3304C2×C4711yescharacteristic, complemented
317#3287C2×C4711yescharacteristic, complemented
318#1721C2×C4711yescharacteristic, complemented
319#1688C2×C4711yescharacteristic, complemented
320#1681C2×C4711yescharacteristic, complemented
321#1667C2×C4711yescharacteristic, complemented
322#1666C2×C4711yescharacteristic, complemented
323#1642C2×C4711yescharacteristic, complemented
324#1628C2×C4711yescharacteristic, complemented
325#1621C2×C4711yescharacteristic, complemented
326#1621C2×C4711yescharacteristic, complemented
327#1584C2×C4711yescharacteristic, complemented
328#1580C2×C4711yescharacteristic, complemented
329C2×C47—81no—
330C2×C47—81no—
331C2×C47—161no—
332C2×C47—161no—
333C2×C47—161no—
334C2×C47#326411yescharacteristic
335C2×C47#328711yescharacteristic
336C2×C47#328811yescharacteristic
337C2×#135—21no—
338#556C22×C4711yescharacteristic
339#550C22×C4711yescharacteristic
340#550C22×C4711yescharacteristic
341#550C22×C4711yescharacteristic
342#496—21no—
343#493—21no—
344#493C22×C4711yescharacteristic
345#493C22×C4711yescharacteristic
346#493C22×C4711yescharacteristic
347#492C22×C4711yescharacteristic
348#486C22×C4711yescharacteristic
349#486C22×C4711yescharacteristic
350#482—21no—
351#475C22×C4711yescharacteristic
352#466—21no—
353#465—21no—
354#463—21no—
355#461C22×C4711yescharacteristic
356#460—21no—
357#460C22×C4711yescharacteristic
358#460C22×C4711yescharacteristic
359#460C22×C4711yescharacteristic
360#459—21no—
361#459—21no—
362#459C22×C4711yescharacteristic
363#459C22×C4711yescharacteristic
364#459C22×C4711yescharacteristic
365#452—21no—
366#449—21no—
367#449—21no—
368#446C22×C4711yescharacteristic
369C8×Q8C22×C4711yescharacteristic
370#440—21no—
371C4×Q16C22×C4711yescharacteristic
372C4×D8—21no—
373C4×D8C22×C4711yescharacteristic
374C8×D4C22×C4711yescharacteristic
375C8×D4C22×C4711yescharacteristic
376#416—21no—
377#335C22×C4711yescharacteristic
378#334C22×C4711yescharacteristic
379#332C22×C4711yescharacteristic
380#331C22×C4711yescharacteristic
381#331C22×C4711yescharacteristic
382#331C22×C4711yescharacteristic
383Q8⋊C8C22×C4711yescharacteristic
384Q8⋊C8C22×C4711yescharacteristic
385D4⋊C8C22×C4711yescharacteristic
386D4⋊C8C22×C4711yescharacteristic
387#322C22×C4711yescharacteristic
388C47#1986711yescharacteristic, direct product factor, Sylow
389C2×#41—21no—
390C2×#41—21no—
391C2×#41—21no—
392#135—44no—
393C2×Q16D4×C4711yescharacteristic
394C2×SD16—21no—
395C2×SD16—21no—
396C2×SD16—21no—
397C2×SD16—21no—
398C2×SD16—21no—
399C2×SD16—21no—
400C2×D8—21no—
401C2×D8—21no—
402C2×D8—21no—
403C2×D8—21no—
404C2×D8—21no—
405C2×D8—21no—
406C2×D8—21no—
407C2×D8—41no—
408C2×D8D4×C4711yescharacteristic
409C22×C8—21no—
410C22×C8—21no—
411C4⋊Q8C23×C4711yescharacteristic
412#127—21no—
413#127—21no—
414#127—21no—
415#127C23×C4711yescharacteristic
416#127C23×C4711yescharacteristic
417#127C23×C4711yescharacteristic
418#126—41no—
419#124—21no—
420C22⋊Q8—21no—
421#121—21no—
422#121—21no—
423#121—21no—
424#121—41no—
425#121—41no—
426#121—41no—
427C4×Q8—21no—
428C4×Q8C23×C4711yescharacteristic
429C4×Q8C23×C4711yescharacteristic
430C4×D4—21no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4C23×C4711yescharacteristic
434C4×D4C23×C4711yescharacteristic
435#117—21no—
436#107D4×C4711yescharacteristic
437#107D4×C4711yescharacteristic
438#106—21no—
439C4⋊C8—21no—
440C4⋊C8—21no—
441C4⋊C8C23×C4711yescharacteristic
442C4⋊C8C23×C4711yescharacteristic
443C4⋊C8C23×C4711yescharacteristic
444C4⋊C8C23×C4711yescharacteristic
445#103—21no—
446#103—21no—
447#103D4×C4711yescharacteristic
448#103D4×C4711yescharacteristic
449#102—21no—
450#102—21no—
451#102—21no—
452#102—21no—
453#102—21no—
454#102—21no—
455#102—21no—
456#102—21no—
457#102—41no—
458#102D4×C4711yescharacteristic
459#102D4×C4711yescharacteristic
460C22⋊C8—21no—
461C22⋊C8—21no—
462#97—21no—
463C4×C8—21no—
464C4×C8C23×C4711yescharacteristic
465C4×C8C23×C4711yescharacteristic
466C4×C8C23×C4711yescharacteristic
467#41—42no—
468#41—42no—
469#41—42no—
470#41—81no—
471#41—81no—
472#41—81no—
473C2×Q8—21no—
474C2×Q8C2×D4×C4711yescharacteristic
475C2×Q8C2×D4×C4711yescharacteristic
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—41no—
486C2×D4—41no—
487C2×D4—41no—
488C2×D4—41no—
489C2×D4—41no—
490C2×D4—41no—
491C2×D4C2×D4×C4711yescharacteristic
492C2×D4C2×D4×C4711yescharacteristic
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—41no—
498Q16—22no—
499SD16—42no—
500SD16—42no—
501SD16—42no—
502SD16—81no—
503SD16—81no—
504SD16—81no—
505D8—22no—
506D8—42no—
507D8—42no—
508D8—81no—
509D8—81no—
510D8—81no—
511D8—81no—
512D8—81no—
513D8—82no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—42no—
520C2×C8—81no—
521C2×C8C2×D4×C4711yescharacteristic
522C2×C8C2×D4×C4711yescharacteristic
523C2×C8C2×D4×C4711yescharacteristic
524C2×C8C2×D4×C4711yescharacteristic
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—41no—
530C4⋊C4C2×D4×C4711yescharacteristic
531C4⋊C4C2×D4×C4711yescharacteristic
532C4⋊C4C2×D4×C4711yescharacteristic
533C4⋊C4C2×D4×C4711yescharacteristic
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—41no—
541C42—21no—
542C42—21no—
543C42—41no—
544C42C24×C4711yescharacteristic
545C23—21no—
546C23—21no—
547C23—41no—
548C23—41no—
549C23—41no—
550Q8—21no—
551Q8—21no—
552Q8—21no—
553Q8—41no—
554Q8—41no—
555Q8AGL(1,ℤ/8)×C4712yes—
556D4—21no—
557D4—21no—
558D4—21no—
559D4—21no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—82no—
573D4—82no—
574D4—81no—
575D4—81no—
576D4—81no—
577D4—82no—
578D4—161no—
579D4—161no—
580D4—161no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4C22×D4×C4711yescharacteristic
596C2×C4C22×D4×C4711yescharacteristic
597C2×C4H(5,𝔽2)×C4711yescharacteristic
598C8—21no—
599C8—21no—
600C8—21no—
601C8—41no—
602C8—41no—
603C8—41no—
604C8—41no—
605C8—41no—
606C8#135×C4712yescomplemented
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22D42×C4711yescharacteristic
623C4—21no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—81no—
631C4C2×#135×C4711yescharacteristic
632C4C2×AGL(1,ℤ/8)×C4711yescharacteristic
633C4C47×#57611yescharacteristic
634C4C47×#57611yescharacteristic
635C2—81no—
636C2—81no—
637C2—161no—
638C2—161no—
639C2—161no—
640C2C47×#326411yescharacteristic
641C2C47×#328711yescharacteristic
642C2C47×#328811yescharacteristic
subgroupG/H
center, socleC22×C47D42
commutator, Frattini, cosocleC42C24×C47
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C47 ◁ C42×C47 ◁ G

Complements

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

NH ≅ G/Ntype
1. C47×#3397C2semidirect
2. C47×#3391C2semidirect
3. C47×#3304C2semidirect
4. C47×#3287C2semidirect
5. C47×#1721C2semidirect
6. C47×#1688C2semidirect
7. C47×#1681C2semidirect
8. C47×#1667C2semidirect
9. C47×#1666C2semidirect
10. C47×#1642C2semidirect
11. C47×#1628C2semidirect
12. C47×#1621C2semidirect
13. C47×#1621C2semidirect
14. C47×#1584C2semidirect
15. C47×#1580C2semidirect
284. C8×C47#135semidirect
285. #19867C47direct (N × H)
314. #3397C2×C47semidirect
315. #3391C2×C47semidirect
316. #3304C2×C47semidirect
317. #3287C2×C47semidirect
318. #1721C2×C47semidirect
319. #1688C2×C47semidirect
320. #1681C2×C47semidirect
321. #1667C2×C47semidirect
322. #1666C2×C47semidirect
323. #1642C2×C47semidirect
324. #1628C2×C47semidirect
325. #1621C2×C47semidirect
326. #1621C2×C47semidirect
327. #1584C2×C47semidirect
328. #1580C2×C47semidirect
388. C47#19867direct (N × H)
606. C8#135×C47semidirect

Maximal subgroups

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

subgroupindex
C47×#33972
C47×#33912
C47×#33042
C47×#32872
C47×#17212
C47×#16882
C47×#16812
C47×#16672
C47×#16662
C47×#16422
C47×#16282
C47×#16212
C47×#15842
C47×#15802
#1986747