Finite Group Databasebeta

← C47×#19805 #26059503

Subgroups — C47×#19805 #26059503

Order 12032.

640 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×SD16×C47C211yesmaximal, characteristic, complemented
2Q8×SD16×C47C211yesmaximal, characteristic, complemented
3C47×#3298C211yesmaximal, characteristic
4C47×#3276C211yesmaximal, characteristic
5C47×#1716C211yesmaximal, characteristic, complemented
6C47×#1691C211yesmaximal, characteristic, complemented
7C47×#1680C211yesmaximal, characteristic, complemented
8C47×#1671C211yesmaximal, characteristic, complemented
9C47×#1670C211yesmaximal, characteristic, complemented
10C47×#1651C211yesmaximal, characteristic, complemented
11C47×#1651C211yesmaximal, characteristic, complemented
12C47×#1645C211yesmaximal, characteristic, complemented
13C47×#1635C211yesmaximal, characteristic
14C47×#1583C211yesmaximal, characteristic, complemented
15C47×SD16⋊C8C211yesmaximal, characteristic, complemented
16C2×#137×C47—21no—
17Q82×C47C2211yescharacteristic
18D4×Q8×C47C2211yescharacteristic
19D4×Q8×C47C2211yescharacteristic
20C47×#547C2211yescharacteristic
21C47×#498—21no—
22C47×#498—21no—
23C47×#498C2211yescharacteristic
24C47×#498C2211yescharacteristic
25C47×#498C2211yescharacteristic
26C47×#494C2211yescharacteristic
27C47×#487C2211yescharacteristic
28C47×#487C2211yescharacteristic
29C47×#483—21no—
30C47×#476—21no—
31C47×#476—21no—
32C47×#476C2211yescharacteristic
33C47×#476C2211yescharacteristic
34C47×#475—21no—
35C47×#475—21no—
36C47×#475C2211yescharacteristic
37C47×#475C2211yescharacteristic
38C47×#475C2211yescharacteristic
39C47×#470—21no—
40C47×#465—21no—
41C47×#462C2211yescharacteristic
42C47×#461C2211yescharacteristic
43C47×#461C2211yescharacteristic
44C47×#452—21no—
45C47×#451—21no—
46C47×#448—21no—
47C8×Q8×C47C2211yescharacteristic
48C8×Q8×C47C2211yescharacteristic
49C4×SD16×C47—21no—
50C4×SD16×C47—21no—
51C4×SD16×C47C2211yescharacteristic
52C4×SD16×C47C2211yescharacteristic
53C47×#436C2211yescharacteristic
54C8×D4×C47C2211yescharacteristic
55C2×#103×C47—21no—
56C47×#335C2211yescharacteristic
57C47×#334C2211yescharacteristic
58C47×#333C2211yescharacteristic
59C47×#332C2211yescharacteristic
60C47×#332C2211yescharacteristic
61C47×#332C2211yescharacteristic
62C47×Q8⋊C8C2211yescharacteristic
63C47×Q8⋊C8C2211yescharacteristic
64C47×D4⋊C8C2211yescharacteristic
65C47×D4⋊C8C2211yescharacteristic
66C47×#322C2211yescharacteristic
67C2×#41×C47—21no—
68C22×Q8×C47—21no—
69C22×Q8×C47—21no—
70#137×C47—44no—
71C2×Q16×C47—21no—
72C2×Q16×C47—21no—
73C2×Q16×C47—21no—
74C2×SD16×C47—21no—
75C2×SD16×C47—21no—
76C2×SD16×C47D411yescharacteristic
77C2×SD16×C47D411yescharacteristic
78C2×M16×C47—21no—
79C22×C8×C47—21no—
80C4⋊Q8×C47—21no—
81C4⋊Q8×C47—21no—
82C4⋊Q8×C47—21no—
83C4⋊Q8×C47—21no—
84C4⋊Q8×C47—41no—
85C4⋊Q8×C47C2311yescharacteristic
86C4⋊Q8×C47C2311yescharacteristic
87C4⋊Q8×C47C2311yescharacteristic
88C4⋊Q8×C47C2311yescharacteristic
89#123×C47—41no—
90C22⋊Q8×C47—21no—
91C22⋊Q8×C47—21no—
92C22⋊Q8×C47—21no—
93C22⋊Q8×C47—41no—
94C22⋊Q8×C47—41no—
95#121×C47—21no—
96C4×Q8×C47—21no—
97C4×Q8×C47—21no—
98C4×Q8×C47C2311yescharacteristic
99C4×Q8×C47C2311yescharacteristic
100C4×D4×C47—21no—
101C4×D4×C47—21no—
102C4×D4×C47C2311yescharacteristic
103C4×D4×C47C2311yescharacteristic
104C2×C4⋊C4×C47—21no—
105#106×C47—21no—
106#106×C47—21no—
107#106×C47—21no—
108#106×C47—21no—
109#106×C47—21no—
110#106×C47—21no—
111#106×C47—21no—
112#106×C47—21no—
113#106×C47—21no—
114#106×C47—41no—
115#106×C47D411yescharacteristic
116#106×C47D411yescharacteristic
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×C47—21no—
126#103×C47—21no—
127#103×C47—21no—
128#103×C47—21no—
129#103×C47—21no—
130#103×C47—21no—
131#103×C47—41no—
132#103×C47D411yescharacteristic
133#103×C47D411yescharacteristic
134#102×C47—21no—
135#102×C47—21no—
136#102×C47D411yescharacteristic
137#102×C47D411yescharacteristic
138C22⋊C8×C47—21no—
139C22⋊C8×C47—21no—
140C4×C8×C47—21no—
141C4×C8×C47—21no—
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C4×C8×C47C2311yescharacteristic
145#41×C47—42no—
146#41×C47—81no—
147C2×Q8×C47—21no—
148C2×Q8×C47—21no—
149C2×Q8×C47—21no—
150C2×Q8×C47—21no—
151C2×Q8×C47—41no—
152C2×Q8×C47—41no—
153C2×Q8×C47—41no—
154C2×Q8×C47—42no—
155C2×Q8×C47—42no—
156C2×Q8×C47—81no—
157C2×Q8×C47—81no—
158C2×Q8×C47C2×D411yescharacteristic
159C2×Q8×C47C2×D411yescharacteristic
160C2×D4×C47—41no—
161C2×D4×C47C2×D411yescharacteristic
162C2×D4×C47C2×D411yescharacteristic
163C22×C4×C47—21no—
164C22×C4×C47—21no—
165C22×C4×C47—41no—
166C22×C4×C47—41no—
167C22×C4×C47—41no—
168Q16×C47—42no—
169Q16×C47—42no—
170Q16×C47—81no—
171SD16×C47—22no—
172SD16×C47—42no—
173SD16×C47—81no—
174SD16×C47SD1614yescomplemented
175M16×C47—42no—
176C2×C8×C47—21no—
177C2×C8×C47—21no—
178C2×C8×C47—21no—
179C2×C8×C47—21no—
180C2×C8×C47—21no—
181C2×C8×C47—81no—
182C2×C8×C47C2×D411yescharacteristic
183C2×C8×C47C2×D411yescharacteristic
184C2×C8×C47C2×D411yescharacteristic
185C2×C8×C47C2×D411yescharacteristic
186C4⋊C4×C47—21no—
187C4⋊C4×C47—21no—
188C4⋊C4×C47—21no—
189C4⋊C4×C47—21no—
190C4⋊C4×C47—21no—
191C4⋊C4×C47—21no—
192C4⋊C4×C47—21no—
193C4⋊C4×C47—21no—
194C4⋊C4×C47—21no—
195C4⋊C4×C47—21no—
196C4⋊C4×C47—41no—
197C4⋊C4×C47—41no—
198C4⋊C4×C47—41no—
199C4⋊C4×C47—41no—
200C4⋊C4×C47—41no—
201C4⋊C4×C47C2×D411yescharacteristic
202C4⋊C4×C47C2×D411yescharacteristic
203C4⋊C4×C47C2×D411yescharacteristic
204C4⋊C4×C47C2×D411yescharacteristic
205C22⋊C4×C47—21no—
206C22⋊C4×C47—21no—
207C22⋊C4×C47—41no—
208C22⋊C4×C47—41no—
209C22⋊C4×C47—41no—
210C42×C47—21no—
211C42×C47—21no—
212C42×C47—41no—
213C42×C47C2411yescharacteristic
214C23×C47—21no—
215C23×C47—21no—
216Q8×C47—21no—
217Q8×C47—21no—
218Q8×C47—41no—
219Q8×C47—41no—
220Q8×C47—41no—
221Q8×C47—41no—
222Q8×C47—41no—
223Q8×C47—41no—
224Q8×C47—82no—
225Q8×C47—82no—
226Q8×C47—81no—
227Q8×C47—161no—
228Q8×C47C2×SD1612yes—
229Q8×C47#13712yescomplemented
230D4×C47—21no—
231D4×C47—21no—
232D4×C47—21no—
233D4×C47—82no—
234D4×C47C2×SD1612yes—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—82no—
250C2×C4×C47—82no—
251C2×C4×C47—82no—
252C2×C4×C47C22×D411yescharacteristic
253C2×C4×C47C22×D411yescharacteristic
254C2×C4×C47H(5,𝔽2)11yescharacteristic
255C8×C47—21no—
256C8×C47—21no—
257C8×C47—21no—
258C8×C47—41no—
259C8×C47—41no—
260C8×C47—41no—
261C8×C47—41no—
262C8×C47—41no—
263C8×C47C2×SD1612yes—
264#19805C4711yesmaximal, characteristic, direct product factor, Sylow
265C22×C47—22no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47—41no—
271C22×C47D4211yescharacteristic
272C4×C47—22no—
273C4×C47—22no—
274C4×C47—21no—
275C4×C47—41no—
276C4×C47—41no—
277C4×C47—41no—
278C4×C47—41no—
279C4×C47—81no—
280C4×C47—81no—
281C4×C47—81no—
282C4×C47—81no—
283C4×C47C22×SD1611yescharacteristic
284C4×C47C2×#13711yescharacteristic
285C4×C47#57711yescharacteristic
286C4×C47#57711yescharacteristic
287Q8×SD16C2×C4711yescharacteristic, complemented
288Q8×SD16C2×C4711yescharacteristic, complemented
289#3298C2×C4711yescharacteristic
290#3276C2×C4711yescharacteristic
291#1716C2×C4711yescharacteristic, complemented
292#1691C2×C4711yescharacteristic, complemented
293#1680C2×C4711yescharacteristic, complemented
294#1671C2×C4711yescharacteristic, complemented
295#1670C2×C4711yescharacteristic, complemented
296#1651C2×C4711yescharacteristic, complemented
297#1651C2×C4711yescharacteristic, complemented
298#1645C2×C4711yescharacteristic, complemented
299#1635C2×C4711yescharacteristic
300#1583C2×C4711yescharacteristic, complemented
301SD16⋊C8C2×C4711yescharacteristic, complemented
302C2×C47—42no—
303C2×C47—81no—
304C2×C47#326611yescharacteristic
305C2×C47D4×SD1611yescharacteristic
306C2×C47#328911yescharacteristic
307C2×#137—21no—
308Q82C22×C4711yescharacteristic
309D4×Q8C22×C4711yescharacteristic
310D4×Q8C22×C4711yescharacteristic
311#547C22×C4711yescharacteristic
312#498—21no—
313#498—21no—
314#498C22×C4711yescharacteristic
315#498C22×C4711yescharacteristic
316#498C22×C4711yescharacteristic
317#494C22×C4711yescharacteristic
318#487C22×C4711yescharacteristic
319#487C22×C4711yescharacteristic
320#483—21no—
321#476—21no—
322#476—21no—
323#476C22×C4711yescharacteristic
324#476C22×C4711yescharacteristic
325#475—21no—
326#475—21no—
327#475C22×C4711yescharacteristic
328#475C22×C4711yescharacteristic
329#475C22×C4711yescharacteristic
330#470—21no—
331#465—21no—
332#462C22×C4711yescharacteristic
333#461C22×C4711yescharacteristic
334#461C22×C4711yescharacteristic
335#452—21no—
336#451—21no—
337#448—21no—
338C8×Q8C22×C4711yescharacteristic
339C8×Q8C22×C4711yescharacteristic
340C4×SD16—21no—
341C4×SD16—21no—
342C4×SD16C22×C4711yescharacteristic
343C4×SD16C22×C4711yescharacteristic
344#436C22×C4711yescharacteristic
345C8×D4C22×C4711yescharacteristic
346C2×#103—21no—
347#335C22×C4711yescharacteristic
348#334C22×C4711yescharacteristic
349#333C22×C4711yescharacteristic
350#332C22×C4711yescharacteristic
351#332C22×C4711yescharacteristic
352#332C22×C4711yescharacteristic
353Q8⋊C8C22×C4711yescharacteristic
354Q8⋊C8C22×C4711yescharacteristic
355D4⋊C8C22×C4711yescharacteristic
356D4⋊C8C22×C4711yescharacteristic
357#322C22×C4711yescharacteristic
358C47#1980511yescharacteristic, direct product factor, Sylow
359C2×#41—21no—
360C22×Q8—21no—
361C22×Q8—21no—
362#137—44no—
363C2×Q16—21no—
364C2×Q16—21no—
365C2×Q16—21no—
366C2×SD16—21no—
367C2×SD16—21no—
368C2×SD16D4×C4711yescharacteristic
369C2×SD16D4×C4711yescharacteristic
370C2×M16—21no—
371C22×C8—21no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8—21no—
376C4⋊Q8—41no—
377C4⋊Q8C23×C4711yescharacteristic
378C4⋊Q8C23×C4711yescharacteristic
379C4⋊Q8C23×C4711yescharacteristic
380C4⋊Q8C23×C4711yescharacteristic
381#123—41no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8—41no—
386C22⋊Q8—41no—
387#121—21no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8C23×C4711yescharacteristic
391C4×Q8C23×C4711yescharacteristic
392C4×D4—21no—
393C4×D4—21no—
394C4×D4C23×C4711yescharacteristic
395C4×D4C23×C4711yescharacteristic
396C2×C4⋊C4—21no—
397#106—21no—
398#106—21no—
399#106—21no—
400#106—21no—
401#106—21no—
402#106—21no—
403#106—21no—
404#106—21no—
405#106—21no—
406#106—41no—
407#106D4×C4711yescharacteristic
408#106D4×C4711yescharacteristic
409C4⋊C8—21no—
410C4⋊C8—21no—
411C4⋊C8C23×C4711yescharacteristic
412C4⋊C8C23×C4711yescharacteristic
413C4⋊C8C23×C4711yescharacteristic
414C4⋊C8C23×C4711yescharacteristic
415#103—21no—
416#103—21no—
417#103—21no—
418#103—21no—
419#103—21no—
420#103—21no—
421#103—21no—
422#103—21no—
423#103—41no—
424#103D4×C4711yescharacteristic
425#103D4×C4711yescharacteristic
426#102—21no—
427#102—21no—
428#102D4×C4711yescharacteristic
429#102D4×C4711yescharacteristic
430C22⋊C8—21no—
431C22⋊C8—21no—
432C4×C8—21no—
433C4×C8—21no—
434C4×C8C23×C4711yescharacteristic
435C4×C8C23×C4711yescharacteristic
436C4×C8C23×C4711yescharacteristic
437#41—42no—
438#41—81no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—41no—
444C2×Q8—41no—
445C2×Q8—41no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—81no—
449C2×Q8—81no—
450C2×Q8C2×D4×C4711yescharacteristic
451C2×Q8C2×D4×C4711yescharacteristic
452C2×D4—41no—
453C2×D4C2×D4×C4711yescharacteristic
454C2×D4C2×D4×C4711yescharacteristic
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460Q16—42no—
461Q16—42no—
462Q16—81no—
463SD16—22no—
464SD16—42no—
465SD16—81no—
466SD16SD16×C4714yescomplemented
467M16—42no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—81no—
474C2×C8C2×D4×C4711yescharacteristic
475C2×C8C2×D4×C4711yescharacteristic
476C2×C8C2×D4×C4711yescharacteristic
477C2×C8C2×D4×C4711yescharacteristic
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4C2×D4×C4711yescharacteristic
494C4⋊C4C2×D4×C4711yescharacteristic
495C4⋊C4C2×D4×C4711yescharacteristic
496C4⋊C4C2×D4×C4711yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C42—21no—
503C42—21no—
504C42—41no—
505C42C24×C4711yescharacteristic
506C23—21no—
507C23—21no—
508Q8—21no—
509Q8—21no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—82no—
517Q8—82no—
518Q8—81no—
519Q8—161no—
520Q8C2×SD16×C4712yes—
521Q8#137×C4712yescomplemented
522D4—21no—
523D4—21no—
524D4—21no—
525D4—82no—
526D4C2×SD16×C4712yes—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4C22×D4×C4711yescharacteristic
545C2×C4C22×D4×C4711yescharacteristic
546C2×C4H(5,𝔽2)×C4711yescharacteristic
547C8—21no—
548C8—21no—
549C8—21no—
550C8—41no—
551C8—41no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8C2×SD16×C4712yes—
556C22—22no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22D42×C4711yescharacteristic
563C4—21no—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4C22×SD16×C4711yescharacteristic
575C4C2×#137×C4711yescharacteristic
576C4C47×#57711yescharacteristic
577C4C47×#57711yescharacteristic
578C2—42no—
579C2—81no—
580C2C47×#326611yescharacteristic
581C2D4×SD16×C4711yescharacteristic
582C2C47×#328911yescharacteristic
subgroupG/H
center, socleC22×C47D42
commutator, Frattini, cosocleC42C24×C47
prime radicalC47×#547C22
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. Q8×SD16×C47C2semidirect
2. Q8×SD16×C47C2semidirect
5. C47×#1716C2semidirect
6. C47×#1691C2semidirect
7. C47×#1680C2semidirect
8. C47×#1671C2semidirect
9. C47×#1670C2semidirect
10. C47×#1651C2semidirect
11. C47×#1651C2semidirect
12. C47×#1645C2semidirect
14. C47×#1583C2semidirect
15. C47×SD16⋊C8C2semidirect
174. SD16×C47SD16semidirect
229. Q8×C47#137semidirect
264. #19805C47direct (N × H)
287. Q8×SD16C2×C47semidirect
288. Q8×SD16C2×C47semidirect
291. #1716C2×C47semidirect
292. #1691C2×C47semidirect
293. #1680C2×C47semidirect
294. #1671C2×C47semidirect
295. #1670C2×C47semidirect
296. #1651C2×C47semidirect
297. #1651C2×C47semidirect
298. #1645C2×C47semidirect
300. #1583C2×C47semidirect
301. SD16⋊C8C2×C47semidirect
358. C47#19805direct (N × H)
466. SD16SD16×C47semidirect
521. Q8#137×C47semidirect

Maximal subgroups

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

subgroupindex
Q8×SD16×C472
C47×#32982
C47×#32762
C47×#17162
C47×#16912
C47×#16802
C47×#16712
C47×#16702
C47×#16512
C47×#16452
C47×#16352
C47×#15832
C47×SD16⋊C82
#1980547