Finite Group Databasebeta

← D47×#3249 #27089087

Subgroups — D47×#3249 #27089087

Order 12032.

1898 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D47C212yesmaximal
2#20254733C211yesmaximal, characteristic, complemented
3#20254732C214yesmaximal, complemented
4#20254731C214yesmaximal, complemented
5#20254730C212yesmaximal, complemented
6#20254729C214yesmaximal, complemented
7#477×D47C214yesmaximal, complemented
8#467×D47C214yesmaximal
9#462×D47C214yesmaximal, complemented
10C2×C4⋊C8×D47C211yesmaximal, characteristic
11C47×#3249C211yesmaximal, characteristic, complemented
12C2×Q16×D47C2214yes—
13C22×C8×D47C2212yes—
14#15689680C2212yes—
15#15689679—22no—
16#15689678C2212yes—
17#15689678C2212yes—
18#15689677—22no—
19#15689677C2214yes—
20#15689676—22no—
21#15689675—22no—
22#15689675C2214yes—
23C4⋊Q8×D47C2212yes—
24C4⋊Q8×D47C2212yes—
25#125×D47—22no—
26C22⋊Q8×D47—22no—
27C22⋊Q8×D47C2214yes—
28C4×Q8×D47—22no—
29C4×Q8×D47C2214yes—
30#117×D47—22no—
31C2×C42×D47C2211yescharacteristic
32#15689419C2214yescomplemented
33#15689418C2214yescomplemented
34#15689417C2218yescomplemented
35#15689416C2218yescomplemented
36#15689415C2214yescomplemented
37#15689365C2214yes—
38#15689364C2214yes—
39#15689363C2218yes—
40#15689362C2218yes—
41#15689361C2214yes—
42#15689334C2214yescomplemented
43#15689333C2214yescomplemented
44#15689332C2218yescomplemented
45#15689331C2218yescomplemented
46#15689330C2214yescomplemented
47#107×D47C2214yes—
48C2×C4⋊(C47⋊C8)C2211yescharacteristic
49C2×Dic47⋊C8C2212yes—
50C4⋊C8×D47C2214yes—
51C4⋊C8×D47C2214yes—
52#103×D47C2218yes—
53C47×#533C2212yes—
54C47×#477C2214yescomplemented
55C47×#467C2214yes—
56C47×#462C2214yescomplemented
57C2×C4⋊C8×C47C2211yescharacteristic
58C2×Q8×D47—22no—
59C2×Q8×D47C2314yes—
60C22×C4×D47D411yescharacteristic
61C22×C4×D47D411yescharacteristic
62C22×C4×D47C2311yescharacteristic
63C2×Dic188C2314yes—
64C2×#10625337C2318yes—
65Q16×D47—48no—
66C22×C47⋊C8C2312yes—
67C2×C8×D47—44no—
68C2×C8×D47C2314yes—
69C2×C8×D47C2314yes—
70#13094024C2312yes—
71#13094024C2312yes—
72#13094023—22no—
73#13094023—22no—
74#13094023—22no—
75#13094023C2314yes—
76#13094022—22no—
77#13094022C2314yes—
78#13094016—22no—
79#13094015—22no—
80#13094015—22no—
81#13094014—24no—
82#13094014—24no—
83#13094004—22no—
84#13094004C2314yes—
85#13094003—22no—
86#13094003C2314yes—
87#13094002—24no—
88#13094002—24no—
89#13094001—24no—
90#13094001—24no—
91#13094000—22no—
92#13094000C2314yes—
93Q8×Dic47—22no—
94Q8×Dic47C2314yes—
95C4×Dic94—22no—
96C4×Dic94—24no—
97C4×Dic94—24no—
98C4×Dic94C2314yes—
99#13093989—22no—
100#13093989—24no—
101#13093989—24no—
102#13093989C2314yes—
103#13093983—22no—
104#13093982—22no—
105#13093982—22no—
106#13093981—24no—
107#13093980—24no—
108C4⋊C4×D47—22no—
109C4⋊C4×D47—22no—
110C4⋊C4×D47—22no—
111C4⋊C4×D47—24no—
112C4⋊C4×D47—24no—
113C4⋊C4×D47C2314yes—
114C22⋊C4×D47—24no—
115C2×C4×Dic47D411yescharacteristic
116C2×C4×Dic47D411yescharacteristic
117C2×C4×Dic47C2311yescharacteristic
118C42×D47—24no—
119C42×D47D412yes—
120C42×D47D412yes—
121C42×D47C2312yes—
122C42×D47C2312yes—
123#13093958C2314yes—
124#13093957C2318yes—
125C4⋊(C47⋊C8)C2314yes—
126Dic47⋊C8C2318yes—
127#13093949C2318yes—
128#13093948C2318yes—
129#13093947C2318yes—
130C2×Q16×C47C2314yes—
131C22×C8×C47C2312yes—
132C4⋊Q8×C47C2312yes—
133C4⋊Q8×C47C2312yes—
134#125×C47—22no—
135C22⋊Q8×C47—22no—
136C22⋊Q8×C47C2314yes—
137C4×Q8×C47—22no—
138C4×Q8×C47C2314yes—
139#117×C47—22no—
140C2×C42×C47C2311yescharacteristic
141#107×C47C2314yes—
142C4⋊C8×C47C2314yes—
143#103×C47C2318yes—
144C23×D47C2×D411yescharacteristic
145C2×Q8×C47—22no—
146C2×Q8×C47C2414yes—
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2411yescharacteristic
150C22×Dic47C2×D411yescharacteristic
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153Q8×D47—28no—
154Q8×D47—28no—
155Q8×D47—48no—
156C2×C4×D47—22no—
157C2×C4×D47—22no—
158C2×C4×D47—24no—
159C2×C4×D47—24no—
160C2×C4×D47—42no—
161C2×C4×D47Q1618yescomplemented
162C2×C4×D47C2×D411yescharacteristic
163C2×C4×D47C2×D411yescharacteristic
164C2×C4×D47C2×D412yes—
165C2×C4×D47C2×D412yes—
166C2×C4×D47C2×D412yes—
167C2×C4×D47C2×D412yes—
168C2×C4×D47C2411yescharacteristic
169C2×C4×D47C2411yescharacteristic
170C2×Dic94—22no—
171C2×Dic94—24no—
172C2×Dic94—24no—
173C2×Dic94C2414yes—
174Q16×C47—44no—
175C2×C8×C47—42no—
176C2×C8×C47C2414yes—
177C4⋊C4×C47—22no—
178C4⋊C4×C47—22no—
179C4⋊C4×C47—22no—
180C4⋊C4×C47—24no—
181C4⋊C4×C47—24no—
182C4⋊C4×C47C2414yes—
183C22⋊C4×C47—24no—
184C42×C47—24no—
185C42×C47C2×D412yes—
186C42×C47C2412yes—
187C22⋊Dic47—24no—
188#10625337—48no—
189(C2×D47)⋊C4—24no—
190(C2×D47)⋊C4—24no—
191C4⋊Dic47—22no—
192C4⋊Dic47—22no—
193C4⋊Dic47—22no—
194C4⋊Dic47—24no—
195C4⋊Dic47—24no—
196C4⋊Dic47C2414yes—
197Dic47⋊C4—24no—
198Dic47⋊C4—24no—
199Dic47⋊C4—24no—
200Dic47⋊C4—24no—
201Dic47⋊C4—24no—
202Dic47⋊C4—24no—
203Dic47⋊C4—24no—
204Dic47⋊C4—24no—
205C4×Dic47—24no—
206C4×Dic47—24no—
207C4×Dic47—24no—
208C4×Dic47C2×D412yes—
209C4×Dic47C2×D412yes—
210C4×Dic47C2×D412yes—
211C4×Dic47C2×D412yes—
212C4×Dic47C2×D412yes—
213C4×Dic47C2412yes—
214C2×C47⋊C8—42no—
215C2×C47⋊C8C2414yes—
216Dic188—44no—
217C8×D47—28no—
218C8×D47—44no—
219C23×C47C22×D411yescharacteristic
220C22×D47—22no—
221C22×D47—22no—
222C22×D47C2×Q1614yes—
223C22×D47C22×D411yescharacteristic
224C22×D47C22×D411yescharacteristic
225Q8×C47—24no—
226Q8×C47—24no—
227Q8×C47—44no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—41no—
233C2×C4×C47C2×Q1614yescomplemented
234C2×C4×C47C22×D411yescharacteristic
235C2×C4×C47C22×D412yes—
236C2×C4×C47C22×D412yes—
237C2×C4×C47C2511yescharacteristic
238C2×Dic47—22no—
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—41no—
242C2×Dic47C2×Q1614yescomplemented
243C2×Dic47C22×D411yescharacteristic
244C2×Dic47C22×D412yes—
245C2×Dic47C22×D411yescharacteristic
246C2×Dic47C22×D412yes—
247C4×D47—22no—
248C4×D47—24no—
249C4×D47—21no—
250C4×D47—21no—
251C4×D47—22no—
252C4×D47—42no—
253C4×D47—48no—
254C4×D47—48no—
255C4×D47C2×Q1618yes—
256C4×D47C2×Q1618yescomplemented
257C4×D47H(5,𝔽2)12yes—
258C4×D472−1+412yes—
259Dic94—24no—
260Dic94—24no—
261Dic94—48no—
262Dic94—44no—
263Dic94—48no—
264C8×C47—24no—
265C47⋊C8—24no—
266C2×#3249—471nomaximal, Sylow
267C22×C47—21no—
268C22×C47—21no—
269C22×C47C22×Q1612yes—
270C22×C47C23×D411yescharacteristic
271C2×D47—21no—
272C2×D47—21no—
273C2×D47—28no—
274C2×D47#53712yes—
275C2×D47C22×Q1612yes—
276C2×D47C22×Q1612yes—
277C2×D47#57512yes—
278C4×C47—21no—
279C4×C47—22no—
280C4×C47—44no—
281C4×C47—44no—
282C4×C47C22×Q1614yes—
283C4×C47C2×H(5,𝔽2)11yescharacteristic
284C4×C47C2×2−1+411yescharacteristic
285Dic47—21no—
286Dic47—22no—
287Dic47—44no—
288Dic47—44no—
289Dic47#53712yes—
290Dic47C22×Q1614yes—
291C2×#533—472no—
292#3249—474no—
293#3249—474no—
294#3249—474no—
295#3249—471no—
296#3249—472no—
297#3249D4711yescharacteristic, direct product factor
298C2×#477—474no—
299C2×#467—474no—
300C2×#462—474no—
301C22×C4⋊C8—471no—
302C2×C47—22no—
303C2×C47C2×#53711yescharacteristic
304C2×C47C23×Q1611yescharacteristic
305C2×C47C2×#57511yescharacteristic
306D47—22no—
307D47#324914yesdirect product factor
308C22×Q16—474no—
309C23×C8—472no—
310#533—472no—
311#533—472no—
312#533—474no—
313#533—474no—
314#533—472no—
315#533—942no—
316#533—942no—
317#533—942no—
318#533—942no—
319#533C2×D4712yes—
320C2×C4⋊Q8—472no—
321C2×C4⋊Q8—472no—
322C2×#125—942no—
323C2×C22⋊Q8—474no—
324C2×C22⋊Q8—942no—
325C2×C4×Q8—474no—
326C2×C4×Q8—942no—
327C2×#117—942no—
328C22×C42—471no—
329#477—474no—
330#477—474no—
331#477—478no—
332#477—474no—
333#477—478no—
334#477C2×D4714yescomplemented
335#467—474no—
336#467—474no—
337#467—478no—
338#467—474no—
339#467—478no—
340#467C2×D4714yes—
341#462—478no—
342#462—474no—
343#462—474no—
344#462—474no—
345#462—478no—
346#462C2×D4714yescomplemented
347C2×#107—474no—
348C2×C4⋊C8—472no—
349C2×C4⋊C8—471no—
350C2×C4⋊C8—474no—
351C2×C4⋊C8—474no—
352C2×C4⋊C8C2×D4711yescharacteristic
353C2×#103—478no—
354C47C2×#324911yescharacteristic, complemented, Sylow
355C22×Q8—474no—
356C22×Q8—942no—
357C23×C4—471no—
358C23×C4—471no—
359C23×C4—471no—
360C2×Q16—478no—
361C2×Q16—474no—
362C2×Q16—1888no—
363C2×Q16C22×D4714yes—
364C22×C8—472no—
365C22×C8—474no—
366C22×C8—474no—
367C22×C8—1884no—
368C22×C8C22×D4712yes—
369C4⋊Q8—474no—
370C4⋊Q8—474no—
371C4⋊Q8—472no—
372C4⋊Q8—472no—
373C4⋊Q8—942no—
374C4⋊Q8—942no—
375C4⋊Q8—942no—
376C4⋊Q8—942no—
377C4⋊Q8C22×D4712yes—
378C4⋊Q8C22×D4712yes—
379#125—22no—
380#125—944no—
381#125—944no—
382#125—942no—
383#125—942no—
384#125—942no—
385C22⋊Q8—22no—
386C22⋊Q8—474no—
387C22⋊Q8—474no—
388C22⋊Q8—474no—
389C22⋊Q8—944no—
390C22⋊Q8—944no—
391C22⋊Q8—944no—
392C22⋊Q8—944no—
393C22⋊Q8—942no—
394C22⋊Q8—942no—
395C22⋊Q8—942no—
396C22⋊Q8C22×D4714yes—
397C4×Q8—22no—
398C4×Q8—474no—
399C4×Q8—474no—
400C4×Q8—474no—
401C4×Q8—944no—
402C4×Q8—944no—
403C4×Q8—944no—
404C4×Q8—942no—
405C4×Q8—944no—
406C4×Q8—942no—
407C4×Q8—942no—
408C4×Q8C22×D4714yes—
409#117—22no—
410#117—944no—
411#117—944no—
412#117—942no—
413#117—942no—
414#117—942no—
415C2×C4⋊C4—474no—
416C2×C4⋊C4—942no—
417C2×C4⋊C4—942no—
418C2×C4⋊C4—944no—
419C2×C4⋊C4—944no—
420C2×C4⋊C4—942no—
421C2×C22⋊C4—944no—
422C2×C42—471no—
423C2×C42—471no—
424C2×C42—471no—
425C2×C42—472no—
426C2×C42—472no—
427C2×C42—472no—
428C2×C42—472no—
429C2×C42—944no—
430C2×C42C22×D4711yescharacteristic
431#107—474no—
432#107—478no—
433#107C22×D4714yes—
434C4⋊C8—478no—
435C4⋊C8—474no—
436C4⋊C8C22×D4714yes—
437#103—478no—
438#103—478no—
439#103—478no—
440#103C22×D4718yes—
441C24—471no—
442C2×Q8—22no—
443C2×Q8—474no—
444C2×Q8—944no—
445C2×Q8—944no—
446C2×Q8—948no—
447C2×Q8—948no—
448C2×Q8—942no—
449C2×Q8—1888no—
450C2×Q8C23×D4714yes—
451C22×C4—471no—
452C22×C4—478no—
453C22×C4—472no—
454C22×C4—472no—
455C22×C4—471no—
456C22×C4—471no—
457C22×C4—471no—
458C22×C4—471no—
459C22×C4—472no—
460C22×C4—471no—
461C22×C4—471no—
462C22×C4—472no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—944no—
466C22×C4—944no—
467C22×C4—1882no—
468C22×C4D4×D4711yescharacteristic
469C22×C4D4×D4711yescharacteristic
470C22×C4C23×D4711yescharacteristic
471Q16—44no—
472Q16—1884no—
473Q16—1888no—
474C2×C8—42no—
475C2×C8—474no—
476C2×C8—948no—
477C2×C8—1884no—
478C2×C8—1882no—
479C2×C8C23×D4714yes—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—474no—
486C4⋊C4—944no—
487C4⋊C4—944no—
488C4⋊C4—944no—
489C4⋊C4—944no—
490C4⋊C4—944no—
491C4⋊C4—944no—
492C4⋊C4—944no—
493C4⋊C4—944no—
494C4⋊C4—944no—
495C4⋊C4—944no—
496C4⋊C4—942no—
497C4⋊C4—942no—
498C4⋊C4—942no—
499C4⋊C4C23×D4714yes—
500C22⋊C4—24no—
501C22⋊C4—944no—
502C22⋊C4—944no—
503C22⋊C4—944no—
504C42—24no—
505C42—472no—
506C42—472no—
507C42—472no—
508C42—472no—
509C42—472no—
510C42—472no—
511C42—944no—
512C42—944no—
513C42—944no—
514C42D4×D4712yes—
515C42C23×D4712yes—
516C23—471no—
517C23—474no—
518C23—471no—
519C23—942no—
520C23—942no—
521C23C2×D4×D4711yescharacteristic
522Q8—24no—
523Q8—24no—
524Q8—44no—
525Q8—944no—
526Q8—944no—
527Q8—1884no—
528Q8—1888no—
529Q8—1888no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—41no—
535C2×C4—474no—
536C2×C4—472no—
537C2×C4—471no—
538C2×C4—478no—
539C2×C4—478no—
540C2×C4—472no—
541C2×C4—472no—
542C2×C4—471no—
543C2×C4—472no—
544C2×C4—942no—
545C2×C4—942no—
546C2×C4—944no—
547C2×C4—942no—
548C2×C4—941no—
549C2×C4—941no—
550C2×C4—944no—
551C2×C4—944no—
552C2×C4—1882no—
553C2×C4—1888no—
554C2×C4—1888no—
555C2×C4—1881no—
556C2×C4Q16×D4714yescomplemented
557C2×C4C2×D4×D4711yescharacteristic
558C2×C4C2×D4×D4712yes—
559C2×C4C2×D4×D4712yes—
560C2×C4C24×D4711yescharacteristic
561C8—24no—
562C8—944no—
563C22—21no—
564C22—21no—
565C22—472no—
566C22—472no—
567C22—472no—
568C22—472no—
569C22—948no—
570C22—941no—
571C22—941no—
572C22C2×Q16×D4712yes—
573C22C22×D4×D4711yescharacteristic
574C4—22no—
575C4—21no—
576C4—44no—
577C4—44no—
578C4—474no—
579C4—472no—
580C4—942no—
581C4—941no—
582C4—1884no—
583C4—1884no—
584C4C2×Q16×D4714yes—
585C4H(5,𝔽2)×D4711yescharacteristic
586C42−1+4×D4711yescharacteristic
587C2—22no—
588C2—474no—
589C2—942no—
590C2#537×D4711yescharacteristic
591C2C22×Q16×D4711yescharacteristic
592C2#575×D4711yescharacteristic
subgroupG/H
centerC22C22×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC47×#3249C2
socleC22×C47C23×D4
prime radicalC23×D47C2×D4
hypercenter#3249D47
hypocenterC47C2×#3249
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
2. #20254733C2semidirect
3. #20254732C2semidirect
4. #20254731C2semidirect
5. #20254730C2semidirect
6. #20254729C2semidirect
7. #477×D47C2semidirect
9. #462×D47C2semidirect
11. C47×#3249C2semidirect
32. #15689419C22semidirect
33. #15689418C22semidirect
34. #15689417C22semidirect
35. #15689416C22semidirect
36. #15689415C22semidirect
42. #15689334C22semidirect
43. #15689333C22semidirect
44. #15689332C22semidirect
45. #15689331C22semidirect
46. #15689330C22semidirect
54. C47×#477C22semidirect
56. C47×#462C22semidirect
161. C2×C4×D47Q16semidirect
233. C2×C4×C47C2×Q16semidirect
242. C2×Dic47C2×Q16semidirect
256. C4×D47C2×Q16semidirect
297. #3249D47direct (N × H)
307. D47#3249direct (N × H)
334. #477C2×D47semidirect
346. #462C2×D47semidirect
354. C47C2×#3249semidirect
556. C2×C4Q16×D47semidirect

Maximal subgroups

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

subgroupindex
#533×D472
#202547332
#202547322
#202547312
#202547302
#202547292
#477×D472
#467×D472
#462×D472
C2×C4⋊C8×D472
C47×#32492
C2×#324947