Finite Group Databasebeta

← C47×#23472 #26063170

Subgroups — C47×#23472 #26063170

Order 12032.

1148 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#3458C211yesmaximal, characteristic, complemented
2C2×C47×#421C211yesmaximal, characteristic
3C2×C47×#421C211yesmaximal, characteristic
4C2×C47×#410C211yesmaximal, characteristic
5C47×#2031C218yesmaximal, complemented
6C2×C47×#344C211yesmaximal, characteristic, complemented
7C47×#1733C211yesmaximal, characteristic, complemented
8C2×C47×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
9C22×#41×C47C2211yescharacteristic
10C22×M16×C47C2211yescharacteristic
11C22×M16×C47C2211yescharacteristic
12C47×#544—22no—
13C47×#544—22no—
14C47×#544—22no—
15C47×#543—22no—
16C47×#533—22no—
17C47×#533—22no—
18C47×#533C2214yescomplemented
19C2×#117×C47C2211yescharacteristic
20C2×#117×C47C2211yescharacteristic
21C2×#117×C47C2211yescharacteristic
22C2×#117×C47C2211yescharacteristic
23C47×#421—22no—
24C47×#421—22no—
25C47×#421—22no—
26C47×#421C2214yes—
27C2×#104×C47—22no—
28C2×#104×C47C2214yes—
29C47×#410—22no—
30C47×#410C2214yes—
31C2×#99×C47—22no—
32C47×#344C2214yescomplemented
33C47×#339C2214yescomplemented
34C47×H(3,ℤ/4)C2214yescomplemented
35C2×#41×C47—21no—
36C2×#41×C47—21no—
37C2×#41×C47—21no—
38C2×#41×C47—21no—
39C2×#41×C47—21no—
40C2×#41×C47—21no—
41C2×#41×C47—21no—
42C2×#41×C47—21no—
43C2×#41×C47—21no—
44C2×#41×C47—21no—
45C2×#41×C47Q812yescomplemented
46C2×#41×C47C2312yes—
47C22×Q8×C47Q811yescharacteristic, complemented
48C22×D4×C47—21no—
49C22×D4×C47Q811yescharacteristic, complemented
50C23×C4×C47—21no—
51C23×C4×C47C2311yescharacteristic
52C2×M16×C47—21no—
53C2×M16×C47—21no—
54C2×M16×C47—21no—
55C2×M16×C47—21no—
56C2×M16×C47—21no—
57C2×M16×C47—21no—
58C2×M16×C47—21no—
59C2×M16×C47—21no—
60C2×M16×C47—21no—
61C2×M16×C47—21no—
62C2×M16×C47C2312yes—
63C2×M16×C47C2312yes—
64C22×C8×C47—21no—
65C22×C8×C47—21no—
66C4⋊Q8×C47—24no—
67C4⋊Q8×C47—24no—
68C4⋊Q8×C47—24no—
69#125×C47—24no—
70#125×C47—24no—
71#125×C47—24no—
72C22⋊Q8×C47—28no—
73C22⋊Q8×C47—28no—
74C22⋊Q8×C47—28no—
75C4×Q8×C47—28no—
76#117×C47—21no—
77#117×C47—21no—
78#117×C47—21no—
79#117×C47—21no—
80#117×C47—21no—
81#117×C47—21no—
82#117×C47—21no—
83#117×C47—21no—
84#117×C47—21no—
85#117×C47D412yes—
86#117×C47D412yes—
87#117×C47D412yes—
88#117×C47D412yes—
89#117×C47C2312yes—
90#117×C47C2312yes—
91#117×C47C2312yes—
92C2×C4⋊C4×C47—21no—
93C2×C4⋊C4×C47—21no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—21no—
96C2×C22⋊C4×C47—21no—
97C2×C22⋊C4×C47—21no—
98C2×C22⋊C4×C47—21no—
99C2×C22⋊C4×C47D412yes—
100C2×C42×C47—21no—
101C2×C42×C47—21no—
102C2×C42×C47D412yes—
103C2×C42×C47C2312yes—
104#104×C47—28no—
105#104×C47—28no—
106#99×C47—28no—
107C24×C47—21no—
108C24×C47C2×D411yescharacteristic
109#41×C47—22no—
110#41×C47—22no—
111#41×C47—22no—
112#41×C47—22no—
113#41×C47—22no—
114#41×C47—22no—
115#41×C47—22no—
116#41×C47—22no—
117#41×C47—41no—
118#41×C47—41no—
119#41×C47—41no—
120#41×C47—41no—
121#41×C47—41no—
122#41×C47—41no—
123#41×C47—41no—
124#41×C47—41no—
125C2×Q8×C47—28no—
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47C2×Q812yes—
132C2×D4×C47—22no—
133C2×D4×C47—22no—
134C2×D4×C47—21no—
135C2×D4×C47—21no—
136C2×D4×C47—21no—
137C2×D4×C47—21no—
138C2×D4×C47—21no—
139C2×D4×C47—41no—
140C2×D4×C47—41no—
141C2×D4×C47—41no—
142C2×D4×C47—41no—
143C2×D4×C47C2×Q812yes—
144C22×C4×C47—21no—
145C22×C4×C47—22no—
146C22×C4×C47—22no—
147C22×C4×C47—22no—
148C22×C4×C47—22no—
149C22×C4×C47—22no—
150C22×C4×C47—41no—
151C22×C4×C47—41no—
152C22×C4×C47—41no—
153C22×C4×C47—41no—
154C22×C4×C47—41no—
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×Q811yescharacteristic
161C22×C4×C47C2×Q811yescharacteristic
162C22×C4×C47#4111yescharacteristic
163C22×C4×C47#4111yescharacteristic
164C22×C4×C47#4111yescharacteristic
165C22×C4×C47#4111yescharacteristic
166C22×C4×C47#4111yescharacteristic
167C22×C4×C47#4111yescharacteristic
168C22×C4×C47C2411yescharacteristic
169M16×C47—22no—
170M16×C47—22no—
171M16×C47—22no—
172M16×C47—22no—
173M16×C47—41no—
174M16×C47—41no—
175M16×C47—41no—
176M16×C47—41no—
177C2×C8×C47—22no—
178C2×C8×C47—22no—
179C2×C8×C47—41no—
180C2×C8×C47—41no—
181C2×C8×C47—41no—
182C2×C8×C47—41no—
183C4⋊C4×C47—28no—
184C4⋊C4×C47—28no—
185C4⋊C4×C47—28no—
186C4⋊C4×C47—22no—
187C4⋊C4×C47—22no—
188C4⋊C4×C47—22no—
189C4⋊C4×C47—22no—
190C4⋊C4×C47—22no—
191C4⋊C4×C47—22no—
192C4⋊C4×C47—22no—
193C4⋊C4×C47—41no—
194C22⋊C4×C47—22no—
195C22⋊C4×C47—22no—
196C22⋊C4×C47—22no—
197C22⋊C4×C47—22no—
198C22⋊C4×C47—22no—
199C22⋊C4×C47—22no—
200C22⋊C4×C47—41no—
201C22⋊C4×C47C2×D414yes—
202C42×C47—22no—
203C42×C47—22no—
204C42×C47—22no—
205C42×C47—22no—
206C42×C47—22no—
207C42×C47—41no—
208C42×C47C2×D414yes—
209C42×C47C2×D414yes—
210C23×C47—22no—
211C23×C47—41no—
212C23×C47—41no—
213C23×C47—81no—
214C23×C47—82no—
215C23×C47#12011yescharacteristic
216C23×C47#12011yescharacteristic
217C23×C47C22⋊Q811yescharacteristic
218C23×C47C22⋊Q811yescharacteristic
219C23×C47#12411yescharacteristic
220C23×C47#12411yescharacteristic
221C23×C47C22×D411yescharacteristic
222Q8×C47—22no—
223Q8×C47—22no—
224Q8×C47—41no—
225Q8×C47—41no—
226Q8×C47—88no—
227D4×C47—22no—
228D4×C47—22no—
229D4×C47—42no—
230D4×C47—42no—
231D4×C47—42no—
232D4×C47—42no—
233D4×C47—41no—
234D4×C47—41no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—24no—
238C2×C4×C47—22no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47—82no—
247C2×C4×C47—82no—
248C2×C4×C47—82no—
249C2×C4×C47—82no—
250C2×C4×C47#12011yescharacteristic
251C2×C4×C47#12011yescharacteristic
252C2×C4×C47C22⋊Q811yescharacteristic
253C2×C4×C47C22⋊Q811yescharacteristic
254C2×C4×C47C22⋊Q811yescharacteristic
255C2×C4×C47C22⋊Q811yescharacteristic
256C2×C4×C47C22⋊Q811yescharacteristic
257C2×C4×C47C22⋊Q811yescharacteristic
258C2×C4×C47C22⋊Q811yescharacteristic
259C2×C4×C47C22⋊Q811yescharacteristic
260C2×C4×C47C22⋊Q811yescharacteristic
261C2×C4×C47C22⋊Q811yescharacteristic
262C2×C4×C47#12411yescharacteristic
263C2×C4×C47#12411yescharacteristic
264C2×C4×C47#12411yescharacteristic
265C2×C4×C47#12411yescharacteristic
266C2×C4×C47#12411yescharacteristic
267C2×C4×C47#12411yescharacteristic
268C2×C4×C47#12411yescharacteristic
269C2×C4×C47#12411yescharacteristic
270C2×C4×C47#12411yescharacteristic
271C2×C4×C47#12411yescharacteristic
272C2×C4×C47C22×D411yescharacteristic
273C2×C4×C47C22×D411yescharacteristic
274C2×C4×C47C22×Q811yescharacteristic
275C2×C4×C47C2×#4111yescharacteristic
276C2×C4×C47C2×#4111yescharacteristic
277C2×C4×C47C2×#4111yescharacteristic
278C8×C47—42no—
279C8×C47—42no—
280#23472C4711yesmaximal, characteristic, direct product factor, Sylow
281C22×C47—41no—
282C22×C47—41no—
283C22×C47—41no—
284C22×C47—41no—
285C22×C47—41no—
286C22×C47—41no—
287C22×C47—41no—
288C22×C47—42no—
289C22×C47—82no—
290C22×C47—82no—
291C22×C47—82no—
292C22×C47#39311yescharacteristic
293C22×C47#39311yescharacteristic
294C22×C47#39311yescharacteristic
295C22×C47#39311yescharacteristic
296C22×C47C2×#12011yescharacteristic
297C22×C47C2×C22⋊Q811yescharacteristic
298C22×C47C2×#12411yescharacteristic
299C4×C47—44no—
300C4×C47—44no—
301C4×C47—44no—
302C4×C47—42no—
303C4×C47—82no—
304C4×C47#39311yescharacteristic
305C4×C47#39311yescharacteristic
306C4×C47#39311yescharacteristic
307C4×C47#39311yescharacteristic
308C4×C47C2×C22⋊Q811yescharacteristic
309C4×C47C2×C22⋊Q811yescharacteristic
310C4×C47C2×#12411yescharacteristic
311C4×C47C2×#12411yescharacteristic
312#3458C2×C4711yescharacteristic, complemented
313C2×#421C2×C4711yescharacteristic
314C2×#421C2×C4711yescharacteristic
315C2×#410C2×C4711yescharacteristic
316#2031C2×C4718yescomplemented
317C2×#344C2×C4711yescharacteristic, complemented
318#1733C2×C4711yescharacteristic, complemented
319C2×H(3,ℤ/4)C2×C4711yescharacteristic, complemented
320C2×C47—21no—
321C2×C47—21no—
322C2×C47—21no—
323C2×C47—21no—
324C2×C47—21no—
325C2×C47—21no—
326C2×C47—21no—
327C2×C47—82no—
328C2×C47C2×#39311yescharacteristic
329C22×#41C22×C4711yescharacteristic
330C22×M16C22×C4711yescharacteristic
331C22×M16C22×C4711yescharacteristic
332#544—22no—
333#544—22no—
334#544—22no—
335#543—22no—
336#533—22no—
337#533—22no—
338#533C22×C4714yescomplemented
339C2×#117C22×C4711yescharacteristic
340C2×#117C22×C4711yescharacteristic
341C2×#117C22×C4711yescharacteristic
342C2×#117C22×C4711yescharacteristic
343#421—22no—
344#421—22no—
345#421—22no—
346#421C22×C4714yes—
347C2×#104—22no—
348C2×#104C22×C4714yes—
349#410—22no—
350#410C22×C4714yes—
351C2×#99—22no—
352#344C22×C4714yescomplemented
353#339C22×C4714yescomplemented
354H(3,ℤ/4)C22×C4714yescomplemented
355C47#2347211yescharacteristic, direct product factor, Sylow
356C2×#41—21no—
357C2×#41—21no—
358C2×#41—21no—
359C2×#41—21no—
360C2×#41—21no—
361C2×#41—21no—
362C2×#41—21no—
363C2×#41—21no—
364C2×#41—21no—
365C2×#41—21no—
366C2×#41Q8×C4712yescomplemented
367C2×#41C23×C4712yes—
368C22×Q8Q8×C4711yescharacteristic, complemented
369C22×D4—21no—
370C22×D4Q8×C4711yescharacteristic, complemented
371C23×C4—21no—
372C23×C4C23×C4711yescharacteristic
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16—21no—
379C2×M16—21no—
380C2×M16—21no—
381C2×M16—21no—
382C2×M16—21no—
383C2×M16C23×C4712yes—
384C2×M16C23×C4712yes—
385C22×C8—21no—
386C22×C8—21no—
387C4⋊Q8—24no—
388C4⋊Q8—24no—
389C4⋊Q8—24no—
390#125—24no—
391#125—24no—
392#125—24no—
393C22⋊Q8—28no—
394C22⋊Q8—28no—
395C22⋊Q8—28no—
396C4×Q8—28no—
397#117—21no—
398#117—21no—
399#117—21no—
400#117—21no—
401#117—21no—
402#117—21no—
403#117—21no—
404#117—21no—
405#117—21no—
406#117D4×C4712yes—
407#117D4×C4712yes—
408#117D4×C4712yes—
409#117D4×C4712yes—
410#117C23×C4712yes—
411#117C23×C4712yes—
412#117C23×C4712yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4D4×C4712yes—
421C2×C42—21no—
422C2×C42—21no—
423C2×C42D4×C4712yes—
424C2×C42C23×C4712yes—
425#104—28no—
426#104—28no—
427#99—28no—
428C24—21no—
429C24C2×D4×C4711yescharacteristic
430#41—22no—
431#41—22no—
432#41—22no—
433#41—22no—
434#41—22no—
435#41—22no—
436#41—22no—
437#41—22no—
438#41—41no—
439#41—41no—
440#41—41no—
441#41—41no—
442#41—41no—
443#41—41no—
444#41—41no—
445#41—41no—
446C2×Q8—28no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8C2×Q8×C4712yes—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4C2×Q8×C4712yes—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4C2×D4×C4711yescharacteristic
477C22×C4C2×D4×C4711yescharacteristic
478C22×C4C2×D4×C4711yescharacteristic
479C22×C4C2×D4×C4711yescharacteristic
480C22×C4C2×D4×C4711yescharacteristic
481C22×C4C2×Q8×C4711yescharacteristic
482C22×C4C2×Q8×C4711yescharacteristic
483C22×C4#41×C4711yescharacteristic
484C22×C4#41×C4711yescharacteristic
485C22×C4#41×C4711yescharacteristic
486C22×C4#41×C4711yescharacteristic
487C22×C4#41×C4711yescharacteristic
488C22×C4#41×C4711yescharacteristic
489C22×C4C24×C4711yescharacteristic
490M16—22no—
491M16—22no—
492M16—22no—
493M16—22no—
494M16—41no—
495M16—41no—
496M16—41no—
497M16—41no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—41no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—41no—
522C22⋊C4C2×D4×C4714yes—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—41no—
529C42C2×D4×C4714yes—
530C42C2×D4×C4714yes—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—81no—
535C23—82no—
536C23#120×C4711yescharacteristic
537C23#120×C4711yescharacteristic
538C23C22⋊Q8×C4711yescharacteristic
539C23C22⋊Q8×C4711yescharacteristic
540C23#124×C4711yescharacteristic
541C23#124×C4711yescharacteristic
542C23C22×D4×C4711yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—41no—
546Q8—41no—
547Q8—88no—
548D4—22no—
549D4—22no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—41no—
555D4—41no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4#120×C4711yescharacteristic
572C2×C4#120×C4711yescharacteristic
573C2×C4C22⋊Q8×C4711yescharacteristic
574C2×C4C22⋊Q8×C4711yescharacteristic
575C2×C4C22⋊Q8×C4711yescharacteristic
576C2×C4C22⋊Q8×C4711yescharacteristic
577C2×C4C22⋊Q8×C4711yescharacteristic
578C2×C4C22⋊Q8×C4711yescharacteristic
579C2×C4C22⋊Q8×C4711yescharacteristic
580C2×C4C22⋊Q8×C4711yescharacteristic
581C2×C4C22⋊Q8×C4711yescharacteristic
582C2×C4C22⋊Q8×C4711yescharacteristic
583C2×C4#124×C4711yescharacteristic
584C2×C4#124×C4711yescharacteristic
585C2×C4#124×C4711yescharacteristic
586C2×C4#124×C4711yescharacteristic
587C2×C4#124×C4711yescharacteristic
588C2×C4#124×C4711yescharacteristic
589C2×C4#124×C4711yescharacteristic
590C2×C4#124×C4711yescharacteristic
591C2×C4#124×C4711yescharacteristic
592C2×C4#124×C4711yescharacteristic
593C2×C4C22×D4×C4711yescharacteristic
594C2×C4C22×D4×C4711yescharacteristic
595C2×C4C22×Q8×C4711yescharacteristic
596C2×C4C2×#41×C4711yescharacteristic
597C2×C4C2×#41×C4711yescharacteristic
598C2×C4C2×#41×C4711yescharacteristic
599C8—42no—
600C8—42no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—42no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22C47×#39311yescharacteristic
613C22C47×#39311yescharacteristic
614C22C47×#39311yescharacteristic
615C22C47×#39311yescharacteristic
616C22C2×#120×C4711yescharacteristic
617C22C2×C22⋊Q8×C4711yescharacteristic
618C22C2×#124×C4711yescharacteristic
619C4—44no—
620C4—44no—
621C4—44no—
622C4—42no—
623C4—82no—
624C4C47×#39311yescharacteristic
625C4C47×#39311yescharacteristic
626C4C47×#39311yescharacteristic
627C4C47×#39311yescharacteristic
628C4C2×C22⋊Q8×C4711yescharacteristic
629C4C2×C22⋊Q8×C4711yescharacteristic
630C4C2×#124×C4711yescharacteristic
631C4C2×#124×C4711yescharacteristic
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—82no—
640C2C2×C47×#39311yescharacteristic
subgroupG/H
centerC4×C47C2×#124
commutator, Frattini, cosocleC22×C4C24×C47
socleC2×C47C2×#393
prime radicalC22×#41×C47C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C47 ◁ C23×C4×C47 ◁ G

Complements

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

NH ≅ G/Ntype
1. C47×#3458C2semidirect
5. C47×#2031C2semidirect
6. C2×C47×#344C2semidirect
7. C47×#1733C2semidirect
8. C2×C47×H(3,ℤ/4)C2semidirect
18. C47×#533C22semidirect
32. C47×#344C22semidirect
33. C47×#339C22semidirect
34. C47×H(3,ℤ/4)C22semidirect
45. C2×#41×C47Q8semidirect
47. C22×Q8×C47Q8semidirect
49. C22×D4×C47Q8semidirect
280. #23472C47direct (N × H)
312. #3458C2×C47semidirect
316. #2031C2×C47semidirect
317. C2×#344C2×C47semidirect
318. #1733C2×C47semidirect
319. C2×H(3,ℤ/4)C2×C47semidirect
338. #533C22×C47semidirect
352. #344C22×C47semidirect
353. #339C22×C47semidirect
354. H(3,ℤ/4)C22×C47semidirect
355. C47#23472direct (N × H)
366. C2×#41Q8×C47semidirect
368. C22×Q8Q8×C47semidirect
370. C22×D4Q8×C47semidirect

Maximal subgroups

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

subgroupindex
C47×#34582
C2×C47×#4212
C2×C47×#4102
C47×#20312
C2×C47×#3442
C47×#17332
C2×C47×H(3,ℤ/4)2
#2347247