Finite Group Databasebeta

← C47×#23901 #26063599

Subgroups — C47×#23901 #26063599

Order 12032.

2056 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#3484C211yesmaximal, characteristic
2C22×#123×C47C211yesmaximal, characteristic, complemented
3C2×C47×#458C212yesmaximal, complemented
4C47×#2849C212yesmaximal, complemented
5C47×#2022C214yesmaximal, complemented
6C47×#1898C212yesmaximal, complemented
7C47×#1890C212yesmaximal, complemented
8C47×#1781C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C47C2211yescharacteristic
10C23×D4×C47C2211yescharacteristic
11C24×C4×C47C2211yescharacteristic, complemented
12C47×#546—24no—
13D42×C47—24no—
14C47×#535—21no—
15C47×#534—21no—
16C47×#534C2212yes—
17C47×#525—21no—
18C2×#123×C47—21no—
19C2×#123×C47—21no—
20C2×#123×C47—28no—
21C2×#123×C47C2212yes—
22C2×#123×C47C2212yescomplemented
23C2×#123×C47C2214yescomplemented
24C2×#120×C47—21no—
25C2×#120×C47C2212yes—
26C47×#518C2211yescharacteristic
27C22×C4⋊C4×C47—21no—
28C22×C22⋊C4×C47—21no—
29C22×C22⋊C4×C47C2211yescharacteristic, complemented
30C47×#458—28no—
31C2×#99×C47C2212yes—
32C2×#99×C47C2214yescomplemented
33C2×#99×C47C2212yescomplemented
34C47×#386—24no—
35C47×#385—24no—
36C47×#381C2214yescomplemented
37C47×#379C2212yescomplemented
38C47×#342C2214yescomplemented
39C25×C47D411yescharacteristic
40C25×C47C2311yescharacteristic
41H(5,𝔽2)×C47—24no—
42H(5,𝔽2)×C47—42no—
43C2×#41×C47—21no—
44C2×#41×C47—22no—
45C22×D4×C47—21no—
46C22×D4×C47—22no—
47C22×D4×C47—24no—
48C22×D4×C47—22no—
49C22×D4×C47—24no—
50C22×D4×C47—42no—
51C22×D4×C47D411yescharacteristic
52C22×D4×C47D411yescharacteristic
53C22×D4×C47D414yes—
54C22×D4×C47C2312yes—
55C22×D4×C47C2311yescharacteristic
56C22×D4×C47C2311yescharacteristic
57C23×C4×C47—21no—
58C23×C4×C47—21no—
59C23×C4×C47—21no—
60C23×C4×C47—24no—
61C23×C4×C47—24no—
62C23×C4×C47—42no—
63C23×C4×C47D411yescharacteristic
64C23×C4×C47C2312yes—
65#127×C47—22no—
66#124×C47—22no—
67#123×C47—28no—
68#123×C47—28no—
69#123×C47—22no—
70#123×C47—24no—
71#123×C47—22no—
72#123×C47—22no—
73#123×C47—24no—
74#123×C47—24no—
75#123×C47—28no—
76#123×C47—44no—
77#123×C47D418yescomplemented
78C22⋊Q8×C47—22no—
79#121×C47—22no—
80#121×C47—24no—
81#121×C47—24no—
82#121×C47—24no—
83#120×C47—24no—
84#120×C47—22no—
85#120×C47—22no—
86#120×C47—22no—
87#120×C47—22no—
88#120×C47—24no—
89#120×C47—22no—
90C4×D4×C47—24no—
91C4×D4×C47—24no—
92#117×C47—21no—
93C2×C4⋊C4×C47—28no—
94C2×C4⋊C4×C47—22no—
95C2×C4⋊C4×C47—22no—
96C2×C4⋊C4×C47—24no—
97C2×C22⋊C4×C47—28no—
98C2×C22⋊C4×C47—22no—
99C2×C22⋊C4×C47—22no—
100C2×C22⋊C4×C47—22no—
101C2×C22⋊C4×C47—21no—
102C2×C22⋊C4×C47—21no—
103C2×C22⋊C4×C47—24no—
104C2×C22⋊C4×C47—24no—
105C2×C22⋊C4×C47—24no—
106C2×C22⋊C4×C47D414yes—
107C2×C22⋊C4×C47C2314yes—
108C2×C22⋊C4×C47C2312yes—
109C2×C22⋊C4×C47C2312yes—
110#99×C47—24no—
111#99×C47—28no—
112#99×C47—42no—
113#95×C47—28no—
114C24×C47—22no—
115C24×C47—22no—
116C24×C47—21no—
117C24×C47—22no—
118C24×C47—22no—
119C24×C47—44no—
120C24×C47—41no—
121C24×C47—44no—
122C24×C47C2×D411yescharacteristic
123C24×C47C2×D414yes—
124C24×C47C2×D411yescharacteristic
125C24×C47C2×D412yes—
126C24×C47C2×D411yescharacteristic
127C24×C47C2×D411yescharacteristic
128C24×C47C2411yescharacteristic
129#41×C47—48no—
130#41×C47—42no—
131#41×C47—42no—
132C2×Q8×C47—21no—
133C2×D4×C47—22no—
134C2×D4×C47—22no—
135C2×D4×C47—22no—
136C2×D4×C47—21no—
137C2×D4×C47—24no—
138C2×D4×C47—22no—
139C2×D4×C47—22no—
140C2×D4×C47—22no—
141C2×D4×C47—22no—
142C2×D4×C47—22no—
143C2×D4×C47—22no—
144C2×D4×C47—22no—
145C2×D4×C47—24no—
146C2×D4×C47—24no—
147C2×D4×C47—22no—
148C2×D4×C47—24no—
149C2×D4×C47—24no—
150C2×D4×C47—22no—
151C2×D4×C47—41no—
152C2×D4×C47—41no—
153C2×D4×C47—416no—
154C2×D4×C47—48no—
155C2×D4×C47—42no—
156C2×D4×C47—42no—
157C2×D4×C47—48no—
158C2×D4×C47—84no—
159C2×D4×C47C2×D414yes—
160C22×C4×C47—24no—
161C22×C4×C47—24no—
162C22×C4×C47—21no—
163C22×C4×C47—22no—
164C22×C4×C47—24no—
165C22×C4×C47—22no—
166C22×C4×C47—22no—
167C22×C4×C47—22no—
168C22×C4×C47—24no—
169C22×C4×C47—22no—
170C22×C4×C47—24no—
171C22×C4×C47—22no—
172C22×C4×C47—22no—
173C22×C4×C47—24no—
174C22×C4×C47—41no—
175C22×C4×C47—416no—
176C22×C4×C47—44no—
177C22×C4×C47—42no—
178C22×C4×C47—42no—
179C22×C4×C47—42no—
180C22×C4×C47—42no—
181C22×C4×C47C2×D412yes—
182C22×C4×C47C2×D411yescharacteristic
183C22×C4×C47C2×D411yescharacteristic
184C22×C4×C47C2×D414yes—
185C22×C4×C47C2×D414yescomplemented
186C4⋊C4×C47—28no—
187C4⋊C4×C47—22no—
188C4⋊C4×C47—24no—
189C4⋊C4×C47—28no—
190C4⋊C4×C47—48no—
191C22⋊C4×C47—24no—
192C22⋊C4×C47—28no—
193C22⋊C4×C47—22no—
194C22⋊C4×C47—24no—
195C22⋊C4×C47—24no—
196C22⋊C4×C47—28no—
197C22⋊C4×C47—24no—
198C22⋊C4×C47—24no—
199C22⋊C4×C47—22no—
200C22⋊C4×C47—22no—
201C22⋊C4×C47—22no—
202C22⋊C4×C47—24no—
203C22⋊C4×C47—24no—
204C22⋊C4×C47—48no—
205C22⋊C4×C47—48no—
206C22⋊C4×C47C2×D414yescomplemented
207C42×C47—22no—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47—24no—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47—21no—
214C23×C47—28no—
215C23×C47—28no—
216C23×C47—21no—
217C23×C47—21no—
218C23×C47—21no—
219C23×C47—22no—
220C23×C47—21no—
221C23×C47—24no—
222C23×C47—44no—
223C23×C47—42no—
224C23×C47—42no—
225C23×C47—44no—
226C23×C47—48no—
227C23×C47—48no—
228C23×C47—42no—
229C23×C47—42no—
230C23×C47—48no—
231C23×C47—42no—
232C23×C47—44no—
233C23×C47—44no—
234C23×C47—82no—
235C23×C47—84no—
236C23×C47—82no—
237C23×C47#12012yes—
238C23×C47#12012yes—
239C23×C47#12012yes—
240C23×C47C22×D411yescharacteristic
241C23×C47C22×D412yes—
242C23×C47C22×D412yes—
243C23×C47H(5,𝔽2)12yes—
244Q8×C47—42no—
245D4×C47—28no—
246D4×C47—44no—
247D4×C47—42no—
248D4×C47—44no—
249D4×C47—44no—
250D4×C47—42no—
251D4×C47—42no—
252D4×C47—44no—
253D4×C47—44no—
254D4×C47—48no—
255D4×C47—84no—
256D4×C47—84no—
257C2×C4×C47—21no—
258C2×C4×C47—21no—
259C2×C4×C47—24no—
260C2×C4×C47—24no—
261C2×C4×C47—21no—
262C2×C4×C47—24no—
263C2×C4×C47—22no—
264C2×C4×C47—21no—
265C2×C4×C47—21no—
266C2×C4×C47—28no—
267C2×C4×C47—24no—
268C2×C4×C47—44no—
269C2×C4×C47—44no—
270C2×C4×C47—48no—
271C2×C4×C47—44no—
272C2×C4×C47—48no—
273C2×C4×C47—48no—
274C2×C4×C47—42no—
275C2×C4×C47—42no—
276C2×C4×C47—42no—
277C2×C4×C47—44no—
278C2×C4×C47—44no—
279C2×C4×C47—48no—
280C2×C4×C47—44no—
281C2×C4×C47—82no—
282C2×C4×C47#12012yes—
283#23901C4711yesmaximal, characteristic, direct product factor, Sylow
284C22×C47—24no—
285C22×C47—24no—
286C22×C47—22no—
287C22×C47—24no—
288C22×C47—22no—
289C22×C47—22no—
290C22×C47—22no—
291C22×C47—22no—
292C22×C47—24no—
293C22×C47—42no—
294C22×C47—42no—
295C22×C47—416no—
296C22×C47—42no—
297C22×C47—42no—
298C22×C47—42no—
299C22×C47—42no—
300C22×C47—42no—
301C22×C47—42no—
302C22×C47—41no—
303C22×C47—44no—
304C22×C47—42no—
305C22×C47—81no—
306C22×C47—88no—
307C22×C47—84no—
308C22×C47—81no—
309C22×C47—162no—
310C22×C47#45814yescomplemented
311C22×C47C2×#12011yescharacteristic
312C22×C47C2×#12011yescharacteristic
313C22×C47#53411yescharacteristic
314C22×C47D4214yes—
315C4×C47—22no—
316C4×C47—44no—
317C4×C47—44no—
318C4×C47—41no—
319C4×C47—42no—
320C4×C47—44no—
321C4×C47—84no—
322#3484C2×C4711yescharacteristic
323C22×#123C2×C4711yescharacteristic, complemented
324C2×#458C2×C4712yescomplemented
325#2849C2×C4712yescomplemented
326#2022C2×C4714yescomplemented
327#1898C2×C4712yescomplemented
328#1890C2×C4712yescomplemented
329#1781C2×C4711yescharacteristic, complemented
330C2×C47—21no—
331C2×C47—24no—
332C2×C47—21no—
333C2×C47—24no—
334C2×C47—42no—
335C2×C47—42no—
336C2×C47—81no—
337C2×C47—82no—
338C2×C47#240311yescharacteristic
339C2×C47C2×#45812yes—
340C2×H(5,𝔽2)C22×C4711yescharacteristic
341C23×D4C22×C4711yescharacteristic
342C24×C4C22×C4711yescharacteristic, complemented
343#546—24no—
344D42—24no—
345#535—21no—
346#534—21no—
347#534C22×C4712yes—
348#525—21no—
349C2×#123—21no—
350C2×#123—21no—
351C2×#123—28no—
352C2×#123C22×C4712yes—
353C2×#123C22×C4712yescomplemented
354C2×#123C22×C4714yescomplemented
355C2×#120—21no—
356C2×#120C22×C4712yes—
357#518C22×C4711yescharacteristic
358C22×C4⋊C4—21no—
359C22×C22⋊C4—21no—
360C22×C22⋊C4C22×C4711yescharacteristic, complemented
361#458—28no—
362C2×#99C22×C4712yes—
363C2×#99C22×C4714yescomplemented
364C2×#99C22×C4712yescomplemented
365#386—24no—
366#385—24no—
367#381C22×C4714yescomplemented
368#379C22×C4712yescomplemented
369#342C22×C4714yescomplemented
370C47#2390111yescharacteristic, direct product factor, Sylow
371C25D4×C4711yescharacteristic
372C25C23×C4711yescharacteristic
373H(5,𝔽2)—24no—
374H(5,𝔽2)—42no—
375C2×#41—21no—
376C2×#41—22no—
377C22×D4—21no—
378C22×D4—22no—
379C22×D4—24no—
380C22×D4—22no—
381C22×D4—24no—
382C22×D4—42no—
383C22×D4D4×C4711yescharacteristic
384C22×D4D4×C4711yescharacteristic
385C22×D4D4×C4714yes—
386C22×D4C23×C4712yes—
387C22×D4C23×C4711yescharacteristic
388C22×D4C23×C4711yescharacteristic
389C23×C4—21no—
390C23×C4—21no—
391C23×C4—21no—
392C23×C4—24no—
393C23×C4—24no—
394C23×C4—42no—
395C23×C4D4×C4711yescharacteristic
396C23×C4C23×C4712yes—
397#127—22no—
398#124—22no—
399#123—28no—
400#123—28no—
401#123—22no—
402#123—24no—
403#123—22no—
404#123—22no—
405#123—24no—
406#123—24no—
407#123—28no—
408#123—44no—
409#123D4×C4718yescomplemented
410C22⋊Q8—22no—
411#121—22no—
412#121—24no—
413#121—24no—
414#121—24no—
415#120—24no—
416#120—22no—
417#120—22no—
418#120—22no—
419#120—22no—
420#120—24no—
421#120—22no—
422C4×D4—24no—
423C4×D4—24no—
424#117—21no—
425C2×C4⋊C4—28no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—24no—
429C2×C22⋊C4—28no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—22no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—24no—
436C2×C22⋊C4—24no—
437C2×C22⋊C4—24no—
438C2×C22⋊C4D4×C4714yes—
439C2×C22⋊C4C23×C4714yes—
440C2×C22⋊C4C23×C4712yes—
441C2×C22⋊C4C23×C4712yes—
442#99—24no—
443#99—28no—
444#99—42no—
445#95—28no—
446C24—22no—
447C24—22no—
448C24—21no—
449C24—22no—
450C24—22no—
451C24—44no—
452C24—41no—
453C24—44no—
454C24C2×D4×C4711yescharacteristic
455C24C2×D4×C4714yes—
456C24C2×D4×C4711yescharacteristic
457C24C2×D4×C4712yes—
458C24C2×D4×C4711yescharacteristic
459C24C2×D4×C4711yescharacteristic
460C24C24×C4711yescharacteristic
461#41—48no—
462#41—42no—
463#41—42no—
464C2×Q8—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—24no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—24no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—24no—
482C2×D4—22no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—416no—
486C2×D4—48no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—48no—
490C2×D4—84no—
491C2×D4C2×D4×C4714yes—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—41no—
507C22×C4—416no—
508C22×C4—44no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4C2×D4×C4712yes—
514C22×C4C2×D4×C4711yescharacteristic
515C22×C4C2×D4×C4711yescharacteristic
516C22×C4C2×D4×C4714yes—
517C22×C4C2×D4×C4714yescomplemented
518C4⋊C4—28no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—28no—
522C4⋊C4—48no—
523C22⋊C4—24no—
524C22⋊C4—28no—
525C22⋊C4—22no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4C2×D4×C4714yescomplemented
539C42—22no—
540C23—24no—
541C23—21no—
542C23—21no—
543C23—28no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—28no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—21no—
553C23—24no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—82no—
567C23—84no—
568C23—82no—
569C23#120×C4712yes—
570C23#120×C4712yes—
571C23#120×C4712yes—
572C23C22×D4×C4711yescharacteristic
573C23C22×D4×C4712yes—
574C23C22×D4×C4712yes—
575C23H(5,𝔽2)×C4712yes—
576Q8—42no—
577D4—28no—
578D4—44no—
579D4—42no—
580D4—44no—
581D4—44no—
582D4—48no—
583D4—42no—
584D4—42no—
585D4—44no—
586D4—44no—
587D4—84no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—82no—
614C2×C4#120×C4712yes—
615C22—24no—
616C22—24no—
617C22—22no—
618C22—24no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—41no—
634C22—44no—
635C22—42no—
636C22—81no—
637C22—88no—
638C22—81no—
639C22—84no—
640C22—162no—
641C22C47×#45814yescomplemented
642C22C2×#120×C4711yescharacteristic
643C22C2×#120×C4711yescharacteristic
644C22C47×#53411yescharacteristic
645C22D42×C4714yes—
646C4—22no—
647C4—44no—
648C4—44no—
649C4—41no—
650C4—42no—
651C4—44no—
652C4—84no—
653C2—21no—
654C2—24no—
655C2—24no—
656C2—21no—
657C2—42no—
658C2—42no—
659C2—81no—
660C2—82no—
661C2C47×#240311yescharacteristic
662C2C2×C47×#45812yes—
subgroupG/H
center, socleC22×C47C2×#120
commutator, Frattini, cosocleC24C24×C47
prime radicalC47×#3484C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C47 ◁ C25×C47 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#123×C47C2semidirect
3. C2×C47×#458C2semidirect
4. C47×#2849C2semidirect
5. C47×#2022C2semidirect
6. C47×#1898C2semidirect
7. C47×#1890C2semidirect
8. C47×#1781C2semidirect
11. C24×C4×C47C22semidirect
22. C2×#123×C47C22semidirect
23. C2×#123×C47C22semidirect
29. C22×C22⋊C4×C47C22semidirect
32. C2×#99×C47C22semidirect
33. C2×#99×C47C22semidirect
36. C47×#381C22semidirect
37. C47×#379C22semidirect
38. C47×#342C22semidirect
77. #123×C47D4semidirect
185. C22×C4×C47C2×D4semidirect
206. C22⋊C4×C47C2×D4semidirect
283. #23901C47direct (N × H)
310. C22×C47#458semidirect
323. C22×#123C2×C47semidirect
324. C2×#458C2×C47semidirect
325. #2849C2×C47semidirect
326. #2022C2×C47semidirect
327. #1898C2×C47semidirect
328. #1890C2×C47semidirect
329. #1781C2×C47semidirect
342. C24×C4C22×C47semidirect
353. C2×#123C22×C47semidirect
354. C2×#123C22×C47semidirect
360. C22×C22⋊C4C22×C47semidirect
363. C2×#99C22×C47semidirect
364. C2×#99C22×C47semidirect
367. #381C22×C47semidirect
368. #379C22×C47semidirect
369. #342C22×C47semidirect
370. C47#23901direct (N × H)
409. #123D4×C47semidirect
517. C22×C4C2×D4×C47semidirect
538. C22⋊C4C2×D4×C47semidirect
641. C22C47×#458semidirect

Maximal subgroups

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

subgroupindex
C47×#34842
C22×#123×C472
C2×C47×#4582
C47×#28492
C47×#20222
C47×#18982
C47×#18902
C47×#17812
#2390147