Finite Group Databasebeta

← C47×#27074 #26066772

Subgroups — C47×#27074 #26066772

Order 12032.

740 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#2514C211yesmaximal, characteristic, complemented
2C47×#2367C211yesmaximal, characteristic
3C47×#2097C211yesmaximal, characteristic, complemented
4C47×#2056C211yesmaximal, characteristic, complemented
5C47×#2042C211yesmaximal, characteristic
6C47×#2015C211yesmaximal, characteristic
7C47×#1968C211yesmaximal, characteristic
8C47×#1928C211yesmaximal, characteristic, complemented
9C47×#1925C211yesmaximal, characteristic
10C47×#1920C211yesmaximal, characteristic, complemented
11C47×#1916C211yesmaximal, characteristic, complemented
12C47×#1878C211yesmaximal, characteristic, complemented
13C47×#1877C211yesmaximal, characteristic
14C47×#1804C211yesmaximal, characteristic
15C4×#103×C47C211yesmaximal, characteristic, complemented
16C22×Q16×C47C2211yescharacteristic
17C22×SD16×C47C2211yescharacteristic
18C2×#124×C47C2211yescharacteristic
19C2×C4×Q8×C47C2211yescharacteristic
20C2×C4×D4×C47C2211yescharacteristic
21C2×#107×C47C2211yescharacteristic
22C2×#106×C47C2211yescharacteristic
23C2×C4⋊C8×C47C2211yescharacteristic
24C2×#103×C47C2211yescharacteristic
25C2×#103×C47C2211yescharacteristic
26C2×#103×C47C2211yescharacteristic
27C2×#103×C47C2211yescharacteristic
28C2×#103×C47C2211yescharacteristic
29C2×#102×C47C2211yescharacteristic
30C2×#102×C47C2211yescharacteristic
31C2×#102×C47C2211yescharacteristic
32C2×C4×C8×C47C2211yescharacteristic
33C47×#401—21no—
34C47×#400—21no—
35C47×#391C2211yescharacteristic
36C47×#391C2211yescharacteristic
37C47×#391C2211yescharacteristic
38C47×#391C2211yescharacteristic
39C47×#390C2211yescharacteristic
40C47×#389C2211yescharacteristic
41C47×#388—21no—
42C47×#388—21no—
43C47×#387—21no—
44C47×#385—21no—
45C47×#383—21no—
46C47×#382C2211yescharacteristic
47C4×C4⋊C4×C47C2211yescharacteristic
48C4×C4⋊C4×C47C2211yescharacteristic
49C4×C4⋊C4×C47C2211yescharacteristic
50C4×C22⋊C4×C47—21no—
51C47×#340C2211yescharacteristic
52C47×#340C2211yescharacteristic
53C47×#340C2211yescharacteristic
54C47×#340C2211yescharacteristic
55C47×#336C2211yescharacteristic
56C47×#336C2211yescharacteristic
57C47×#336C2211yescharacteristic
58C47×#336C2211yescharacteristic
59C22×Q8×C47C2311yescharacteristic
60C22×Q8×C47C2311yescharacteristic
61C22×D4×C47C2311yescharacteristic
62C23×C4×C47—21no—
63C2×Q16×C47—22no—
64C2×Q16×C47—41no—
65C2×Q16×C47—41no—
66C2×SD16×C47—22no—
67C2×SD16×C47—22no—
68C2×SD16×C47—41no—
69C22×C8×C47C2311yescharacteristic
70C22×C8×C47C2311yescharacteristic
71C22×C8×C47C2311yescharacteristic
72C22×C8×C47C2311yescharacteristic
73#124×C47—22no—
74#124×C47—22no—
75C4×Q8×C47—22no—
76C4×Q8×C47—41no—
77C4×D4×C47—22no—
78C4×D4×C47—41no—
79C2×C4⋊C4×C47—21no—
80C2×C4⋊C4×C47—21no—
81C2×C4⋊C4×C47D411yescharacteristic
82C2×C4⋊C4×C47D411yescharacteristic
83C2×C4⋊C4×C47C2311yescharacteristic
84C2×C4⋊C4×C47C2311yescharacteristic
85C2×C4⋊C4×C47C2311yescharacteristic
86C2×C4⋊C4×C47C2311yescharacteristic
87C2×C4⋊C4×C47C2311yescharacteristic
88C2×C22⋊C4×C47—21no—
89C2×C22⋊C4×C47—21no—
90C2×C22⋊C4×C47—21no—
91C2×C22⋊C4×C47—21no—
92C2×C22⋊C4×C47—21no—
93C2×C42×C47—21no—
94C2×C42×C47—21no—
95C2×C42×C47—21no—
96C2×C42×C47C2311yescharacteristic
97C2×C42×C47C2311yescharacteristic
98C2×C42×C47C2311yescharacteristic
99#107×C47—41no—
100#106×C47—22no—
101C4⋊C8×C47—41no—
102#103×C47—22no—
103#103×C47—22no—
104#103×C47—22no—
105#103×C47—41no—
106#103×C47—41no—
107#102×C47—22no—
108#102×C47—41no—
109#102×C47—41no—
110C4×C8×C47—41no—
111#95×C47—21no—
112#95×C47—21no—
113#95×C47—21no—
114#95×C47—21no—
115#95×C47—21no—
116#95×C47—21no—
117#95×C47—21no—
118#95×C47—21no—
119#95×C47—21no—
120#95×C47—21no—
121#95×C47—21no—
122#95×C47—21no—
123#95×C47—21no—
124#95×C47D411yescharacteristic
125#95×C47D411yescharacteristic
126C24×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47—21no—
132C2×Q8×C47—21no—
133C2×Q8×C47—21no—
134C2×Q8×C47—21no—
135C2×Q8×C47—21no—
136C2×Q8×C47—21no—
137C2×Q8×C47—21no—
138C2×Q8×C47#4112yes—
139C2×D4×C47—21no—
140C2×D4×C47—21no—
141C2×D4×C47—21no—
142C2×D4×C47—21no—
143C2×D4×C47—21no—
144C2×D4×C47#4112yes—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—21no—
148C22×C4×C47—21no—
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47—21no—
152C22×C4×C47—42no—
153C22×C4×C47—42no—
154C22×C4×C47—81no—
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2411yescharacteristic
162Q16×C47—44no—
163SD16×C47—44no—
164C2×C8×C47—21no—
165C2×C8×C47—21no—
166C2×C8×C47—21no—
167C2×C8×C47—21no—
168C2×C8×C47—21no—
169C2×C8×C47—21no—
170C2×C8×C47—21no—
171C2×C8×C47—21no—
172C2×C8×C47—21no—
173C2×C8×C47—21no—
174C2×C8×C47—21no—
175C2×C8×C47#4112yes—
176C4⋊C4×C47—22no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—21no—
183C4⋊C4×C47—21no—
184C4⋊C4×C47—21no—
185C4⋊C4×C47—21no—
186C4⋊C4×C47—41no—
187C4⋊C4×C47—42no—
188C4⋊C4×C47—42no—
189C4⋊C4×C47#4112yes—
190C22⋊C4×C47—24no—
191C22⋊C4×C47—24no—
192C22⋊C4×C47—42no—
193C22⋊C4×C47—42no—
194C22⋊C4×C47—81no—
195C42×C47—21no—
196C42×C47—21no—
197C42×C47—21no—
198C42×C47—21no—
199C42×C47—21no—
200C42×C47—21no—
201C42×C47—42no—
202C42×C47—42no—
203C42×C47—81no—
204C23×C47—22no—
205C23×C47—41no—
206C23×C47—41no—
207C23×C47—41no—
208C23×C47—41no—
209C23×C47—41no—
210C23×C47—41no—
211C23×C47C22×D411yescharacteristic
212Q8×C47—22no—
213Q8×C47—22no—
214Q8×C47—41no—
215Q8×C47—41no—
216Q8×C47—41no—
217Q8×C47—41no—
218Q8×C47—41no—
219Q8×C47—41no—
220D4×C47—22no—
221D4×C47—22no—
222D4×C47—41no—
223D4×C47—41no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—22no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—22no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
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—21no—
242C2×C4×C47—21no—
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—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—41no—
257C2×C4×C47—41no—
258C2×C4×C47—41no—
259C2×C4×C47—41no—
260C2×C4×C47—41no—
261C2×C4×C47—41no—
262C2×C4×C47—84no—
263C2×C4×C47#13512yes—
264C2×C4×C47C2×#4111yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C2×C4×C47C2×#4111yescharacteristic
267C2×C4×C47C2×#4111yescharacteristic
268C2×C4×C47H(5,𝔽2)11yescharacteristic
269C2×C4×C47H(5,𝔽2)11yescharacteristic
270C8×C47—22no—
271C8×C47—41no—
272C8×C47—41no—
273C8×C47—41no—
274#27074C4711yesmaximal, characteristic, direct product factor, Sylow
275C22×C47—42no—
276C22×C47—42no—
277C22×C47—42no—
278C22×C47—81no—
279C22×C47—81no—
280C22×C47—81no—
281C22×C47—81no—
282C22×C47#52511yescharacteristic
283C22×C47#54611yescharacteristic
284C22×C47#55011yescharacteristic
285C22×C47C2×#13511yescharacteristic
286C22×C47#57511yescharacteristic
287C22×C47#57711yescharacteristic
288C22×C47#57811yescharacteristic
289C4×C47—22no—
290C4×C47—41no—
291C4×C47—41no—
292C4×C47—42no—
293C4×C47—41no—
294C4×C47—41no—
295C4×C47—41no—
296C4×C47—42no—
297C4×C47—81no—
298C4×C47—81no—
299C4×C47—81no—
300C4×C47—81no—
301C4×C47—81no—
302C4×C47#52911yescharacteristic
303C4×C47#53811yescharacteristic
304C4×C47#55111yescharacteristic
305C4×C47#55311yescharacteristic
306#2514C2×C4711yescharacteristic, complemented
307#2367C2×C4711yescharacteristic
308#2097C2×C4711yescharacteristic, complemented
309#2056C2×C4711yescharacteristic, complemented
310#2042C2×C4711yescharacteristic
311#2015C2×C4711yescharacteristic
312#1968C2×C4711yescharacteristic
313#1928C2×C4711yescharacteristic, complemented
314#1925C2×C4711yescharacteristic
315#1920C2×C4711yescharacteristic, complemented
316#1916C2×C4711yescharacteristic, complemented
317#1878C2×C4711yescharacteristic, complemented
318#1877C2×C4711yescharacteristic
319#1804C2×C4711yescharacteristic
320C4×#103C2×C4711yescharacteristic, complemented
321C2×C47—82no—
322C2×C47#246711yescharacteristic
323C2×C47#310211yescharacteristic
324C2×C47#312011yescharacteristic
325C2×C47#330011yescharacteristic
326C2×C47#331111yescharacteristic
327C2×C47#339411yescharacteristic
328C2×C47#340211yescharacteristic
329C22×Q16C22×C4711yescharacteristic
330C22×SD16C22×C4711yescharacteristic
331C2×#124C22×C4711yescharacteristic
332C2×C4×Q8C22×C4711yescharacteristic
333C2×C4×D4C22×C4711yescharacteristic
334C2×#107C22×C4711yescharacteristic
335C2×#106C22×C4711yescharacteristic
336C2×C4⋊C8C22×C4711yescharacteristic
337C2×#103C22×C4711yescharacteristic
338C2×#103C22×C4711yescharacteristic
339C2×#103C22×C4711yescharacteristic
340C2×#103C22×C4711yescharacteristic
341C2×#103C22×C4711yescharacteristic
342C2×#102C22×C4711yescharacteristic
343C2×#102C22×C4711yescharacteristic
344C2×#102C22×C4711yescharacteristic
345C2×C4×C8C22×C4711yescharacteristic
346#401—21no—
347#400—21no—
348#391C22×C4711yescharacteristic
349#391C22×C4711yescharacteristic
350#391C22×C4711yescharacteristic
351#391C22×C4711yescharacteristic
352#390C22×C4711yescharacteristic
353#389C22×C4711yescharacteristic
354#388—21no—
355#388—21no—
356#387—21no—
357#385—21no—
358#383—21no—
359#382C22×C4711yescharacteristic
360C4×C4⋊C4C22×C4711yescharacteristic
361C4×C4⋊C4C22×C4711yescharacteristic
362C4×C4⋊C4C22×C4711yescharacteristic
363C4×C22⋊C4—21no—
364#340C22×C4711yescharacteristic
365#340C22×C4711yescharacteristic
366#340C22×C4711yescharacteristic
367#340C22×C4711yescharacteristic
368#336C22×C4711yescharacteristic
369#336C22×C4711yescharacteristic
370#336C22×C4711yescharacteristic
371#336C22×C4711yescharacteristic
372C47#2707411yescharacteristic, direct product factor, Sylow
373C22×Q8C23×C4711yescharacteristic
374C22×Q8C23×C4711yescharacteristic
375C22×D4C23×C4711yescharacteristic
376C23×C4—21no—
377C2×Q16—22no—
378C2×Q16—41no—
379C2×Q16—41no—
380C2×SD16—22no—
381C2×SD16—22no—
382C2×SD16—41no—
383C22×C8C23×C4711yescharacteristic
384C22×C8C23×C4711yescharacteristic
385C22×C8C23×C4711yescharacteristic
386C22×C8C23×C4711yescharacteristic
387#124—22no—
388#124—22no—
389C4×Q8—22no—
390C4×Q8—41no—
391C4×D4—22no—
392C4×D4—41no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4D4×C4711yescharacteristic
396C2×C4⋊C4D4×C4711yescharacteristic
397C2×C4⋊C4C23×C4711yescharacteristic
398C2×C4⋊C4C23×C4711yescharacteristic
399C2×C4⋊C4C23×C4711yescharacteristic
400C2×C4⋊C4C23×C4711yescharacteristic
401C2×C4⋊C4C23×C4711yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42C23×C4711yescharacteristic
411C2×C42C23×C4711yescharacteristic
412C2×C42C23×C4711yescharacteristic
413#107—41no—
414#106—22no—
415C4⋊C8—41no—
416#103—22no—
417#103—22no—
418#103—22no—
419#103—41no—
420#103—41no—
421#102—22no—
422#102—41no—
423#102—41no—
424C4×C8—41no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95D4×C4711yescharacteristic
439#95D4×C4711yescharacteristic
440C24—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8#41×C4712yes—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4#41×C4712yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—81no—
469C22×C4C2×D4×C4711yescharacteristic
470C22×C4C2×D4×C4711yescharacteristic
471C22×C4C2×D4×C4711yescharacteristic
472C22×C4C2×D4×C4711yescharacteristic
473C22×C4C2×D4×C4711yescharacteristic
474C22×C4C2×D4×C4711yescharacteristic
475C22×C4C24×C4711yescharacteristic
476Q16—44no—
477SD16—44no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8#41×C4712yes—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—41no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4#41×C4712yes—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—81no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—42no—
516C42—42no—
517C42—81no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C22×D4×C4711yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—41no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—84no—
577C2×C4#135×C4712yes—
578C2×C4C2×#41×C4711yescharacteristic
579C2×C4C2×#41×C4711yescharacteristic
580C2×C4C2×#41×C4711yescharacteristic
581C2×C4C2×#41×C4711yescharacteristic
582C2×C4H(5,𝔽2)×C4711yescharacteristic
583C2×C4H(5,𝔽2)×C4711yescharacteristic
584C8—22no—
585C8—41no—
586C8—41no—
587C8—41no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C47×#52511yescharacteristic
596C22C47×#54611yescharacteristic
597C22C47×#55011yescharacteristic
598C22C2×#135×C4711yescharacteristic
599C22C47×#57511yescharacteristic
600C22C47×#57711yescharacteristic
601C22C47×#57811yescharacteristic
602C4—22no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—41no—
608C4—41no—
609C4—42no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4C47×#52911yescharacteristic
616C4C47×#53811yescharacteristic
617C4C47×#55111yescharacteristic
618C4C47×#55311yescharacteristic
619C2—82no—
620C2C47×#246711yescharacteristic
621C2C47×#310211yescharacteristic
622C2C47×#312011yescharacteristic
623C2C47×#330011yescharacteristic
624C2C47×#331111yescharacteristic
625C2C47×#339411yescharacteristic
626C2C47×#340211yescharacteristic
subgroupG/H
center, socleC23×C47C22×D4
commutator, Frattini, cosocleC22×C4C24×C47
prime radicalC22×D4×C47C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C47 ◁ C4×C4⋊C4×C47 ◁ G

Complements

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

NH ≅ G/Ntype
1. C47×#2514C2semidirect
3. C47×#2097C2semidirect
4. C47×#2056C2semidirect
8. C47×#1928C2semidirect
10. C47×#1920C2semidirect
11. C47×#1916C2semidirect
12. C47×#1878C2semidirect
15. C4×#103×C47C2semidirect
274. #27074C47direct (N × H)
306. #2514C2×C47semidirect
308. #2097C2×C47semidirect
309. #2056C2×C47semidirect
313. #1928C2×C47semidirect
315. #1920C2×C47semidirect
316. #1916C2×C47semidirect
317. #1878C2×C47semidirect
320. C4×#103C2×C47semidirect
372. C47#27074direct (N × H)

Maximal subgroups

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

subgroupindex
C47×#25142
C47×#23672
C47×#20972
C47×#20562
C47×#20422
C47×#20152
C47×#19682
C47×#19282
C47×#19252
C47×#19202
C47×#19162
C47×#18782
C47×#18772
C47×#18042
C4×#103×C472
#2707447