Finite Group Databasebeta

← C47×#27054 #26066752

Subgroups — C47×#27054 #26066752

Order 12032.

756 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#2496C211yesmaximal, characteristic, complemented
2C47×#2363C211yesmaximal, characteristic
3C47×#2071C211yesmaximal, characteristic, complemented
4C47×#2067C211yesmaximal, characteristic, complemented
5C47×#2034C211yesmaximal, characteristic
6C47×#2034C211yesmaximal, characteristic
7C47×#1948C211yesmaximal, characteristic, complemented
8C47×#1944C211yesmaximal, characteristic, complemented
9C47×#1924C211yesmaximal, characteristic, complemented
10C47×#1919C211yesmaximal, characteristic, complemented
11C47×#1919C211yesmaximal, characteristic, complemented
12C47×#1864C211yesmaximal, characteristic, complemented
13C47×#1854C211yesmaximal, characteristic, complemented
14C47×#1825C211yesmaximal, characteristic, complemented
15C47×#1762C211yesmaximal, characteristic, complemented
16C2×#126×C47—21no—
17C2×#121×C47C2211yescharacteristic
18C2×C4×D4×C47C2211yescharacteristic
19C2×#117×C47C2211yescharacteristic
20C2×#107×C47C2211yescharacteristic
21C2×#106×C47C2211yescharacteristic
22C2×#106×C47C2211yescharacteristic
23C2×#106×C47C2211yescharacteristic
24C2×C4⋊C8×C47C2211yescharacteristic
25C2×#102×C47C2211yescharacteristic
26C2×#102×C47C2211yescharacteristic
27C2×#102×C47C2211yescharacteristic
28C2×#102×C47C2211yescharacteristic
29C2×C22⋊C8×C47C2211yescharacteristic
30C2×C22⋊C8×C47C2211yescharacteristic
31C2×#97×C47C2211yescharacteristic
32C47×#400—21no—
33C47×#397—21no—
34C47×#390—21no—
35C47×#390C2211yescharacteristic
36C47×#389C2211yescharacteristic
37C47×#389C2211yescharacteristic
38C47×#389C2211yescharacteristic
39C47×#388—21no—
40C47×#387—21no—
41C47×#386—21no—
42C47×#384C2211yescharacteristic
43C47×#380C2211yescharacteristic
44C47×#380C2211yescharacteristic
45C4×C4⋊C4×C47C2211yescharacteristic
46C4×C4⋊C4×C47C2211yescharacteristic
47C4×C22⋊C4×C47—21no—
48C4×C22⋊C4×C47C2211yescharacteristic
49C47×#340C2211yescharacteristic
50C47×#340C2211yescharacteristic
51C47×#340C2211yescharacteristic
52C47×#340C2211yescharacteristic
53C47×#340C2211yescharacteristic
54C47×#340C2211yescharacteristic
55C47×#340C2211yescharacteristic
56C47×#340C2211yescharacteristic
57C47×#336C2211yescharacteristic
58C47×#336C2211yescharacteristic
59C22×D4×C47—21no—
60C22×D4×C47C2311yescharacteristic
61C23×C4×C47—21no—
62C23×C4×C47C2311yescharacteristic
63C22×C8×C47C2311yescharacteristic
64C22×C8×C47C2311yescharacteristic
65C22×C8×C47C2311yescharacteristic
66C22×C8×C47C2311yescharacteristic
67#126×C47—44no—
68#121×C47—22no—
69#121×C47—22no—
70C4×D4×C47—22no—
71C4×D4×C47—41no—
72#117×C47—22no—
73#117×C47—41no—
74C2×C4⋊C4×C47—21no—
75C2×C4⋊C4×C47—21no—
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47—21no—
78C2×C4⋊C4×C47—21no—
79C2×C4⋊C4×C47C2311yescharacteristic
80C2×C4⋊C4×C47C2311yescharacteristic
81C2×C4⋊C4×C47C2311yescharacteristic
82C2×C4⋊C4×C47C2311yescharacteristic
83C2×C4⋊C4×C47C2311yescharacteristic
84C2×C4⋊C4×C47C2311yescharacteristic
85C2×C4⋊C4×C47C2311yescharacteristic
86C2×C22⋊C4×C47—21no—
87C2×C22⋊C4×C47—21no—
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×C22⋊C4×C47D411yescharacteristic
94C2×C22⋊C4×C47D411yescharacteristic
95C2×C42×C47—21no—
96C2×C42×C47—21no—
97C2×C42×C47—21no—
98C2×C42×C47C2311yescharacteristic
99C2×C42×C47C2311yescharacteristic
100#107×C47—41no—
101#106×C47—22no—
102#106×C47—22no—
103#106×C47—22no—
104C4⋊C8×C47—22no—
105#102×C47—22no—
106#102×C47—22no—
107#102×C47—22no—
108#102×C47—41no—
109C22⋊C8×C47—22no—
110C22⋊C8×C47—41no—
111#97×C47—41no—
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×C47D411yescharacteristic
120#95×C47D411yescharacteristic
121C24×C47—21no—
122C24×C47C2×D411yescharacteristic
123C2×D4×C47—24no—
124C2×D4×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—81no—
130C2×D4×C47—81no—
131C2×D4×C47#4112yes—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47—21no—
145C22×C4×C47—42no—
146C22×C4×C47—42no—
147C22×C4×C47—81no—
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2411yescharacteristic
154C2×C8×C47—21no—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C2×C8×C47—21no—
165C2×C8×C47#4112yes—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—42no—
179C4⋊C4×C47—42no—
180C4⋊C4×C47—42no—
181C4⋊C4×C47—42no—
182C4⋊C4×C47—81no—
183C4⋊C4×C47#4112yes—
184C4⋊C4×C47#4112yes—
185C22⋊C4×C47—24no—
186C22⋊C4×C47—22no—
187C22⋊C4×C47—41no—
188C22⋊C4×C47—42no—
189C22⋊C4×C47—42no—
190C22⋊C4×C47—42no—
191C22⋊C4×C47—42no—
192C22⋊C4×C47—42no—
193C22⋊C4×C47—81no—
194C42×C47—21no—
195C42×C47—21no—
196C42×C47—21no—
197C42×C47—21no—
198C42×C47—42no—
199C42×C47—42no—
200C42×C47—81no—
201C23×C47—22no—
202C23×C47—21no—
203C23×C47—21no—
204C23×C47—21no—
205C23×C47—21no—
206C23×C47—21no—
207C23×C47—21no—
208C23×C47—21no—
209C23×C47—41no—
210C23×C47—41no—
211C23×C47—41no—
212C23×C47—41no—
213C23×C47—41no—
214C23×C47—41no—
215C23×C47C22×D411yescharacteristic
216D4×C47—22no—
217D4×C47—22no—
218D4×C47—41no—
219D4×C47—41no—
220D4×C47—84no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—22no—
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—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
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—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—84no—
261C2×C4×C47AGL(1,ℤ/8)12yes—
262C2×C4×C47C2×#4111yescharacteristic
263C2×C4×C47C2×#4111yescharacteristic
264C2×C4×C47C2×#4111yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C2×C4×C47H(5,𝔽2)11yescharacteristic
267C2×C4×C472−1+411yescharacteristic
268C8×C47—22no—
269C8×C47—41no—
270C8×C47—41no—
271C8×C47—41no—
272#27054C4711yesmaximal, characteristic, direct product factor, Sylow
273C22×C47—41no—
274C22×C47—41no—
275C22×C47—41no—
276C22×C47—41no—
277C22×C47—42no—
278C22×C47—41no—
279C22×C47—42no—
280C22×C47—41no—
281C22×C47—42no—
282C22×C47—41no—
283C22×C47—81no—
284C22×C47—81no—
285C22×C47—81no—
286C22×C47—81no—
287C22×C47#52511yescharacteristic
288C22×C47#54611yescharacteristic
289C22×C47#54711yescharacteristic
290C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
291C22×C47#57511yescharacteristic
292C22×C47#57711yescharacteristic
293C22×C47#57711yescharacteristic
294C4×C47—22no—
295C4×C47—41no—
296C4×C47—41no—
297C4×C47—42no—
298C4×C47—41no—
299C4×C47—41no—
300C4×C47—42no—
301C4×C47—81no—
302C4×C47—81no—
303C4×C47—81no—
304C4×C47—81no—
305C4×C47—81no—
306C4×C47#52911yescharacteristic
307C4×C47#53911yescharacteristic
308C4×C47#55311yescharacteristic
309C4×C47#55611yescharacteristic
310#2496C2×C4711yescharacteristic, complemented
311#2363C2×C4711yescharacteristic
312#2071C2×C4711yescharacteristic, complemented
313#2067C2×C4711yescharacteristic, complemented
314#2034C2×C4711yescharacteristic
315#2034C2×C4711yescharacteristic
316#1948C2×C4711yescharacteristic, complemented
317#1944C2×C4711yescharacteristic, complemented
318#1924C2×C4711yescharacteristic, complemented
319#1919C2×C4711yescharacteristic, complemented
320#1919C2×C4711yescharacteristic, complemented
321#1864C2×C4711yescharacteristic, complemented
322#1854C2×C4711yescharacteristic, complemented
323#1825C2×C4711yescharacteristic, complemented
324#1762C2×C4711yescharacteristic, complemented
325C2×C47—81no—
326C2×C47—82no—
327C2×C47#246611yescharacteristic
328C2×C47#311511yescharacteristic
329C2×C47#312311yescharacteristic
330C2×C47#331111yescharacteristic
331C2×C47#333211yescharacteristic
332C2×C47#334511yescharacteristic
333C2×C47#334811yescharacteristic
334C2×#126—21no—
335C2×#121C22×C4711yescharacteristic
336C2×C4×D4C22×C4711yescharacteristic
337C2×#117C22×C4711yescharacteristic
338C2×#107C22×C4711yescharacteristic
339C2×#106C22×C4711yescharacteristic
340C2×#106C22×C4711yescharacteristic
341C2×#106C22×C4711yescharacteristic
342C2×C4⋊C8C22×C4711yescharacteristic
343C2×#102C22×C4711yescharacteristic
344C2×#102C22×C4711yescharacteristic
345C2×#102C22×C4711yescharacteristic
346C2×#102C22×C4711yescharacteristic
347C2×C22⋊C8C22×C4711yescharacteristic
348C2×C22⋊C8C22×C4711yescharacteristic
349C2×#97C22×C4711yescharacteristic
350#400—21no—
351#397—21no—
352#390—21no—
353#390C22×C4711yescharacteristic
354#389C22×C4711yescharacteristic
355#389C22×C4711yescharacteristic
356#389C22×C4711yescharacteristic
357#388—21no—
358#387—21no—
359#386—21no—
360#384C22×C4711yescharacteristic
361#380C22×C4711yescharacteristic
362#380C22×C4711yescharacteristic
363C4×C4⋊C4C22×C4711yescharacteristic
364C4×C4⋊C4C22×C4711yescharacteristic
365C4×C22⋊C4—21no—
366C4×C22⋊C4C22×C4711yescharacteristic
367#340C22×C4711yescharacteristic
368#340C22×C4711yescharacteristic
369#340C22×C4711yescharacteristic
370#340C22×C4711yescharacteristic
371#340C22×C4711yescharacteristic
372#340C22×C4711yescharacteristic
373#340C22×C4711yescharacteristic
374#340C22×C4711yescharacteristic
375#336C22×C4711yescharacteristic
376#336C22×C4711yescharacteristic
377C47#2705411yescharacteristic, direct product factor, Sylow
378C22×D4—21no—
379C22×D4C23×C4711yescharacteristic
380C23×C4—21no—
381C23×C4C23×C4711yescharacteristic
382C22×C8C23×C4711yescharacteristic
383C22×C8C23×C4711yescharacteristic
384C22×C8C23×C4711yescharacteristic
385C22×C8C23×C4711yescharacteristic
386#126—44no—
387#121—22no—
388#121—22no—
389C4×D4—22no—
390C4×D4—41no—
391#117—22no—
392#117—41no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4C23×C4711yescharacteristic
399C2×C4⋊C4C23×C4711yescharacteristic
400C2×C4⋊C4C23×C4711yescharacteristic
401C2×C4⋊C4C23×C4711yescharacteristic
402C2×C4⋊C4C23×C4711yescharacteristic
403C2×C4⋊C4C23×C4711yescharacteristic
404C2×C4⋊C4C23×C4711yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4D4×C4711yescharacteristic
413C2×C22⋊C4D4×C4711yescharacteristic
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42C23×C4711yescharacteristic
418C2×C42C23×C4711yescharacteristic
419#107—41no—
420#106—22no—
421#106—22no—
422#106—22no—
423C4⋊C8—22no—
424#102—22no—
425#102—22no—
426#102—22no—
427#102—41no—
428C22⋊C8—22no—
429C22⋊C8—41no—
430#97—41no—
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—
441C24C2×D4×C4711yescharacteristic
442C2×D4—24no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—81no—
449C2×D4—81no—
450C2×D4#41×C4712yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—81no—
467C22×C4C2×D4×C4711yescharacteristic
468C22×C4C2×D4×C4711yescharacteristic
469C22×C4C2×D4×C4711yescharacteristic
470C22×C4C2×D4×C4711yescharacteristic
471C22×C4C2×D4×C4711yescharacteristic
472C22×C4C24×C4711yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8#41×C4712yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—81no—
502C4⋊C4#41×C4712yes—
503C4⋊C4#41×C4712yes—
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—41no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—81no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—42no—
518C42—42no—
519C42—81no—
520C23—22no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23C22×D4×C4711yescharacteristic
535D4—22no—
536D4—22no—
537D4—41no—
538D4—41no—
539D4—84no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
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—41no—
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—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—84no—
580C2×C4AGL(1,ℤ/8)×C4712yes—
581C2×C4C2×#41×C4711yescharacteristic
582C2×C4C2×#41×C4711yescharacteristic
583C2×C4C2×#41×C4711yescharacteristic
584C2×C4C2×#41×C4711yescharacteristic
585C2×C4H(5,𝔽2)×C4711yescharacteristic
586C2×C42−1+4×C4711yescharacteristic
587C8—22no—
588C8—41no—
589C8—41no—
590C8—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—42no—
597C22—41no—
598C22—42no—
599C22—41no—
600C22—42no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22C47×#52511yescharacteristic
606C22C47×#54611yescharacteristic
607C22C47×#54711yescharacteristic
608C22C2×AGL(1,ℤ/8)×C4711yescharacteristic
609C22C47×#57511yescharacteristic
610C22C47×#57711yescharacteristic
611C22C47×#57711yescharacteristic
612C4—22no—
613C4—41no—
614C4—42no—
615C4—41no—
616C4—41no—
617C4—42no—
618C4—41no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4C47×#52911yescharacteristic
625C4C47×#53911yescharacteristic
626C4C47×#55311yescharacteristic
627C4C47×#55611yescharacteristic
628C2—81no—
629C2—82no—
630C2C47×#246611yescharacteristic
631C2C47×#311511yescharacteristic
632C2C47×#312311yescharacteristic
633C2C47×#331111yescharacteristic
634C2C47×#333211yescharacteristic
635C2C47×#334511yescharacteristic
636C2C47×#334811yescharacteristic
subgroupG/H
center, socleC23×C47C22×D4
commutator, Frattini, cosocleC22×C4C24×C47
prime radicalC2×#121×C47C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C47 ◁ C4×C22⋊C4×C47 ◁ G

Complements

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

NH ≅ G/Ntype
1. C47×#2496C2semidirect
3. C47×#2071C2semidirect
4. C47×#2067C2semidirect
7. C47×#1948C2semidirect
8. C47×#1944C2semidirect
9. C47×#1924C2semidirect
10. C47×#1919C2semidirect
11. C47×#1919C2semidirect
12. C47×#1864C2semidirect
13. C47×#1854C2semidirect
14. C47×#1825C2semidirect
15. C47×#1762C2semidirect
272. #27054C47direct (N × H)
310. #2496C2×C47semidirect
312. #2071C2×C47semidirect
313. #2067C2×C47semidirect
316. #1948C2×C47semidirect
317. #1944C2×C47semidirect
318. #1924C2×C47semidirect
319. #1919C2×C47semidirect
320. #1919C2×C47semidirect
321. #1864C2×C47semidirect
322. #1854C2×C47semidirect
323. #1825C2×C47semidirect
324. #1762C2×C47semidirect
377. C47#27054direct (N × H)

Maximal subgroups

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

subgroupindex
C47×#24962
C47×#23632
C47×#20712
C47×#20672
C47×#20342
C47×#19482
C47×#19442
C47×#19242
C47×#19192
C47×#18642
C47×#18542
C47×#18252
C47×#17622
#2705447