Finite Group Databasebeta

← C47×#25520 #26065218

Subgroups — C47×#25520 #26065218

Order 12032.

1042 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C47×#3493C211yesmaximal, characteristic
2C47×#3056C212yesmaximal, complemented
3C47×#2499C211yesmaximal, characteristic, complemented
4C47×#2061C211yesmaximal, characteristic, complemented
5C47×#2040C211yesmaximal, characteristic, complemented
6C47×#2028C211yesmaximal, characteristic, complemented
7C47×#2007C211yesmaximal, characteristic, complemented
8C47×#1882C212yesmaximal, complemented
9C47×#1871C212yesmaximal, complemented
10C47×#1859C212yesmaximal, complemented
11C47×#1742C211yesmaximal, characteristic, complemented
12C2×#137×C47C2212yes—
13C22×M16×C47C2212yes—
14C47×#556—22no—
15C47×#555—22no—
16C47×#553—22no—
17C47×#552—22no—
18C47×#544C2211yescharacteristic
19C47×#538—22no—
20C47×#537C2211yescharacteristic
21C47×#529—22no—
22C47×#525C2212yes—
23C2×C22⋊Q8×C47C2211yescharacteristic
24C47×#520C2212yes—
25C2×#117×C47C2211yescharacteristic
26C47×#470—22no—
27C47×#468—22no—
28C47×#467—22no—
29C47×#465—22no—
30C47×#428C2212yes—
31C2×#104×C47C2212yes—
32C2×#104×C47C2212yes—
33C47×#419C2212yes—
34C47×#417C2212yes—
35C47×#397—22no—
36C47×#391C2211yescharacteristic
37C47×#391C2211yescharacteristic
38C47×#387—22no—
39C47×#384C2211yescharacteristic
40C47×#382C2211yescharacteristic
41C4×C22⋊C4×C47C2211yescharacteristic
42C47×#343C2211yescharacteristic
43C47×#343C2211yescharacteristic
44C47×#343C2212yes—
45C47×#339C2212yes—
46C47×#339C2212yes—
47C2×#41×C47C2312yes—
48C22×Q8×C47C2311yescharacteristic
49C23×C4×C47C2311yescharacteristic
50#137×C47—24no—
51#137×C47—42no—
52C2×Q16×C47—22no—
53C2×SD16×C47—22no—
54C2×M16×C47—22no—
55C2×M16×C47—24no—
56C2×M16×C47—42no—
57C2×M16×C47C2312yes—
58C2×M16×C47C2312yes—
59C22×C8×C47—22no—
60C4⋊Q8×C47—22no—
61#126×C47—24no—
62#126×C47—24no—
63#125×C47—22no—
64#125×C47—22no—
65#124×C47—22no—
66#124×C47—24no—
67#123×C47—24no—
68#123×C47—22no—
69C22⋊Q8×C47—22no—
70C22⋊Q8×C47—22no—
71C22⋊Q8×C47—22no—
72C22⋊Q8×C47—22no—
73C22⋊Q8×C47D412yes—
74C22⋊Q8×C47D412yes—
75#121×C47—24no—
76#121×C47—22no—
77#121×C47—22no—
78#121×C47D412yes—
79#121×C47D412yes—
80#120×C47—22no—
81C4×Q8×C47—22no—
82C4×Q8×C47—22no—
83C4×D4×C47—22no—
84C4×D4×C47—22no—
85C4×D4×C47—24no—
86C4×D4×C47—22no—
87C4×D4×C47—22no—
88#117×C47—24no—
89#117×C47—22no—
90#117×C47—22no—
91#117×C47—22no—
92#117×C47C2312yes—
93C2×C4⋊C4×C47—22no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—22no—
96C2×C4⋊C4×C47C2311yescharacteristic
97C2×C4⋊C4×C47C2311yescharacteristic
98C2×C4⋊C4×C47C2311yescharacteristic
99C2×C22⋊C4×C47—21no—
100C2×C22⋊C4×C47—21no—
101C2×C22⋊C4×C47D412yes—
102C2×C42×C47D412yes—
103C2×C42×C47C2311yescharacteristic
104C2×C42×C47C2311yescharacteristic
105#107×C47—22no—
106#106×C47—22no—
107#104×C47—24no—
108#104×C47—42no—
109#103×C47—22no—
110#103×C47—22no—
111#103×C47—22no—
112#102×C47—22no—
113#95×C47—21no—
114#95×C47—22no—
115#95×C47—22no—
116#95×C47—22no—
117C24×C47C2×D411yescharacteristic
118#41×C47—22no—
119#41×C47—22no—
120#41×C47—22no—
121#41×C47—22no—
122C2×Q8×C47—22no—
123C2×Q8×C47—22no—
124C2×Q8×C47—22no—
125C2×Q8×C47—21no—
126C2×Q8×C47—21no—
127C2×Q8×C47C2×D412yes—
128C2×D4×C47—22no—
129C2×D4×C47—22no—
130C2×D4×C47—22no—
131C2×D4×C47—22no—
132C2×D4×C47C2×D412yes—
133C22×C4×C47—22no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—24no—
138C22×C4×C47—22no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2×D412yes—
144C22×C4×C47C2×D412yes—
145C22×C4×C47C2×D412yes—
146C22×C4×C47C2×D412yes—
147C22×C4×C47C2411yescharacteristic
148Q16×C47—44no—
149SD16×C47—44no—
150M16×C47—22no—
151M16×C47—22no—
152M16×C47—22no—
153M16×C47—44no—
154M16×C47#4114yes—
155C2×C8×C47—22no—
156C2×C8×C47—22no—
157C2×C8×C47—44no—
158C4⋊C4×C47—24no—
159C4⋊C4×C47—24no—
160C4⋊C4×C47—22no—
161C4⋊C4×C47—22no—
162C4⋊C4×C47—22no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—22no—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—22no—
167C4⋊C4×C47—44no—
168C4⋊C4×C47—41no—
169C4⋊C4×C47—41no—
170C4⋊C4×C47C2×D412yes—
171C4⋊C4×C47C2×D412yes—
172C22⋊C4×C47—24no—
173C22⋊C4×C47—24no—
174C22⋊C4×C47—24no—
175C22⋊C4×C47—22no—
176C22⋊C4×C47—22no—
177C22⋊C4×C47—22no—
178C22⋊C4×C47—22no—
179C22⋊C4×C47—22no—
180C22⋊C4×C47—42no—
181C22⋊C4×C47—42no—
182C42×C47—24no—
183C42×C47—22no—
184C42×C47—22no—
185C42×C47—22no—
186C42×C47—22no—
187C42×C47—21no—
188C42×C47—21no—
189C23×C47—21no—
190C23×C47—21no—
191C23×C47—22no—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47#12012yes—
195C23×C47C22×D411yescharacteristic
196Q8×C47—22no—
197Q8×C47—22no—
198Q8×C47—42no—
199Q8×C47—42no—
200Q8×C47—82no—
201D4×C47—22no—
202D4×C47—22no—
203D4×C47—42no—
204D4×C47—42no—
205D4×C47—82no—
206D4×C47—82no—
207D4×C47—82no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—42no—
223C2×C4×C47—42no—
224C2×C4×C47—42no—
225C2×C4×C47—42no—
226C2×C4×C47—41no—
227C2×C4×C47—42no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—84no—
235C2×C4×C47#12011yescharacteristic
236C2×C4×C47#12011yescharacteristic
237C2×C4×C47C22×D412yes—
238C2×C4×C47C2×#4112yes—
239C2×C4×C47H(5,𝔽2)11yescharacteristic
240C2×C4×C472−1+411yescharacteristic
241C8×C47—24no—
242C8×C47—42no—
243#25520C4711yesmaximal, characteristic, direct product factor, Sylow
244C22×C47—21no—
245C22×C47—21no—
246C22×C47—22no—
247C22×C47—22no—
248C22×C47—22no—
249C22×C47—42no—
250C22×C47—42no—
251C22×C47—42no—
252C22×C47—81no—
253C22×C47—81no—
254C22×C47C2×#12011yescharacteristic
255C22×C47#53211yescharacteristic
256C22×C47#53711yescharacteristic
257C4×C47—21no—
258C4×C47—21no—
259C4×C47—42no—
260C4×C47—42no—
261C4×C47—42no—
262C4×C47—42no—
263C4×C47—42no—
264C4×C47—42no—
265C4×C47—81no—
266C4×C47—81no—
267C4×C47—81no—
268C4×C47#54612yes—
269C4×C47#54712yes—
270#3493C2×C4711yescharacteristic
271#3056C2×C4712yescomplemented
272#2499C2×C4711yescharacteristic, complemented
273#2061C2×C4711yescharacteristic, complemented
274#2040C2×C4711yescharacteristic, complemented
275#2028C2×C4711yescharacteristic, complemented
276#2007C2×C4711yescharacteristic, complemented
277#1882C2×C4712yescomplemented
278#1871C2×C4712yescomplemented
279#1859C2×C4712yescomplemented
280#1742C2×C4711yescharacteristic, complemented
281C2×C47—21no—
282C2×C47—21no—
283C2×C47—42no—
284C2×C47—82no—
285C2×C47#243011yescharacteristic
286C2×C47#301811yescharacteristic
287C2×C47#302211yescharacteristic
288C2×#137C22×C4712yes—
289C22×M16C22×C4712yes—
290#556—22no—
291#555—22no—
292#553—22no—
293#552—22no—
294#544C22×C4711yescharacteristic
295#538—22no—
296#537C22×C4711yescharacteristic
297#529—22no—
298#525C22×C4712yes—
299C2×C22⋊Q8C22×C4711yescharacteristic
300#520C22×C4712yes—
301C2×#117C22×C4711yescharacteristic
302#470—22no—
303#468—22no—
304#467—22no—
305#465—22no—
306#428C22×C4712yes—
307C2×#104C22×C4712yes—
308C2×#104C22×C4712yes—
309#419C22×C4712yes—
310#417C22×C4712yes—
311#397—22no—
312#391C22×C4711yescharacteristic
313#391C22×C4711yescharacteristic
314#387—22no—
315#384C22×C4711yescharacteristic
316#382C22×C4711yescharacteristic
317C4×C22⋊C4C22×C4711yescharacteristic
318#343C22×C4711yescharacteristic
319#343C22×C4712yes—
320#343C22×C4711yescharacteristic
321#339C22×C4712yes—
322#339C22×C4712yes—
323C47#2552011yescharacteristic, direct product factor, Sylow
324C2×#41C23×C4712yes—
325C22×Q8C23×C4711yescharacteristic
326C23×C4C23×C4711yescharacteristic
327#137—24no—
328#137—42no—
329C2×Q16—22no—
330C2×SD16—22no—
331C2×M16—22no—
332C2×M16—24no—
333C2×M16—42no—
334C2×M16C23×C4712yes—
335C2×M16C23×C4712yes—
336C22×C8—22no—
337C4⋊Q8—22no—
338#126—24no—
339#126—24no—
340#125—22no—
341#125—22no—
342#124—22no—
343#124—24no—
344#123—24no—
345#123—22no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8D4×C4712yes—
351C22⋊Q8D4×C4712yes—
352#121—24no—
353#121—22no—
354#121—22no—
355#121D4×C4712yes—
356#121D4×C4712yes—
357#120—22no—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—24no—
363C4×D4—22no—
364C4×D4—22no—
365#117—24no—
366#117—22no—
367#117—22no—
368#117—22no—
369#117C23×C4712yes—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4C23×C4711yescharacteristic
374C2×C4⋊C4C23×C4711yescharacteristic
375C2×C4⋊C4C23×C4711yescharacteristic
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4D4×C4712yes—
379C2×C42D4×C4712yes—
380C2×C42C23×C4711yescharacteristic
381C2×C42C23×C4711yescharacteristic
382#107—22no—
383#106—22no—
384#104—24no—
385#104—42no—
386#103—22no—
387#103—22no—
388#103—22no—
389#102—22no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394C24C2×D4×C4711yescharacteristic
395#41—22no—
396#41—22no—
397#41—22no—
398#41—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8C2×D4×C4712yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4C2×D4×C4712yes—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4C2×D4×C4711yescharacteristic
420C22×C4C2×D4×C4712yes—
421C22×C4C2×D4×C4712yes—
422C22×C4C2×D4×C4712yes—
423C22×C4C2×D4×C4712yes—
424C22×C4C24×C4711yescharacteristic
425Q16—44no—
426SD16—44no—
427M16—22no—
428M16—22no—
429M16—22no—
430M16—44no—
431M16#41×C4714yes—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—44no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—44no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4C2×D4×C4712yes—
448C4⋊C4C2×D4×C4712yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C42—24no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—21no—
465C42—21no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23#120×C4712yes—
472C23C22×D4×C4711yescharacteristic
473Q8—22no—
474Q8—22no—
475Q8—42no—
476Q8—42no—
477Q8—82no—
478D4—22no—
479D4—22no—
480D4—42no—
481D4—42no—
482D4—82no—
483D4—82no—
484D4—82no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—84no—
512C2×C4#120×C4711yescharacteristic
513C2×C4#120×C4711yescharacteristic
514C2×C4C22×D4×C4712yes—
515C2×C4C2×#41×C4712yes—
516C2×C4H(5,𝔽2)×C4711yescharacteristic
517C2×C42−1+4×C4711yescharacteristic
518C8—24no—
519C8—42no—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—81no—
529C22—81no—
530C22C2×#120×C4711yescharacteristic
531C22C47×#53211yescharacteristic
532C22C47×#53711yescharacteristic
533C4—21no—
534C4—21no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—81no—
542C4—81no—
543C4—81no—
544C4C47×#54612yes—
545C4C47×#54712yes—
546C2—21no—
547C2—21no—
548C2—42no—
549C2—82no—
550C2C47×#243011yescharacteristic
551C2C47×#301811yescharacteristic
552C2C47×#302211yescharacteristic
subgroupG/H
center, socleC22×C47C2×#120
commutator, Frattini, cosocleC22×C4C24×C47
prime radicalC47×#3493C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C47 ◁ C23×C4×C47 ◁ G

Complements

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

NH ≅ G/Ntype
2. C47×#3056C2semidirect
3. C47×#2499C2semidirect
4. C47×#2061C2semidirect
5. C47×#2040C2semidirect
6. C47×#2028C2semidirect
7. C47×#2007C2semidirect
8. C47×#1882C2semidirect
9. C47×#1871C2semidirect
10. C47×#1859C2semidirect
11. C47×#1742C2semidirect
243. #25520C47direct (N × H)
271. #3056C2×C47semidirect
272. #2499C2×C47semidirect
273. #2061C2×C47semidirect
274. #2040C2×C47semidirect
275. #2028C2×C47semidirect
276. #2007C2×C47semidirect
277. #1882C2×C47semidirect
278. #1871C2×C47semidirect
279. #1859C2×C47semidirect
280. #1742C2×C47semidirect
323. C47#25520direct (N × H)

Maximal subgroups

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

subgroupindex
C47×#34932
C47×#30562
C47×#24992
C47×#20612
C47×#20402
C47×#20282
C47×#20072
C47×#18822
C47×#18712
C47×#18592
C47×#17422
#2552047