Finite Group Databasebeta

← C2×#20257532 #27123176

Subgroups — C2×#20257532 #27123176

Order 12032.

6554 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257532C2132yesmaximal, direct product factor
2C2×C47⋊#564C211yesmaximal, characteristic, complemented
3C2×#15689876C214yesmaximal, complemented
4C2×#15689874C214yesmaximal, complemented
5C2×#15689873C212yesmaximal, complemented
6C2×#15689868C214yesmaximal, complemented
7C2×#15689679C211yesmaximal, characteristic, complemented
8C2×#15689675C214yesmaximal, complemented
9C2×#15689654C214yesmaximal, complemented
10C2×#15689652C212yesmaximal, complemented
11C2×#15689650C214yesmaximal, complemented
12C2×C47×#562C211yesmaximal, characteristic, complemented
13C47⋊#564C22116yescomplemented
14#15689876C22164yescomplemented
15#15689874C22164yescomplemented
16#15689873C22132yescomplemented
17#15689868C22164yescomplemented
18#15689679C22116yescomplemented
19#15689675C22164yescomplemented
20C2×#13094023C2214yes—
21C2×#13094023C2212yes—
22C2×#13094022C2214yes—
23#15689654C22164yescomplemented
24#15689652C22132yescomplemented
25#15689650C22164yescomplemented
26C2×#13094019C2214yes—
27C2×#13094017C2214yes—
28C2×#13094017C2214yes—
29C2×#13094017C2214yes—
30C2×#13094017C2214yes—
31C2×#13094016C2212yes—
32C2×#13094016C2211yescharacteristic
33C2×#13094015C2214yes—
34C2×#13094014C2214yes—
35C2×#13094014C2214yes—
36C2×#13094014C2214yes—
37C2×#13094011C2214yes—
38C2×#13094010C2214yes—
39C2×#13094007C2214yes—
40C2×#13094006C2214yes—
41C2×#13094005C2214yes—
42C2×#13094005C2214yes—
43C2×#13094003C2214yes—
44C2×#13094003C2214yes—
45C2×#13094002C2214yes—
46C2×#13094001C2214yes—
47C2×#13094001C2214yes—
48C2×#13094001C2214yes—
49C2×#13094000C2212yes—
50C2×#13093996C2214yes—
51C2×#13093995C2212yes—
52C2×Q8×Dic47C2212yes—
53C2×#13093989C2214yes—
54C2×#13093989C2214yes—
55C2×D4×Dic47C2214yes—
56C2×D4×Dic47C2212yes—
57C2×(C22⋊D47)⋊C4C2214yes—
58C2×#13093982C2212yes—
59C2×#13093980C2214yes—
60C2×#13093980C2214yes—
61C22×C4×Dic47C2214yes—
62C2×C42×D47C2211yescharacteristic
63C47×#562C22116yescomplemented
64C2×#125×C47C2211yescharacteristic
65C2×#124×C47C2212yes—
66C2×#123×C47C2214yes—
67C2×C22⋊Q8×C47C2214yes—
68C2×#121×C47C2214yes—
69C22×Dic94C2314yes—
70C22×Dic94C2314yes—
71C22×C22⋊D47C2314yes—
72C23×Dic47C2314yes—
73C23×Dic47C2314yes—
74C22×C4×D47C2311yescharacteristic
75C22×C4×D47C2312yes—
76#13094023C23132yes—
77#13094023C23116yes—
78#13094022C23132yes—
79#13094019C23132yes—
80#13094017C23132yes—
81#13094017C23132yes—
82#13094017C23132yes—
83#13094017C23132yes—
84#13094016C2318yes—
85#13094016C23116yes—
86#13094015C23132yes—
87#13094014C23132yes—
88#13094014C23132yes—
89#13094014C23132yes—
90#13094011C23132yes—
91#13094010C23132yes—
92#13094007C23132yes—
93#13094006C23132yes—
94#13094005C23132yes—
95#13094005C23132yes—
96#13094003C23132yes—
97#13094003C23132yes—
98#13094002C23132yes—
99#13094001C23132yes—
100#13094001C23132yes—
101#13094001C23132yes—
102#13094000C23116yes—
103#13093996C23132yes—
104#13093995C23116yes—
105Q8×Dic47C23116yes—
106#13093989C23132yes—
107#13093989C23132yes—
108D4×Dic47C23132yes—
109D4×Dic47C23116yes—
110(C22⋊D47)⋊C4C23132yes—
111#13093982C23116yes—
112#13093980C23132yes—
113#13093980C23132yes—
114C2×C4⋊Dic47C2312yes—
115C2×C4⋊Dic47C2312yes—
116C2×C4⋊Dic47C2314yes—
117C2×C4⋊Dic47C2312yes—
118C2×C4⋊Dic47C2314yes—
119C2×Dic47⋊C4C2314yes—
120C2×Dic47⋊C4C2314yes—
121C2×Dic47⋊C4C2314yes—
122C2×Dic47⋊C4C2314yes—
123C2×Dic47⋊C4C2314yes—
124C2×Dic47⋊C4C2314yes—
125C2×Dic47⋊C4C2314yes—
126C2×Dic47⋊C4C2314yes—
127C2×Dic47⋊C4C2314yes—
128C2×C22⋊Dic47C2314yes—
129C2×C22⋊Dic47C2314yes—
130C2×C22⋊Dic47C2314yes—
131C2×C22⋊Dic47C2314yes—
132C2×C22⋊Dic47C2314yes—
133C2×(C2×D47)⋊C4C2314yes—
134C2×(C2×D47)⋊C4C2314yes—
135C2×C4×Dic47C2312yes—
136C2×C4×Dic47C2311yescharacteristic
137C2×C4×Dic47C2314yes—
138C2×C4×Dic47C2314yes—
139C2×C4×Dic47C2314yes—
140C2×C4×Dic47C2314yes—
141C2×C4×Dic47C23132yes—
142C42×D47C2318yes—
143C22×Q8×C47C2312yes—
144C22×D4×C47C2314yes—
145C22×D4×C47C2312yes—
146C23×C4×C47C2314yes—
147#125×C47C2318yes—
148#124×C47C23116yes—
149#123×C47C23132yes—
150C22⋊Q8×C47C23132yes—
151#121×C47C23132yes—
152C2×C4⋊C4×C47C2314yes—
153C2×C4⋊C4×C47C2314yes—
154C2×C4⋊C4×C47C2312yes—
155C2×C22⋊C4×C47C2314yes—
156C2×C22⋊C4×C47C2314yes—
157C2×C22⋊C4×C47C2314yes—
158C2×C42×C47C2311yescharacteristic
159C24×C47C2414yes—
160C23×D47C2411yescharacteristic
161C2×Q8×C47—44no—
162C2×Q8×C47C2418yes—
163C2×D4×C47—48no—
164C2×D4×C47—44no—
165C2×D4×C47C24116yes—
166C2×D4×C47C2418yes—
167C22×C4×C47—48no—
168C22×C4×C47C2411yescharacteristic
169C22×C4×C47C2412yes—
170C22×C4×C47C2414yes—
171C22×C4×C47C2414yes—
172C22×C4×C47C24116yes—
173C2×C22⋊D47—48no—
174C2×C22⋊D47C24116yes—
175C22×Dic47—216no—
176C22×Dic47—48no—
177C22×Dic47C2414yes—
178C22×Dic47C2414yes—
179C22×Dic47C2412yes—
180C22×Dic47C2414yes—
181C22×Dic47C24116yes—
182C22×Dic47C24116yes—
183C22×Dic47C2411yescharacteristic
184C2×C4×D47—24no—
185C2×C4×D47—44no—
186C2×C4×D47C2414yes—
187C2×C4×D47C2418yes—
188C2×Dic94—48no—
189C2×Dic94—48no—
190C2×Dic94C24116yes—
191C2×Dic94C24116yes—
192C4⋊C4×C47C24116yes—
193C4⋊C4×C47C24116yes—
194C4⋊C4×C47C2418yes—
195C22⋊C4×C47C24116yes—
196C22⋊C4×C47C24116yes—
197C22⋊C4×C47C24116yes—
198C42×C47C2414yes—
199C22⋊Dic47C24116yes—
200C22⋊Dic47C24116yes—
201C22⋊Dic47C24116yes—
202C22⋊Dic47C24116yes—
203C22⋊Dic47C24116yes—
204(C2×D47)⋊C4C24116yes—
205(C2×D47)⋊C4C24116yes—
206C4⋊Dic47C2418yes—
207C4⋊Dic47C2418yes—
208C4⋊Dic47C24116yes—
209C4⋊Dic47C2418yes—
210C4⋊Dic47C24116yes—
211Dic47⋊C4C24116yes—
212Dic47⋊C4C24116yes—
213Dic47⋊C4C24116yes—
214Dic47⋊C4C24116yes—
215Dic47⋊C4C24116yes—
216Dic47⋊C4C24116yes—
217Dic47⋊C4C24116yes—
218Dic47⋊C4C24116yes—
219Dic47⋊C4C24116yes—
220C4×Dic47C2418yes—
221C4×Dic47C2414yes—
222C4×Dic47C24116yes—
223C4×Dic47C24116yes—
224C4×Dic47C24116yes—
225C4×Dic47C24116yes—
226C23×C47—28no—
227C23×C47—28no—
228C23×C47—28no—
229C23×C47C2518yes—
230C23×C47C2511yescharacteristic
231C22×D47—22no—
232C22×D47—24no—
233C22×D47C2512yes—
234Q8×C47—48no—
235D4×C47—416no—
236D4×C47—48no—
237C2×C4×C47—22no—
238C2×C4×C47—28no—
239C2×C4×C47—28no—
240C2×C4×C47—24no—
241C2×C4×C47—416no—
242C2×C4×C47C2518yes—
243C2×C4×C47C2518yes—
244C2×C4×C47C2512yes—
245C2×C4×C47C2514yes—
246C22⋊D47—416no—
247C2×Dic47—28no—
248C2×Dic47—24no—
249C2×Dic47—28no—
250C2×Dic47—28no—
251C2×Dic47—232no—
252C2×Dic47—416no—
253C2×Dic472−1+414yes—
254C2×Dic47C2518yes—
255C2×Dic47C2518yes—
256C2×Dic47C2518yes—
257C2×Dic47C2514yes—
258C2×Dic47C2512yes—
259C4×D47—28no—
260C4×D47—48no—
261Dic94—416no—
262Dic94—416no—
263C2×#3531—471nomaximal, Sylow
264C22×C47—28no—
265C22×C47—28no—
266C22×C47—28no—
267C22×C47—416no—
268C22×C47C2×2−1+412yes—
269C22×C47#58514yes—
270C22×C47C2611yescharacteristic
271C2×D47—22no—
272C2×D47—24no—
273C2×D47—44no—
274C4×C47—22no—
275C4×C47—28no—
276C4×C47—28no—
277C4×C47—24no—
278Dic47—28no—
279Dic47—24no—
280Dic47—28no—
281Dic47—28no—
282Dic47C2×2−1+414yes—
283#3531—4732no—
284C2×#564—471no—
285C2×#563—472no—
286C2×#563—474no—
287C2×#563—474no—
288C2×#562—474no—
289C2×#562D4711yescharacteristic, complemented
290C2×#533—471no—
291C2×#533—474no—
292C2×#529—472no—
293C2×#529—474no—
294C2×#529—474no—
295C2×C47—48no—
296C2×C47#353114yescomplemented
297C2×C47C22×2−1+411yescharacteristic
298C2×C47C2×#58512yes—
299D47—42no—
300#564—4716no—
301#563—4732no—
302#563—4764no—
303#563—4764no—
304#562—4764no—
305#562C2×D47116yescomplemented
306#533—4716no—
307#533—4764no—
308C2×C4⋊Q8—474no—
309C2×C4⋊Q8—474no—
310C2×C4⋊Q8—472no—
311#529—4732no—
312#529—4764no—
313#529—4764no—
314C2×#126—474no—
315C2×#126—474no—
316C2×#126—474no—
317C2×#126—474no—
318C2×#126—474no—
319C2×#125—474no—
320C2×#125—474no—
321C2×#125—474no—
322C2×#125—474no—
323C2×#125—472no—
324C2×#125—471no—
325C2×#125C2×D4711yescharacteristic
326C2×#124—474no—
327C2×#124—474no—
328C2×#124C2×D4712yes—
329C2×#123—474no—
330C2×#123—474no—
331C2×#123—474no—
332C2×#123—474no—
333C2×#123C2×D4714yes—
334C2×C22⋊Q8—472no—
335C2×C22⋊Q8—474no—
336C2×C22⋊Q8—474no—
337C2×C22⋊Q8—474no—
338C2×C22⋊Q8—474no—
339C2×C22⋊Q8—474no—
340C2×C22⋊Q8—474no—
341C2×C22⋊Q8C2×D4714yes—
342C2×#121—472no—
343C2×#121—474no—
344C2×#121C2×D4714yes—
345C2×C4×Q8—472no—
346C2×C4×Q8—474no—
347C2×C4×Q8—474no—
348C2×C4×D4—472no—
349C2×C4×D4—474no—
350C2×C4×D4—474no—
351C2×#117—472no—
352C2×#117—474no—
353C2×#117—474no—
354C22×C42—474no—
355C22×C42—471no—
356C47C2×#353111yescharacteristic, complemented, Sylow
357C22×Q8—474no—
358C22×Q8—474no—
359C22×Q8C22×D4712yes—
360C22×D4—474no—
361C22×D4C22×D4712yes—
362C22×D4C22×D4714yes—
363C23×C4—474no—
364C23×C4—474no—
365C23×C4—471no—
366C23×C4—472no—
367C23×C4C22×D4714yes—
368C4⋊Q8—4732no—
369C4⋊Q8—4732no—
370C4⋊Q8—4716no—
371#126—4732no—
372#126—4732no—
373#126—4732no—
374#126—4732no—
375#126—4732no—
376#125—4732no—
377#125—4732no—
378#125—4732no—
379#125—4732no—
380#125—4716no—
381#125—478no—
382#125C22×D4718yes—
383#124—4732no—
384#124—4732no—
385#124C22×D47116yes—
386#123—4732no—
387#123—4732no—
388#123—4732no—
389#123—4732no—
390#123C22×D47132yes—
391C22⋊Q8—4716no—
392C22⋊Q8—4732no—
393C22⋊Q8—4732no—
394C22⋊Q8—4732no—
395C22⋊Q8—4732no—
396C22⋊Q8—4732no—
397C22⋊Q8—4732no—
398C22⋊Q8C22×D47132yes—
399#121—4716no—
400#121—4732no—
401#121C22×D47132yes—
402C4×Q8—4716no—
403C4×Q8—4732no—
404C4×Q8—4732no—
405C4×D4—4716no—
406C4×D4—4732no—
407C4×D4—4732no—
408#117—4716no—
409#117—4732no—
410#117—4732no—
411C2×C4⋊C4—474no—
412C2×C4⋊C4—474no—
413C2×C4⋊C4—472no—
414C2×C4⋊C4—474no—
415C2×C4⋊C4—474no—
416C2×C4⋊C4—474no—
417C2×C4⋊C4—474no—
418C2×C4⋊C4—472no—
419C2×C4⋊C4—474no—
420C2×C4⋊C4—472no—
421C2×C4⋊C4—474no—
422C2×C4⋊C4—474no—
423C2×C4⋊C4—474no—
424C2×C4⋊C4—474no—
425C2×C4⋊C4C22×D4712yes—
426C2×C4⋊C4C22×D4714yes—
427C2×C4⋊C4C22×D4714yes—
428C2×C22⋊C4—474no—
429C2×C22⋊C4—474no—
430C2×C22⋊C4—474no—
431C2×C22⋊C4—474no—
432C2×C22⋊C4—474no—
433C2×C22⋊C4—474no—
434C2×C22⋊C4—474no—
435C2×C22⋊C4C22×D4714yes—
436C2×C22⋊C4C22×D4714yes—
437C2×C22⋊C4C22×D4714yes—
438C2×C42—474no—
439C2×C42—472no—
440C2×C42—474no—
441C2×C42—474no—
442C2×C42—474no—
443C2×C42—471no—
444C2×C42—4732no—
445C2×C42—478no—
446C2×C42C22×D4711yescharacteristic
447C24—471no—
448C24C23×D4714yes—
449C2×Q8—44no—
450C2×Q8—4716no—
451C2×Q8—4716no—
452C2×Q8—1888no—
453C2×Q8—1888no—
454C2×Q8C23×D4718yes—
455C2×D4—48no—
456C2×D4—44no—
457C2×D4—4716no—
458C2×D4—1888no—
459C2×D4C23×D4718yes—
460C2×D4C23×D47116yes—
461C22×C4—48no—
462C22×C4—471no—
463C22×C4—4716no—
464C22×C4—474no—
465C22×C4—474no—
466C22×C4—472no—
467C22×C4—474no—
468C22×C4—4716no—
469C22×C4—474no—
470C22×C4—478no—
471C22×C4—944no—
472C22×C4—9416no—
473C22×C4—1884no—
474C22×C4—1888no—
475C22×C4C23×D4714yes—
476C22×C4C23×D4714yes—
477C22×C4C23×D4711yescharacteristic
478C22×C4C23×D4712yes—
479C22×C4C23×D47116yes—
480C4⋊C4—4716no—
481C4⋊C4—4716no—
482C4⋊C4—4716no—
483C4⋊C4—4716no—
484C4⋊C4—4716no—
485C4⋊C4—4716no—
486C4⋊C4—4716no—
487C4⋊C4—4716no—
488C4⋊C4—4716no—
489C4⋊C4—478no—
490C4⋊C4—4716no—
491C4⋊C4—478no—
492C4⋊C4—478no—
493C4⋊C4—4716no—
494C4⋊C4C23×D4718yes—
495C4⋊C4C23×D47116yes—
496C4⋊C4C23×D47116yes—
497C22⋊C4—4716no—
498C22⋊C4—4716no—
499C22⋊C4—4716no—
500C22⋊C4—4716no—
501C22⋊C4—4716no—
502C22⋊C4—4716no—
503C22⋊C4—4716no—
504C22⋊C4C23×D47116yes—
505C22⋊C4C23×D47116yes—
506C22⋊C4C23×D47116yes—
507C42—4716no—
508C42—478no—
509C42—4716no—
510C42—4716no—
511C42—4716no—
512C42—474no—
513C42C23×D4714yes—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—472no—
518C23—942no—
519C23—944no—
520C23C24×D4718yes—
521C23C24×D4711yescharacteristic
522Q8—48no—
523Q8—18816no—
524Q8—18816no—
525D4—416no—
526D4—48no—
527D4—18816no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—416no—
533C2×C4—474no—
534C2×C4—472no—
535C2×C4—478no—
536C2×C4—478no—
537C2×C4—474no—
538C2×C4—478no—
539C2×C4—948no—
540C2×C4—9432no—
541C2×C4—948no—
542C2×C4—948no—
543C2×C4—944no—
544C2×C4—948no—
545C2×C4—1888no—
546C2×C4—18816no—
547C2×C4C24×D4718yes—
548C2×C4C24×D4718yes—
549C2×C4C24×D4712yes—
550C2×C4C24×D4714yes—
551C22—28no—
552C22—28no—
553C22—28no—
554C22—416no—
555C22—942no—
556C22—944no—
557C22—1884no—
558C222−1+4×D4712yes—
559C22#1568995814yes—
560C22C25×D4711yescharacteristic
561C4—22no—
562C4—28no—
563C4—28no—
564C4—24no—
565C4—474no—
566C4—948no—
567C4—948no—
568C4—944no—
569C4—948no—
570C2—48no—
571C2—1882no—
572C2#2025753214yesdirect product factor
573C2C2×2−1+4×D4711yescharacteristic
574C2C2×#1568995812yes—
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×C47×#562C2
socleC23×C47C25
hypercenterC2×#562D47
hypocenterC47C2×#3531
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #20257532C2direct (N × H)
2. C2×C47⋊#564C2semidirect
3. C2×#15689876C2semidirect
4. C2×#15689874C2semidirect
5. C2×#15689873C2semidirect
6. C2×#15689868C2semidirect
7. C2×#15689679C2semidirect
8. C2×#15689675C2semidirect
9. C2×#15689654C2semidirect
10. C2×#15689652C2semidirect
11. C2×#15689650C2semidirect
12. C2×C47×#562C2semidirect
13. C47⋊#564C22semidirect
14. #15689876C22semidirect
15. #15689874C22semidirect
16. #15689873C22semidirect
17. #15689868C22semidirect
18. #15689679C22semidirect
19. #15689675C22semidirect
23. #15689654C22semidirect
24. #15689652C22semidirect
25. #15689650C22semidirect
63. C47×#562C22semidirect
289. C2×#562D47semidirect
296. C2×C47#3531semidirect
305. #562C2×D47semidirect
356. C47C2×#3531semidirect
572. C2#20257532direct (N × H)

Maximal subgroups

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

subgroupindex
#202575322
C2×C47⋊#5642
C2×#156898762
C2×#156898742
C2×#156898732
C2×#156898682
C2×#156896792
C2×#156896752
C2×#156896542
C2×#156896522
C2×#156896502
C2×C47×#5622
C2×#353147