Finite Group Databasebeta

← C2×#20253371 #27084132

Subgroups — C2×#20253371 #27084132

Order 12032.

2234 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689704C211yesmaximal, characteristic
2C2×#15689604C211yesmaximal, characteristic, complemented
3#20253371C2116yesmaximal, direct product factor
4C2×#15689460C212yesmaximal, complemented
5C2×#15689466C212yesmaximal, complemented
6C2×#15689450C212yesmaximal, complemented
7C2×#15689438C212yesmaximal, complemented
8C2×#15689163C211yesmaximal, characteristic, complemented
9C2×#15689122C211yesmaximal, characteristic, complemented
10C2×#15689120C211yesmaximal, characteristic, complemented
11C2×#15689089C211yesmaximal, characteristic, complemented
12C22×#107×C47C211yesmaximal, characteristic, complemented
13C22×#10625350C2211yescharacteristic
14C22×#10625350C2211yescharacteristic
15C22×#10625329C2211yescharacteristic
16#15689704—24no—
17#15689704C2218yes—
18C2×#13094004—21no—
19C2×#13094004C2212yes—
20C2×#13093999—21no—
21C2×#13093999C2212yes—
22C2×#13093998—22no—
23#15689604—24no—
24#15689604C2218yescomplemented
25C2×#13093989—21no—
26C2×#13093986—21no—
27C2×C4×C22⋊D47—22no—
28C2×#13093983C2211yescharacteristic
29C2×#13093982C2212yes—
30C2×C4⋊C4×D47C2212yes—
31#15689466C22116yescomplemented
32#15689460C22116yescomplemented
33#15689450C22116yescomplemented
34#15689438C22116yescomplemented
35#15689163C2218yescomplemented
36C2×#13093957C2212yes—
37C2×#13093955C2212yes—
38#15689122C2218yescomplemented
39#15689120C2218yescomplemented
40C2×#13093949C2211yescharacteristic
41C2×#13093949C2211yescharacteristic
42C2×#13093947C2212yes—
43C2×#13093946C2211yescharacteristic
44C2×#13093946C2211yescharacteristic
45C2×#13093944C2212yes—
46#15689089C2218yescomplemented
47C2×(C2×D47)⋊C8C2212yes—
48C2×(C2×D47)⋊C8C2212yes—
49C23×C8×C47C2211yescharacteristic
50C22×C4⋊C4×C47C2211yescharacteristic
51C22×C4⋊C4×C47C2211yescharacteristic
52C2×#107×C47C2212yes—
53C2×#107×C47C2212yes—
54C2×#107×C47C2218yescomplemented
55C2×#10625350—24no—
56C2×#10625350—24no—
57C2×#10625350—22no—
58C2×#10625350—24no—
59C2×#10625350—24no—
60C2×#10625350D414yes—
61C2×#10625350C2314yes—
62C2×#10625350C2314yes—
63C22×Dic94D411yescharacteristic
64C22×Dic94C2311yescharacteristic
65C22×D188—22no—
66C22×D188D411yescharacteristic
67C22×D188C2311yescharacteristic
68C22×C22⋊D47—21no—
69C22×C22⋊D47D412yes—
70C22×C4×D47—21no—
71C22×C4×D47—22no—
72C22×C4×D47C2312yes—
73C2×#10625329—24no—
74C2×#10625329C2314yes—
75C22×C47⋊C8C2312yes—
76#13094004—24no—
77#13094004—28no—
78#13094004C2318yes—
79#13093999—28no—
80#13093999—24no—
81#13093999C2318yes—
82#13093998—216no—
83#13093989—28no—
84#13093986—28no—
85C4×C22⋊D47—216no—
86#13093983—22no—
87#13093983C2314yes—
88#13093982—24no—
89#13093982C2318yes—
90C2×C4⋊Dic47—22no—
91C2×C4⋊Dic47C2311yescharacteristic
92C2×C4⋊Dic47C2311yescharacteristic
93C2×Dic47⋊C4—22no—
94C4⋊C4×D47—24no—
95C4⋊C4×D47C2318yes—
96C2×C22⋊Dic47—21no—
97C2×(C2×D47)⋊C4—22no—
98C2×(C2×D47)⋊C4—22no—
99C2×(C2×D47)⋊C4—22no—
100C2×C4×Dic47—21no—
101C2×C4×Dic47—22no—
102#13093957C2318yes—
103#13093955C2318yes—
104#13093949C2314yes—
105#13093949C2314yes—
106#13093947C2318yes—
107#13093946C2314yes—
108#13093946C2314yes—
109#13093944C2318yes—
110(C2×D47)⋊C8C2318yes—
111(C2×D47)⋊C8C2318yes—
112C23×C4×C47—21no—
113C23×C4×C47—21no—
114C23×C4×C47C2311yescharacteristic
115C22×C8×C47—24no—
116C22×C8×C47C2311yescharacteristic
117C22×C8×C47C2311yescharacteristic
118C22×C8×C47C2314yes—
119C2×C4⋊C4×C47—21no—
120C2×C4⋊C4×C47—22no—
121C2×C4⋊C4×C47—21no—
122C2×C4⋊C4×C47—22no—
123C2×C4⋊C4×C47C2312yes—
124C2×C4⋊C4×C47C2314yes—
125C2×C4⋊C4×C47C2312yes—
126C2×C4⋊C4×C47C2314yes—
127#107×C47C2318yes—
128#107×C47C2318yes—
129C24×C47C2×D411yescharacteristic
130C23×D47—21no—
131C23×D47C2×D412yes—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—24no—
135C22×C4×C47—24no—
136C22×C4×C47—82no—
137C22×C4×C47—82no—
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D412yes—
140C22×C4×C47#4114yes—
141C22×C4×C47#4114yes—
142C22×C4×C47C2411yescharacteristic
143C22×C4×C47C2412yes—
144C2×C22⋊D47—28no—
145C2×C22⋊D47—24no—
146C2×C22⋊D47—44no—
147C2×C22⋊D47C2×D418yes—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47C2×D412yes—
152#10625350—216no—
153#10625350—28no—
154#10625350—28no—
155#10625350—28no—
156#10625350—28no—
157#10625350—48no—
158C2×D188—24no—
159C2×D188—22no—
160C2×D188—22no—
161C2×D188—21no—
162C2×D188—28no—
163C2×D188—84no—
164C2×D188C2×D414yes—
165C2×D188C2412yes—
166C2×C4×D47—24no—
167C2×C4×D47—24no—
168C2×C4×D47—24no—
169C2×C4×D47—28no—
170C2×C4×D47—44no—
171C2×C4×D47—84no—
172C2×C4×D47C2×D414yes—
173C2×C4×D47C2414yes—
174C2×Dic94—24no—
175C2×Dic94—22no—
176C2×Dic94—22no—
177C2×Dic94—21no—
178C2×Dic94C2×D414yes—
179C2×Dic94C2412yes—
180C2×C8×C47—22no—
181C2×C8×C47—22no—
182C2×C8×C47—28no—
183C2×C8×C47C2412yes—
184C2×C8×C47C2412yes—
185C4⋊C4×C47—22no—
186C4⋊C4×C47—24no—
187C4⋊C4×C47—24no—
188C4⋊C4×C47—22no—
189C4⋊C4×C47C2414yes—
190C4⋊C4×C47C2414yes—
191C22⋊Dic47—24no—
192(C2×D47)⋊C4—28no—
193(C2×D47)⋊C4—28no—
194(C2×D47)⋊C4—28no—
195C4⋊Dic47—21no—
196C4⋊Dic47—21no—
197C4⋊Dic47—28no—
198C4⋊Dic47C2412yes—
199C4⋊Dic47C2412yes—
200Dic47⋊C4—28no—
201C4×Dic47—24no—
202C4×Dic47—28no—
203#10625329—28no—
204C2×C47⋊C8—24no—
205C2×C47⋊C8C2414yes—
206C23×C47—22no—
207C23×C47—22no—
208C23×C47#12314yes—
209C23×C47C22×D412yes—
210C23×C47C22×D411yescharacteristic
211C22×D47—24no—
212C22×D47—24no—
213C22×D47—24no—
214C22×D47—22no—
215C22×D47—42no—
216C22×D47—42no—
217C22×D47—42no—
218C22×D47C22×D414yes—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—44no—
222C2×C4×C47—44no—
223C2×C4×C47—84no—
224C2×C4×C47—84no—
225C2×C4×C47#12314yes—
226C2×C4×C47#12318yes—
227C2×C4×C47C22×D412yes—
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C2×#4112yes—
230C2×C4×C47C2×#4114yes—
231C2×C4×C47C2×#4112yes—
232C2×C4×C47C2×#4114yes—
233C2×C4×C47C2511yescharacteristic
234C22⋊D47—216no—
235C22⋊D47—48no—
236C2×Dic47—24no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—22no—
240C2×Dic47—42no—
241C2×Dic47—42no—
242C2×Dic47—42no—
243C2×Dic47C22×D414yes—
244D188—28no—
245D188—22no—
246D188—22no—
247D188—42no—
248D188—88no—
249C4×D47—28no—
250C4×D47—24no—
251C4×D47—24no—
252C4×D47—48no—
253C4×D47—88no—
254Dic94—28no—
255Dic94—22no—
256Dic94—22no—
257Dic94—42no—
258C8×C47—22no—
259C8×C47—22no—
260C47⋊C8—24no—
261C2×#3098—471nomaximal, Sylow
262C22×C47—28no—
263C22×C47—22no—
264C22×C47—22no—
265C22×C47C2×#12314yes—
266C22×C47C2×#12312yes—
267C22×C47#57612yes—
268C22×C47#57812yes—
269C22×C47C23×D411yescharacteristic
270C2×D47—24no—
271C2×D47—24no—
272C2×D47—24no—
273C2×D47—48no—
274C2×D47—42no—
275C2×D47—42no—
276C2×D47—42no—
277C2×D47—84no—
278C4×C47—44no—
279C4×C47—44no—
280C4×C47C2×#12314yes—
281C4×C47C2×#12312yes—
282C4×C47C22×#4111yescharacteristic
283C4×C47C22×#4111yescharacteristic
284Dic47—24no—
285Dic47—42no—
286Dic47—42no—
287Dic47—42no—
288C2×#537—471no—
289C2×#520—471no—
290#3098—4716no—
291C2×#484—472no—
292C2×#485—472no—
293C2×#482—472no—
294C2×#480—472no—
295C2×#428—471no—
296C22×#107D4711yescharacteristic, complemented
297C2×#417—471no—
298C2×#416—471no—
299C2×#408—471no—
300C2×C47—24no—
301C2×C47#309814yescomplemented
302C2×C47C22×#12311yescharacteristic
303C2×C47C2×#57611yescharacteristic
304C2×C47C2×#57811yescharacteristic
305D47—44no—
306D47—82no—
307C22×#41—471no—
308C22×#41—471no—
309C22×M16—471no—
310C23×C8C2×D4711yescharacteristic
311#537—478no—
312#537—944no—
313C2×C22⋊Q8—472no—
314C2×C22⋊Q8—941no—
315C2×#121—472no—
316C2×#121—941no—
317C2×#121—942no—
318#520—478no—
319#520—944no—
320C2×C4×Q8—941no—
321C2×C4×D4—941no—
322C2×C4×D4—942no—
323C2×#117—471no—
324C2×#117—472no—
325C22×C4⋊C4—472no—
326C22×C4⋊C4C2×D4711yescharacteristic
327C22×C4⋊C4C2×D4711yescharacteristic
328#485—4716no—
329#484—4716no—
330#482—4716no—
331#480—4716no—
332#428—478no—
333C2×#107—472no—
334C2×#107C2×D4712yes—
335C2×#107C2×D4712yes—
336C2×#107C2×D4718yescomplemented
337C2×#106—472no—
338#417—478no—
339#416—478no—
340C2×#103—471no—
341C2×#103—471no—
342C2×#103—472no—
343C2×#102—471no—
344C2×#102—471no—
345C2×#102—472no—
346#408—478no—
347C2×C22⋊C8—472no—
348C2×C22⋊C8—472no—
349C47C2×#309811yescharacteristic, complemented, Sylow
350C2×#41—474no—
351C2×#41—474no—
352C2×#41—474no—
353C2×#41—944no—
354C2×#41—942no—
355C2×#41—944no—
356C2×#41—944no—
357C2×#41—944no—
358C22×Q8—471no—
359C22×Q8—471no—
360C22×D4—472no—
361C22×D4—471no—
362C22×D4—471no—
363C22×D4—942no—
364C22×D4—941no—
365C23×C4—21no—
366C23×C4—21no—
367C23×C4—472no—
368C23×C4—942no—
369C23×C4—941no—
370C23×C4C22×D4711yescharacteristic
371C2×M16—474no—
372C2×M16—944no—
373C22×C8—24no—
374C22×C8—472no—
375C22×C8C22×D4711yescharacteristic
376C22×C8C22×D4711yescharacteristic
377C22×C8C22×D4714yes—
378C22⋊Q8—478no—
379C22⋊Q8—944no—
380C22⋊Q8—948no—
381#121—478no—
382#121—944no—
383#121—948no—
384#121—9416no—
385C4×Q8—948no—
386C4×D4—948no—
387C4×D4—9416no—
388#117—474no—
389#117—478no—
390#117—944no—
391#117—942no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—471no—
397C2×C4⋊C4—471no—
398C2×C4⋊C4—478no—
399C2×C4⋊C4—944no—
400C2×C4⋊C4—942no—
401C2×C4⋊C4—942no—
402C2×C4⋊C4C22×D4712yes—
403C2×C4⋊C4C22×D4714yes—
404C2×C4⋊C4C22×D4712yes—
405C2×C4⋊C4C22×D4714yes—
406C2×C22⋊C4—942no—
407C2×C22⋊C4—942no—
408C2×C22⋊C4—942no—
409C2×C22⋊C4—941no—
410C2×C42—942no—
411C2×C42—941no—
412#107—478no—
413#107C22×D4718yes—
414#107C22×D4718yes—
415#106—478no—
416#103—474no—
417#103—474no—
418#103—478no—
419#102—474no—
420#102—474no—
421#102—478no—
422C22⋊C8—478no—
423C22⋊C8—478no—
424C24—472no—
425C24—941no—
426C24D4×D4711yescharacteristic
427#41—9416no—
428#41—948no—
429#41—948no—
430#41—948no—
431#41—948no—
432#41—1888no—
433C2×Q8—474no—
434C2×Q8—472no—
435C2×Q8—944no—
436C2×Q8—941no—
437C2×Q8—942no—
438C2×Q8—942no—
439C2×D4—478no—
440C2×D4—474no—
441C2×D4—472no—
442C2×D4—948no—
443C2×D4—948no—
444C2×D4—944no—
445C2×D4—944no—
446C2×D4—941no—
447C2×D4—942no—
448C2×D4—942no—
449C2×D4—3764no—
450C2×D4—1884no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—472no—
458C22×C4—474no—
459C22×C4—474no—
460C22×C4—948no—
461C22×C4—944no—
462C22×C4—944no—
463C22×C4—941no—
464C22×C4—941no—
465C22×C4—941no—
466C22×C4—944no—
467C22×C4—3764no—
468C22×C4—1884no—
469C22×C4#1062535014yes—
470C22×C4D4×D4712yes—
471C22×C4D4×D4711yescharacteristic
472C22×C4#1062535414yes—
473C22×C4C23×D4712yes—
474C22×C4C23×D4711yescharacteristic
475M16—948no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—28no—
479C2×C8—474no—
480C2×C8—944no—
481C2×C8C23×D4712yes—
482C2×C8C23×D4712yes—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—472no—
488C4⋊C4—472no—
489C4⋊C4—948no—
490C4⋊C4—948no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4C23×D4714yes—
494C4⋊C4C23×D4714yes—
495C22⋊C4—948no—
496C22⋊C4—948no—
497C22⋊C4—948no—
498C22⋊C4—944no—
499C42—948no—
500C42—944no—
501C23—22no—
502C23—22no—
503C23—474no—
504C23—944no—
505C23—944no—
506C23—944no—
507C23—942no—
508C23—1882no—
509C23—1882no—
510C23—1882no—
511C23#1309400914yes—
512C23C2×D4×D4712yes—
513C23C2×D4×D4711yescharacteristic
514Q8—948no—
515Q8—942no—
516Q8—942no—
517Q8—1882no—
518D4—9416no—
519D4—948no—
520D4—942no—
521D4—942no—
522D4—3768no—
523D4—1888no—
524D4—1882no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4—474no—
532C2×C4—944no—
533C2×C4—948no—
534C2×C4—944no—
535C2×C4—944no—
536C2×C4—942no—
537C2×C4—942no—
538C2×C4—942no—
539C2×C4—3768no—
540C2×C4—1882no—
541C2×C4—1882no—
542C2×C4—1882no—
543C2×C4—1888no—
544C2×C4#1309400914yes—
545C2×C4#1309400918yes—
546C2×C4C2×D4×D4712yes—
547C2×C4C2×D4×D4711yescharacteristic
548C2×C4C2×#1062535012yes—
549C2×C4C2×#1062535014yes—
550C2×C4C2×#1062535412yes—
551C2×C4C2×#1062535414yes—
552C2×C4C24×D4711yescharacteristic
553C8—22no—
554C8—22no—
555C8—944no—
556C22—22no—
557C22—22no—
558C22—28no—
559C22—944no—
560C22—944no—
561C22—944no—
562C22—3764no—
563C22—1882no—
564C22—1882no—
565C22—1882no—
566C22—1888no—
567C22C2×#1309400914yes—
568C22C2×#1309400912yes—
569C22#1568992412yes—
570C22#1568993612yes—
571C22C22×D4×D4711yescharacteristic
572C4—44no—
573C4—44no—
574C4—944no—
575C4—1882no—
576C4—1882no—
577C4—1882no—
578C4C2×#1309400912yes—
579C4C2×#1309400914yes—
580C4C22×#1062535011yescharacteristic
581C4C22×#1062535411yescharacteristic
582C2—24no—
583C2—3762no—
584C2—1884no—
585C2#2025337114yesdirect product factor
586C2C22×#1309400911yescharacteristic
587C2C2×#1568992411yescharacteristic
588C2C2×#1568993611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC22×#107×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689704C2
hypercenterC22×#107D47
hypocenterC47C2×#3098
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#107

Complements

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

NH ≅ G/Ntype
2. C2×#15689604C2semidirect
3. #20253371C2direct (N × H)
4. C2×#15689460C2semidirect
5. C2×#15689466C2semidirect
6. C2×#15689450C2semidirect
7. C2×#15689438C2semidirect
8. C2×#15689163C2semidirect
9. C2×#15689122C2semidirect
10. C2×#15689120C2semidirect
11. C2×#15689089C2semidirect
12. C22×#107×C47C2semidirect
24. #15689604C22semidirect
31. #15689466C22semidirect
32. #15689460C22semidirect
33. #15689450C22semidirect
34. #15689438C22semidirect
35. #15689163C22semidirect
38. #15689122C22semidirect
39. #15689120C22semidirect
46. #15689089C22semidirect
54. C2×#107×C47C22semidirect
296. C22×#107D47semidirect
301. C2×C47#3098semidirect
336. C2×#107C2×D47semidirect
349. C47C2×#3098semidirect
585. C2#20253371direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156897042
C2×#156896042
#202533712
C2×#156894602
C2×#156894662
C2×#156894502
C2×#156894382
C2×#156891632
C2×#156891222
C2×#156891202
C2×#156890892
C22×#107×C472
C2×#309847