Finite Group Databasebeta

← C2×#20245059 #26369066

Subgroups — C2×#20245059 #26369066

Order 12032.

1422 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689687C211yesmaximal, characteristic
2C2×#15689097C211yesmaximal, characteristic, complemented
3#20245059C218yesmaximal, direct product factor
4C2×#15688915C214yesmaximal, complemented
5C2×C47×#410C211yesmaximal, characteristic, complemented
6C23×D188C411yescharacteristic, complemented
7#15689687—22no—
8#15689687—22no—
9#15689687C414yescomplemented
10#15689687C2214yes—
11C2×#13094020C2212yes—
12C2×#13094013C412yescomplemented
13C2×#13093998—21no—
14C2×#13093998—21no—
15C2×#13093992—21no—
16C2×#13093992—21no—
17C2×#13093983—21no—
18C2×#13093983C411yescharacteristic, complemented
19#15689097—22no—
20#15689097C2214yescomplemented
21C2×#13093939—21no—
22C2×#13093939C2212yescomplemented
23#15688915C22116yescomplemented
24C22×#41×C47C2211yescharacteristic
25C2×#117×C47—21no—
26C47×#410—22no—
27C47×#410C2214yescomplemented
28C2×#99×C47—21no—
29C2×#99×C47C2212yescomplemented
30C24×D47—21no—
31C22×D188—21no—
32C22×D188—22no—
33C22×D188—42no—
34C22×D188—41no—
35C22×D188—41no—
36C22×D188C2×C412yes—
37C22×D188C2×C414yescomplemented
38C22×C22⋊D47—22no—
39C22×C22⋊D47—22no—
40#13094020—22no—
41#13094020—22no—
42#13094020C2×C414yes—
43#13094020C2314yes—
44#13094013—24no—
45#13094013C2×C418yescomplemented
46#13093998—24no—
47#13093998—24no—
48#13093998—24no—
49#13093998—24no—
50#13093992—24no—
51#13093992—24no—
52#13093992—24no—
53#13093992—24no—
54#13093983—22no—
55#13093983—24no—
56#13093983—42no—
57#13093983C2×C414yescomplemented
58C2×C4⋊Dic47—21no—
59C2×C4⋊Dic47—41no—
60C2×C22⋊Dic47—21no—
61C2×C22⋊Dic47—22no—
62C2×(C2×D47)⋊C4—22no—
63C2×(C2×D47)⋊C4—22no—
64C2×C4×Dic47—41no—
65C2×C4×Dic47C2×C412yes—
66#13093939—24no—
67#13093939—22no—
68#13093939C2314yescomplemented
69C2×#41×C47—41no—
70C2×#41×C47—41no—
71C2×#41×C47—42no—
72C2×#41×C47C2×C412yes—
73C2×#41×C47D412yes—
74C2×#41×C47D412yes—
75C2×#41×C47C2312yes—
76C22×Q8×C47C2×C411yescharacteristic
77C22×D4×C47D411yescharacteristic
78C22×D4×C47D411yescharacteristic
79C22×D4×C47C2311yescharacteristic
80C23×C4×C47C2×C411yescharacteristic
81C23×C4×C47D411yescharacteristic
82C23×C4×C47D411yescharacteristic
83#117×C47—24no—
84#117×C47—42no—
85C2×C4⋊C4×C47—41no—
86C2×C22⋊C4×C47—22no—
87C2×C42×C47—41no—
88#99×C47—24no—
89#99×C47—22no—
90#99×C47C2314yescomplemented
91C24×C47C22⋊C411yescharacteristic
92C24×C47C2×D411yescharacteristic
93C24×C47C2×D411yescharacteristic
94C23×D47—22no—
95C23×D47—24no—
96C23×D47—42no—
97C23×D47—42no—
98C23×D47—84no—
99#41×C47—42no—
100#41×C47—42no—
101#41×C47—42no—
102#41×C47—42no—
103#41×C47—42no—
104#41×C47—42no—
105#41×C47—42no—
106#41×C47—42no—
107C2×Q8×C47—41no—
108C2×Q8×C47—41no—
109C2×Q8×C47C22⋊C412yes—
110C2×Q8×C47C22×C412yes—
111C2×D4×C47—41no—
112C2×D4×C47—41no—
113C2×D4×C47—41no—
114C2×D4×C47—41no—
115C2×D4×C47—41no—
116C2×D4×C47—41no—
117C2×D4×C47C22⋊C412yes—
118C2×D4×C47C22⋊C412yes—
119C2×D4×C47C22×C411yescharacteristic
120C2×D4×C47C2×D412yes—
121C2×D4×C47C2×D411yescharacteristic
122C2×D4×C47C2×D412yes—
123C2×D4×C47C2×D411yescharacteristic
124C2×D4×C47C2411yescharacteristic
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—22no—
134C22×C4×C47—42no—
135C22×C4×C47C22⋊C412yes—
136C22×C4×C47C22⋊C411yescharacteristic
137C22×C4×C47C22⋊C411yescharacteristic
138C22×C4×C47C22⋊C412yes—
139C22×C4×C47C22⋊C412yes—
140C22×C4×C47C22⋊C411yescharacteristic
141C22×C4×C47C22×C411yescharacteristic
142C22×C4×C47C22×C412yes—
143C22×C4×C47C2×D412yes—
144C22×C4×C47C2×D412yes—
145C2×C22⋊D47—24no—
146C2×C22⋊D47—24no—
147C2×C22⋊D47—24no—
148C2×C22⋊D47—24no—
149C2×C22⋊D47—84no—
150C2×C22⋊D47—84no—
151C22×Dic47—22no—
152C22×Dic47—42no—
153C2×D188—24no—
154C2×D188—22no—
155C2×D188—44no—
156C2×D188—44no—
157C2×D188—42no—
158C2×D188—42no—
159C2×D188—42no—
160C2×D188—41no—
161C2×D188—41no—
162C2×D188—44no—
163C2×D188—44no—
164C2×D188C22×C414yes—
165C4⋊C4×C47—44no—
166C22⋊C4×C47—24no—
167C22⋊C4×C47—42no—
168C42×C47—44no—
169C22⋊Dic47—24no—
170C22⋊Dic47—24no—
171C22⋊Dic47—42no—
172(C2×D47)⋊C4—28no—
173(C2×D47)⋊C4—28no—
174C4⋊Dic47—24no—
175C4⋊Dic47—44no—
176C4×Dic47—22no—
177C4×Dic47—44no—
178C4×Dic47C22×C414yes—
179C23×C47—21no—
180C23×C47—21no—
181C23×C47—21no—
182C23×C47—21no—
183C23×C47—21no—
184C23×C47—21no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47—42no—
188C23×C47C2×C22⋊C411yescharacteristic
189C23×C47C2×C22⋊C412yes—
190C23×C47C2×C22⋊C411yescharacteristic
191C23×C47C2×C22⋊C411yescharacteristic
192C23×C47C22×D411yescharacteristic
193C23×C47C22×D411yescharacteristic
194C22×D47—24no—
195C22×D47—48no—
196C22×D47—44no—
197C22×D47—44no—
198C22×D47—42no—
199C22×D47—42no—
200C22×D47—88no—
201C22×D47—88no—
202C22×D47—88no—
203Q8×C47—42no—
204Q8×C47—42no—
205D4×C47—42no—
206D4×C47—42no—
207D4×C47—41no—
208D4×C47—41no—
209D4×C47—42no—
210D4×C47—41no—
211D4×C47—42no—
212D4×C47—41no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—44no—
232C2×C4×C47—84no—
233C2×C4×C47#9912yes—
234C2×C4×C47#9911yescharacteristic
235C2×C4×C47#9911yescharacteristic
236C2×C4×C47C2×C22⋊C411yescharacteristic
237C2×C4×C47C2×C22⋊C412yes—
238C2×C4×C47C2×C22⋊C412yes—
239C2×C4×C47C2×C22⋊C412yes—
240C2×C4×C47C23×C411yescharacteristic
241C22⋊D47—84no—
242C22⋊D47—84no—
243C22⋊D47—84no—
244C22⋊D47—84no—
245C2×Dic47—24no—
246C2×Dic47—44no—
247C2×Dic47—44no—
248C2×Dic47—84no—
249D188—44no—
250D188—44no—
251D188—44no—
252D188—44no—
253C2×#2128—471nomaximal, Sylow
254C22×C47—22no—
255C22×C47—21no—
256C22×C47—21no—
257C22×C47—22no—
258C22×C47—21no—
259C22×C47—21no—
260C22×C47—41no—
261C22×C47—41no—
262C22×C47—41no—
263C22×C47—41no—
264C22×C47—42no—
265C22×C47—42no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—81no—
270C22×C47—81no—
271C22×C47C2×#9911yescharacteristic
272C22×C47C2×#9911yescharacteristic
273C22×C47C22×C22⋊C411yescharacteristic
274C2×D47—44no—
275C2×D47—44no—
276C2×D47—44no—
277C2×D47—84no—
278C2×D47—816no—
279C2×D47—84no—
280C4×C47—21no—
281C4×C47—21no—
282C4×C47—22no—
283C4×C47—22no—
284C4×C47—84no—
285C4×C47C2×#9912yes—
286C4×C47C2×#9912yes—
287Dic47—44no—
288Dic47—84no—
289C2×#535—471no—
290C2×#410—471no—
291C2×#410D4711yescharacteristic, complemented
292#2128—478no—
293C2×#353—474no—
294C2×C47—21no—
295C2×C47—21no—
296C2×C47—41no—
297C2×C47—41no—
298C2×C47—42no—
299C2×C47—41no—
300C2×C47—41no—
301C2×C47#212812yescomplemented
302C2×C47C22×#9911yescharacteristic
303D47—84no—
304C22×#41C2×D4711yescharacteristic
305C23×D4—471no—
306#535—474no—
307#535—474no—
308#535—942no—
309#535—942no—
310C2×#127—472no—
311C2×#124—472no—
312C2×#121—941no—
313C2×#121—941no—
314C2×#120—941no—
315C2×#120—941no—
316C2×#117—21no—
317C2×#117—471no—
318C2×#117—941no—
319#410—22no—
320#410—474no—
321#410—942no—
322#410C2×D4714yescomplemented
323C2×#99—21no—
324C2×#99—472no—
325C2×#99—941no—
326C2×#99C2×D4712yescomplemented
327#353—4716no—
328C47C2×#212811yescharacteristic, complemented, Sylow
329C25—941no—
330C2×#41—41no—
331C2×#41—41no—
332C2×#41—42no—
333C2×#41C4×D4712yes—
334C2×#41D18812yes—
335C2×#41C22⋊D4712yes—
336C2×#41C22×D4712yes—
337C22×Q8C4×D4711yescharacteristic
338C22×D4—472no—
339C22×D4—474no—
340C22×D4—942no—
341C22×D4—942no—
342C22×D4—942no—
343C22×D4—941no—
344C22×D4—1882no—
345C22×D4—1881no—
346C22×D4—1881no—
347C22×D4D18811yescharacteristic
348C22×D4C22⋊D4711yescharacteristic
349C22×D4C22×D4711yescharacteristic
350C23×C4C4×D4711yescharacteristic
351C23×C4D18811yescharacteristic
352C23×C4C22⋊D4711yescharacteristic
353#127—474no—
354#127—474no—
355#127—942no—
356#127—942no—
357#124—478no—
358#124—944no—
359#121—944no—
360#121—944no—
361#121—944no—
362#121—944no—
363#120—944no—
364#120—944no—
365#120—944no—
366#120—944no—
367#117—24no—
368#117—42no—
369#117—474no—
370#117—942no—
371#117—944no—
372#117—1882no—
373C2×C4⋊C4—41no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—1881no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—941no—
378C2×C22⋊C4—942no—
379C2×C22⋊C4—942no—
380C2×C22⋊C4—942no—
381C2×C42—41no—
382C2×C42—472no—
383C2×C42—1881no—
384#99—24no—
385#99—22no—
386#99—474no—
387#99—942no—
388#99—944no—
389#99C22×D4714yescomplemented
390C24—944no—
391C24—942no—
392C24—3764no—
393C24—1882no—
394C24—1882no—
395C24(C2×D47)⋊C411yescharacteristic
396C24C2×D18811yescharacteristic
397C24C2×C22⋊D4711yescharacteristic
398#41—42no—
399#41—42no—
400#41—42no—
401#41—42no—
402#41—42no—
403#41—42no—
404#41—42no—
405#41—42no—
406C2×Q8—41no—
407C2×Q8—41no—
408C2×Q8(C2×D47)⋊C412yes—
409C2×Q8C2×C4×D4712yes—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—474no—
417C2×D4—944no—
418C2×D4—944no—
419C2×D4—944no—
420C2×D4—944no—
421C2×D4—944no—
422C2×D4—942no—
423C2×D4—3764no—
424C2×D4—3764no—
425C2×D4—1882no—
426C2×D4—1882no—
427C2×D4—1884no—
428C2×D4—1884no—
429C2×D4—1884no—
430C2×D4—1884no—
431C2×D4—1882no—
432C2×D4—1881no—
433C2×D4—1881no—
434C2×D4(C2×D47)⋊C412yes—
435C2×D4(C2×D47)⋊C412yes—
436C2×D4C2×C4×D4711yescharacteristic
437C2×D4C2×D18812yes—
438C2×D4C2×D18811yescharacteristic
439C2×D4C2×C22⋊D4712yes—
440C2×D4C2×C22⋊D4711yescharacteristic
441C2×D4C23×D4711yescharacteristic
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—942no—
453C22×C4—1882no—
454C22×C4(C2×D47)⋊C411yescharacteristic
455C22×C4(C2×D47)⋊C411yescharacteristic
456C22×C4(C2×D47)⋊C412yes—
457C22×C4(C2×D47)⋊C412yes—
458C22×C4(C2×D47)⋊C412yes—
459C22×C4(C2×D47)⋊C411yescharacteristic
460C22×C4C2×C4×D4711yescharacteristic
461C22×C4C2×C4×D4712yes—
462C22×C4C2×D18812yes—
463C22×C4C2×C22⋊D4712yes—
464C4⋊C4—44no—
465C4⋊C4—944no—
466C4⋊C4—1884no—
467C22⋊C4—24no—
468C22⋊C4—42no—
469C22⋊C4—944no—
470C22⋊C4—948no—
471C22⋊C4—948no—
472C22⋊C4—944no—
473C22⋊C4—1882no—
474C42—44no—
475C42—474no—
476C42—942no—
477C42—1884no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—944no—
488C23—3768no—
489C23—3768no—
490C23—3768no—
491C23—1884no—
492C23—1884no—
493C23—1888no—
494C23—1882no—
495C23—1882no—
496C23C2×(C2×D47)⋊C411yescharacteristic
497C23C2×(C2×D47)⋊C412yes—
498C23C2×(C2×D47)⋊C411yescharacteristic
499C23C2×(C2×D47)⋊C411yescharacteristic
500C23C22×C22⋊D4711yescharacteristic
501C23C22×D18811yescharacteristic
502Q8—42no—
503Q8—42no—
504D4—42no—
505D4—42no—
506D4—41no—
507D4—41no—
508D4—42no—
509D4—41no—
510D4—42no—
511D4—41no—
512D4—3764no—
513D4—3764no—
514D4—3764no—
515D4—3764no—
516D4—1884no—
517D4—1884no—
518D4—1884no—
519D4—1884no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—44no—
539C2×C4—84no—
540C2×C4—944no—
541C2×C4—3764no—
542C2×C4—1884no—
543C2×C4—1884no—
544C2×C4#1309393812yes—
545C2×C4#1309393811yescharacteristic
546C2×C4#1309393811yescharacteristic
547C2×C4C2×(C2×D47)⋊C411yescharacteristic
548C2×C4C2×(C2×D47)⋊C412yes—
549C2×C4C2×(C2×D47)⋊C412yes—
550C2×C4C2×(C2×D47)⋊C412yes—
551C2×C4C22×C4×D4711yescharacteristic
552C22—22no—
553C22—21no—
554C22—21no—
555C22—22no—
556C22—21no—
557C22—21no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—81no—
568C22—81no—
569C22—3764no—
570C22—37616no—
571C22—3764no—
572C22—1884no—
573C22—1884no—
574C22—1884no—
575C22C2×#1309393811yescharacteristic
576C22C2×#1309393811yescharacteristic
577C22C22×(C2×D47)⋊C411yescharacteristic
578C4—21no—
579C4—22no—
580C4—22no—
581C4—21no—
582C4—84no—
583C4—3764no—
584C4—1884no—
585C4C2×#1309393812yes—
586C4C2×#1309393812yes—
587C2—21no—
588C2—21no—
589C2—41no—
590C2—41no—
591C2—42no—
592C2—41no—
593C2—41no—
594C2—3764no—
595C2#2024505912yesdirect product factor
596C2C22×#1309393811yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC2×C4×C47C23×C4
FrattiniC2×D4C23×D47
FittingC2×C47×#410C2
socleC22×C47C2×#99
prime radicalC2×#15689687C2
cosocleC2×D4×C47C24
hypercenterC2×#410D47
hypocenterC47C2×#2128
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. C2×#15689097C2semidirect
3. #20245059C2direct (N × H)
4. C2×#15688915C2semidirect
5. C2×C47×#410C2semidirect
6. C23×D188C4semidirect
9. #15689687C4semidirect
12. C2×#13094013C4semidirect
18. C2×#13093983C4semidirect
20. #15689097C22semidirect
22. C2×#13093939C22semidirect
23. #15688915C22semidirect
27. C47×#410C22semidirect
29. C2×#99×C47C22semidirect
37. C22×D188C2×C4semidirect
45. #13094013C2×C4semidirect
57. #13093983C2×C4semidirect
68. #13093939C23semidirect
90. #99×C47C23semidirect
291. C2×#410D47semidirect
301. C2×C47#2128semidirect
322. #410C2×D47semidirect
326. C2×#99C2×D47semidirect
328. C47C2×#2128semidirect
389. #99C22×D47semidirect
595. C2#20245059direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896872
C2×#156890972
#202450592
C2×#156889152
C2×C47×#4102
C2×#212847