Finite Group Databasebeta

← C4×#15689300 #26222161

Subgroups — C4×#15689300 #26222161

Order 12032.

1442 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13094069C211yesmaximal, characteristic
2C2×#15689300C211yesmaximal, characteristic
3C4×#13094055C212yesmaximal, complemented
4#20243777C211yesmaximal, characteristic, complemented
5#20243760C212yesmaximal, complemented
6#20243620C212yesmaximal, complemented
7#20243597C211yesmaximal, characteristic, complemented
8#20243341C211yesmaximal, characteristic
9C4×#13093952C212yesmaximal, complemented
10C4×#13093941C211yesmaximal, characteristic, complemented
11C4×#124×C47C211yesmaximal, characteristic, complemented
12C2×#13094069C2211yescharacteristic
13C2×#13094055C2212yes—
14#15689604C2212yes—
15#15689602—22no—
16#15689598C2211yescharacteristic
17#15689596—22no—
18#15689595—22no—
19C4×#10625350C2212yes—
20C4×#10625352—22no—
21C2×C4×C22⋊D47—22no—
22C4×D4×D47—22no—
23#15689300C418yesdirect product factor
24#15689228—22no—
25#15689221—22no—
26C4×#10625337—22no—
27C4×#10625336—22no—
28#15689163C2212yescomplemented
29C2×#13093952C2212yes—
30C2×#13093952C2212yes—
31#15689132C2212yescomplemented
32#15689122C2212yes—
33C2×#13093941C2211yescharacteristic
34C2×#13093941C2211yescharacteristic
35C4×#10625329C2212yescomplemented
36#15688892C2212yescomplemented
37#15688883C2212yescomplemented
38#15688883C2212yescomplemented
39C2×#124×C47C2211yescharacteristic
40C2×C4×Q8×C47C2211yescharacteristic, complemented
41C2×C4×D4×C47C2211yescharacteristic
42C47×#391C2211yescharacteristic, complemented
43C47×#390C2211yescharacteristic
44C47×#388—22no—
45C47×#382C2211yescharacteristic, complemented
46C4×C22⋊C4×C47—22no—
47C43×C47C2211yescharacteristic, complemented
48#13094069—22no—
49#13094069—24no—
50#13094069C2×C414yes—
51C2×#10625350C2312yes—
52C2×#10625352—22no—
53C22×C22⋊D47—22no—
54C2×D4×D47—22no—
55C22×C4×D47—22no—
56#13094055—24no—
57#13094055C2×C418yescomplemented
58C2×#10625337—22no—
59C2×#10625336—22no—
60C2×#10625329C2312yes—
61C2×#10625329C2312yes—
62C4×Dic94D412yes—
63C4×Dic94D412yes—
64#13093989—22no—
65C4×D188D412yes—
66C4×D188D412yes—
67D4×Dic47—22no—
68D4×Dic47—22no—
69#13093986—22no—
70C4×C22⋊D47—24no—
71C4×C22⋊D47—24no—
72C4×C22⋊D47—22no—
73C4×C22⋊D47—24no—
74C4×C22⋊D47—22no—
75(C22⋊D47)⋊C4—24no—
76(C22⋊D47)⋊C4—24no—
77#13093983C2312yes—
78#13093982—22no—
79#13093981—22no—
80#13093980—24no—
81C2×Dic47⋊C4—22no—
82C4⋊C4×D47—22no—
83C2×C22⋊Dic47—22no—
84C2×(C2×D47)⋊C4—22no—
85C22⋊C4×D47—24no—
86C2×C4×Dic47—22no—
87C42×D47—22no—
88#13093957—22no—
89#13093955—22no—
90#13093952—24no—
91#13093952C2×C418yescomplemented
92#13093948—22no—
93#13093948—22no—
94#13093947—22no—
95#13093944—22no—
96#13093941—22no—
97#13093941C2×C414yescomplemented
98#13093935—22no—
99C4×C47⋊C8—22no—
100C22×Q8×C47C2311yescharacteristic
101C22×D4×C47C2311yescharacteristic
102C23×C4×C47D412yes—
103#124×C47—22no—
104#124×C47C2×C414yescomplemented
105C4×Q8×C47—22no—
106C4×Q8×C47—22no—
107C4×D4×C47—22no—
108C4×D4×C47—22no—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C22⋊C4×C47—22no—
113C2×C22⋊C4×C47—22no—
114C2×C22⋊C4×C47D412yes—
115C2×C42×C47—21no—
116C2×C42×C47—21no—
117C2×C42×C47—21no—
118C2×C42×C47C2311yescharacteristic
119C2×C42×C47C2311yescharacteristic
120C2×C42×C47C2311yescharacteristic
121#95×C47—22no—
122#95×C47—22no—
123C24×C47C2×D412yes—
124C23×D47—22no—
125C2×Q8×C47—21no—
126C2×Q8×C47—22no—
127C2×Q8×C47—22no—
128C2×Q8×C47C22×C412yescomplemented
129C2×D4×C47—22no—
130C2×D4×C47—22no—
131C2×D4×C47C22×C412yes—
132C2×D4×C47#4112yes—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—22no—
137C22×C4×C47—22no—
138C22×C4×C47—21no—
139C22×C4×C47—44no—
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D412yes—
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2411yescharacteristic
144C2×C22⋊D47—24no—
145C2×C22⋊D47—28no—
146C2×C22⋊D47—28no—
147C2×C22⋊D47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151#10625352—28no—
152#10625352—28no—
153D4×D47—28no—
154D4×D47—28no—
155#10625350—22no—
156#10625350—22no—
157#10625350C22×C414yes—
158#10625350#4114yes—
159C2×D188C2×D412yes—
160C2×C4×D47—22no—
161C2×C4×D47—22no—
162C2×C4×D47—24no—
163C2×C4×D47—22no—
164C2×C4×D47—84no—
165C2×Dic94C2×D412yes—
166C4⋊C4×C47—22no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—42no—
169C22⋊C4×C47—28no—
170C22⋊C4×C47—24no—
171C22⋊C4×C47—44no—
172C42×C47—21no—
173C42×C47—22no—
174C42×C47—42no—
175C42×C47—42no—
176C42×C47—41no—
177C42×C47—41no—
178C42×C47C22×C412yescomplemented
179C42×C47C2×D412yes—
180C42×C47C2×D412yes—
181C22⋊Dic47—24no—
182C22⋊Dic47—22no—
183#10625337—28no—
184#10625336—28no—
185(C2×D47)⋊C4—24no—
186(C2×D47)⋊C4—22no—
187(C2×D47)⋊C4—22no—
188C4⋊Dic47C2×D412yes—
189C4⋊Dic47C2×D412yes—
190Dic47⋊C4—24no—
191Dic47⋊C4—22no—
192Dic47⋊C4—22no—
193C4×Dic47—24no—
194C4×Dic47—22no—
195C4×Dic47—22no—
196C4×Dic47—22no—
197#10625329—22no—
198#10625329—22no—
199#10625329—22no—
200#10625329C22×C414yescomplemented
201C2×C47⋊C8—22no—
202C2×C47⋊C8—22no—
203C23×C47—22no—
204C23×C47—22no—
205C23×C47—42no—
206C23×C47—42no—
207C23×C47C4×D414yes—
208C23×C47C22×D411yescharacteristic
209C22×D47—22no—
210C22×D47—28no—
211C22×D47—82no—
212C22×D47—82no—
213Q8×C47—24no—
214Q8×C47—42no—
215D4×C47—24no—
216D4×C47—24no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—44no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—41no—
233C2×C4×C47—82no—
234C2×C4×C47#12012yes—
235C2×C4×C47#12011yescharacteristic
236C2×C4×C47#12011yescharacteristic
237C2×C4×C47C23×C411yescharacteristic
238C2×C4×C47C22×D412yes—
239C2×C4×C47C2×#4112yes—
240C2×C4×C47C2×#4111yescharacteristic
241C22⋊D47—216no—
242C22⋊D47—24no—
243C22⋊D47—24no—
244C2×Dic47—28no—
245C2×Dic47—22no—
246C2×Dic47—22no—
247C2×Dic47—22no—
248C2×Dic47—82no—
249C2×Dic47—82no—
250D188—22no—
251D188C4×D414yes—
252C4×D47—24no—
253C4×D47—24no—
254C4×D47—44no—
255C4×D47—82no—
256C4×D47—84no—
257Dic94—22no—
258Dic94C4×D414yes—
259C47⋊C8—24no—
260C47⋊C8—42no—
261C4×#455—471nomaximal, Sylow
262C22×C47—21no—
263C22×C47—21no—
264C22×C47—24no—
265C22×C47—44no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—82no—
269C22×C47C2×C4×D411yescharacteristic
270C22×C47C2×#12011yescharacteristic
271C22×C47#52511yescharacteristic
272C2×D47—24no—
273C2×D47—42no—
274C2×D47—42no—
275C2×D47—88no—
276C4×C47—21no—
277C4×C47—22no—
278C4×C47—42no—
279C4×C47—42no—
280C4×C47—42no—
281C4×C47—42no—
282C4×C47—81no—
283C4×C47#45512yescomplemented
284C4×C47C2×C4×D412yes—
285C4×C47#52512yes—
286Dic47—24no—
287Dic47—42no—
288Dic47—42no—
289C4×H(5,𝔽2)—471no—
290C2×#455—471no—
291C4×#137—472no—
292C4×#124D4711yescharacteristic, complemented
293#1885—471no—
294#1882—472no—
295#1860—472no—
296#1856—471no—
297#1792—471no—
298C4×#104—472no—
299C4×#100—471no—
300C2×C47—21no—
301C2×C47—21no—
302C2×C47—44no—
303C2×C47C4×#12011yescharacteristic
304C2×C47C2×#45511yescharacteristic
305C2×C47#301811yescharacteristic
306D47—44no—
307C2×H(5,𝔽2)—471no—
308C2×#137—472no—
309C2×#124C2×D4711yescharacteristic
310#520—472no—
311#520—942no—
312#518—471no—
313#518—942no—
314#518—942no—
315C4×#41—472no—
316C4×#41—942no—
317C2×C4×Q8C2×D4711yescharacteristic, complemented
318C2×C4×D4—942no—
319C2×C4×D4—942no—
320C2×C4×D4C2×D4711yescharacteristic
321#455—478no—
322#441—942no—
323#440—942no—
324C4×Q16—942no—
325C4×SD16—942no—
326#428—472no—
327C2×#104—472no—
328C2×#104—472no—
329#419—472no—
330#417—472no—
331C2×#100—471no—
332C2×#100—471no—
333C4×M16—472no—
334#391C2×D4711yescharacteristic, complemented
335#390C2×D4711yescharacteristic
336#388—22no—
337#382C2×D4711yescharacteristic, complemented
338C4×C22⋊C4—22no—
339C43C2×D4711yescharacteristic, complemented
340#343—472no—
341#339—472no—
342#339—472no—
343C47C4×#45511yescharacteristic, complemented, Sylow
344H(5,𝔽2)—474no—
345H(5,𝔽2)—944no—
346H(5,𝔽2)—942no—
347C2×#41—472no—
348C2×#41—942no—
349C22×Q8C22×D4711yescharacteristic
350C22×D4—942no—
351C22×D4—942no—
352C22×D4C22×D4711yescharacteristic
353C23×C4—942no—
354C23×C4C22⋊D4712yes—
355#137—478no—
356#137—944no—
357C2×Q16—942no—
358C2×SD16—942no—
359C2×M16—472no—
360C2×M16—472no—
361#124—22no—
362#124C4×D4714yescomplemented
363C4×Q8—22no—
364C4×Q8—22no—
365C4×Q8—472no—
366C4×Q8—472no—
367C4×Q8—942no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—472no—
371C4×D4—472no—
372C4×D4—944no—
373C4×D4—944no—
374C4×D4—942no—
375C4×D4—942no—
376C4×D4—942no—
377C4×D4—944no—
378C4×D4—944no—
379C4×D4—942no—
380C4×D4—944no—
381C4×D4—942no—
382#117—472no—
383#117—944no—
384#117—942no—
385#117—942no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—942no—
388C2×C4⋊C4—942no—
389C2×C4⋊C4C22×D4711yescharacteristic
390C2×C4⋊C4C22×D4711yescharacteristic
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—944no—
394C2×C22⋊C4—942no—
395C2×C22⋊C4—942no—
396C2×C22⋊C4C22⋊D4712yes—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—942no—
401C2×C42—942no—
402C2×C42C22×D4711yescharacteristic
403C2×C42C22×D4711yescharacteristic
404C2×C42C22×D4711yescharacteristic
405#107—942no—
406#106—942no—
407#104—478no—
408#104—944no—
409#103—942no—
410#103—942no—
411#103—942no—
412#102—942no—
413#100—474no—
414#100—942no—
415#97—942no—
416C4×C8—942no—
417#95—22no—
418#95—22no—
419C24—942no—
420C24C2×C22⋊D4712yes—
421#41—474no—
422#41—474no—
423#41—948no—
424#41—948no—
425#41—942no—
426#41—942no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—472no—
431C2×Q8C2×C4×D4712yescomplemented
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—472no—
435C2×D4—948no—
436C2×D4—948no—
437C2×D4—944no—
438C2×D4—948no—
439C2×D4—948no—
440C2×D4—942no—
441C2×D4C2×C4×D4712yes—
442C2×D4#1062535012yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—44no—
450C22×C4—944no—
451C22×C4—942no—
452C22×C4—942no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—942no—
457C22×C4—3764no—
458C22×C4C2×C22⋊D4711yescharacteristic
459C22×C4C2×C22⋊D4712yes—
460C22×C4C2×C22⋊D4711yescharacteristic
461C22×C4C23×D4711yescharacteristic
462Q16—948no—
463SD16—948no—
464M16—474no—
465M16—942no—
466M16—942no—
467M16—942no—
468C2×C8—942no—
469C2×C8—942no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—42no—
473C4⋊C4—472no—
474C4⋊C4—472no—
475C4⋊C4—944no—
476C4⋊C4—942no—
477C4⋊C4—942no—
478C22⋊C4—28no—
479C22⋊C4—24no—
480C22⋊C4—44no—
481C22⋊C4—944no—
482C22⋊C4—944no—
483C22⋊C4—942no—
484C22⋊C4—942no—
485C22⋊C4—942no—
486C42—21no—
487C42—22no—
488C42—42no—
489C42—42no—
490C42—41no—
491C42—41no—
492C42—944no—
493C42—942no—
494C42—942no—
495C42—942no—
496C42C2×C4×D4712yescomplemented
497C42D4×D4712yes—
498C42D4×D4712yes—
499C23—22no—
500C23—22no—
501C23—42no—
502C23—42no—
503C23—948no—
504C23—942no—
505C23—3762no—
506C23—3762no—
507C23C4×C22⋊D4714yes—
508C23C22×C22⋊D4711yescharacteristic
509Q8—24no—
510Q8—42no—
511Q8—474no—
512Q8—942no—
513D4—24no—
514D4—24no—
515D4—474no—
516D4—9416no—
517D4—944no—
518D4—944no—
519D4—942no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—82no—
537C2×C4—948no—
538C2×C4—942no—
539C2×C4—944no—
540C2×C4—944no—
541C2×C4—942no—
542C2×C4—942no—
543C2×C4—3762no—
544C2×C4—3764no—
545C2×C4—3762no—
546C2×C4—3762no—
547C2×C4—1884no—
548C2×C4#1309399211yescharacteristic
549C2×C4#1309399212yes—
550C2×C4#1309399211yescharacteristic
551C2×C4C22×C4×D4711yescharacteristic
552C2×C4C2×D4×D4712yes—
553C2×C4#41×D4712yes—
554C2×C4C2×#1062535011yescharacteristic
555C8—944no—
556C8—1882no—
557C22—21no—
558C22—21no—
559C22—24no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—82no—
564C22—944no—
565C22—3768no—
566C22—1882no—
567C22—1882no—
568C22C2×C4×C22⋊D4711yescharacteristic
569C22C2×#1309399211yescharacteristic
570C22#1568963011yescharacteristic
571C4—21no—
572C4—22no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—81no—
578C4—944no—
579C4—1882no—
580C4—1882no—
581C4#1568930012yesdirect product factor
582C4C4×D4×D4712yes—
583C4#1568963212yes—
584C2—21no—
585C2—21no—
586C2—44no—
587C2—1884no—
588C2C4×#1309399211yescharacteristic
589C2C2×#1568930011yescharacteristic
590C2#2025273611yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×#124×C47C2
socleC22×C47C2×#120
prime radicalC2×#13094069C22
cosocleC22×C4×C47C24
hypercenterC4×#124D47
hypocenterC47C4×#455
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
3. C4×#13094055C2semidirect
4. #20243777C2semidirect
5. #20243760C2semidirect
6. #20243620C2semidirect
7. #20243597C2semidirect
9. C4×#13093952C2semidirect
10. C4×#13093941C2semidirect
11. C4×#124×C47C2semidirect
23. #15689300C4direct (N × H)
28. #15689163C22semidirect
31. #15689132C22semidirect
35. C4×#10625329C22semidirect
36. #15688892C22semidirect
37. #15688883C22semidirect
38. #15688883C22semidirect
40. C2×C4×Q8×C47C22semidirect
42. C47×#391C22semidirect
45. C47×#382C22semidirect
47. C43×C47C22semidirect
57. #13094055C2×C4semidirect
91. #13093952C2×C4semidirect
97. #13093941C2×C4semidirect
104. #124×C47C2×C4semidirect
128. C2×Q8×C47C22×C4semidirect
178. C42×C47C22×C4semidirect
200. #10625329C22×C4semidirect
283. C4×C47#455semidirect
292. C4×#124D47semidirect
317. C2×C4×Q8C2×D47semidirect
334. #391C2×D47semidirect
337. #382C2×D47semidirect
339. C43C2×D47semidirect
343. C47C4×#455semidirect
362. #124C4×D47semidirect
431. C2×Q8C2×C4×D47semidirect
496. C42C2×C4×D47semidirect
581. C4#15689300direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#130940692
C2×#156893002
C4×#130940552
#202437772
#202437602
#202436202
#202435972
#202433412
C4×#130939522
C4×#130939412
C4×#124×C472
C4×#45547