Finite Group Databasebeta

← C4×#15689291 #26222152

Subgroups — C4×#15689291 #26222152

Order 12032.

1578 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689291C211yesmaximal, characteristic
2C4×#13094049C212yesmaximal, complemented
3C4×#13094020C211yesmaximal, characteristic, complemented
4#20243786C211yesmaximal, characteristic, complemented
5#20243764C212yesmaximal, complemented
6#20243614C212yesmaximal, complemented
7#20243598C211yesmaximal, characteristic, complemented
8#20243342C211yesmaximal, characteristic, complemented
9C4×#13093951C212yesmaximal, complemented
10C4×#13093941C211yesmaximal, characteristic, complemented
11C4×H(5,𝔽2)×C47C211yesmaximal, characteristic, complemented
12C2×#13094049C2212yes—
13C2×#13094020C2211yescharacteristic
14C2×C4×D188C2211yescharacteristic, complemented
15C2×C4×C22⋊D47—22no—
16#15689291C418yesdirect product factor
17#15689233—22no—
18#15689221—22no—
19C4×#10625336—22no—
20C4×#10625334—22no—
21#15689166C2212yescomplemented
22C2×#13093951C2212yes—
23C2×#13093951C2212yes—
24#15689127C2212yescomplemented
25#15689124C2212yes—
26C2×#13093941C2211yescharacteristic
27C2×#13093941C2211yescharacteristic
28C4×#10625329C2212yescomplemented
29#15689026C2211yescharacteristic
30#15689024C2211yescharacteristic, complemented
31#15689023—22no—
32#15688992C2211yescharacteristic, complemented
33C42×Dic47C2211yescharacteristic, complemented
34#15688892C2212yescomplemented
35#15688882C2212yescomplemented
36#15688882C2212yescomplemented
37C2×H(5,𝔽2)×C47C2211yescharacteristic
38C47×#520—22no—
39C47×#520C2212yes—
40C47×#518—22no—
41C47×#518—22no—
42C47×#518C2211yescharacteristic
43C4×#41×C47—22no—
44C4×#41×C47C2212yes—
45C2×C4×D4×C47—22no—
46C2×C4×D4×C47—22no—
47C2×C4×D4×C47C2211yescharacteristic
48C22×D188C2311yescharacteristic
49C22×C22⋊D47—22no—
50C22×C4×D47—21no—
51#13094049—24no—
52#13094049C2×C418yescomplemented
53C2×#10625336—22no—
54C2×#10625334—22no—
55C2×#10625329C2312yes—
56C2×#10625329C2312yes—
57#13094020—22no—
58#13094020C2×C414yescomplemented
59C4×D188—22no—
60C4×D188—22no—
61C4×C22⋊D47—48no—
62C2×C4⋊Dic47C2311yescharacteristic
63C2×Dic47⋊C4—22no—
64C2×C22⋊Dic47—22no—
65C2×(C2×D47)⋊C4—21no—
66C2×(C2×D47)⋊C4—22no—
67C2×C4×Dic47—21no—
68C2×C4×Dic47—21no—
69C2×C4×Dic47C2311yescharacteristic
70C2×C4×Dic47C2311yescharacteristic
71#13093957—22no—
72#13093955—22no—
73#13093951—24no—
74#13093951C2×C418yescomplemented
75#13093948—22no—
76#13093945—22no—
77#13093944—22no—
78#13093944—22no—
79#13093941—22no—
80#13093941C2×C414yescomplemented
81#13093935—22no—
82C4×C47⋊C8—22no—
83H(5,𝔽2)×C47—24no—
84H(5,𝔽2)×C47—22no—
85H(5,𝔽2)×C47C2×C414yescomplemented
86C2×#41×C47—22no—
87C2×#41×C47C2312yes—
88C22×D4×C47—22no—
89C22×D4×C47—22no—
90C22×D4×C47C2311yescharacteristic
91C23×C4×C47—22no—
92C23×C4×C47D412yes—
93C4×Q8×C47—22no—
94C4×Q8×C47D412yescomplemented
95C4×Q8×C47D412yescomplemented
96C4×D4×C47—24no—
97C4×D4×C47—24no—
98C4×D4×C47—24no—
99C4×D4×C47—24no—
100C4×D4×C47—22no—
101C4×D4×C47—22no—
102C4×D4×C47—24no—
103C4×D4×C47—22no—
104C4×D4×C47—22no—
105C4×D4×C47—22no—
106C4×D4×C47—22no—
107C4×D4×C47—22no—
108C4×D4×C47D412yescomplemented
109C4×D4×C47D412yescomplemented
110#117×C47—24no—
111#117×C47—22no—
112#117×C47—22no—
113#117×C47C2312yes—
114C2×C4⋊C4×C47—22no—
115C2×C4⋊C4×C47—22no—
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C22⋊C4×C47—22no—
118C2×C22⋊C4×C47—24no—
119C2×C22⋊C4×C47—22no—
120C2×C22⋊C4×C47D412yes—
121C2×C42×C47—22no—
122C2×C42×C47—22no—
123C2×C42×C47C2311yescharacteristic
124C24×C47—22no—
125C24×C47C2×D412yes—
126C23×D47—21no—
127#41×C47—28no—
128#41×C47—28no—
129#41×C47—22no—
130#41×C47—22no—
131#41×C47C22×C414yes—
132#41×C47#4114yes—
133C2×Q8×C47C2×D412yes—
134C2×D4×C47—24no—
135C2×D4×C47—28no—
136C2×D4×C47—28no—
137C2×D4×C47—28no—
138C2×D4×C47—22no—
139C2×D4×C47—28no—
140C2×D4×C47—22no—
141C2×D4×C47—22no—
142C2×D4×C47C22×C412yes—
143C2×D4×C47C2×D412yes—
144C2×D4×C47#4112yes—
145C22×C4×C47—22no—
146C22×C4×C47—22no—
147C22×C4×C47—22no—
148C22×C4×C47—24no—
149C22×C4×C47—22no—
150C22×C4×C47—22no—
151C22×C4×C47—22no—
152C22×C4×C47—22no—
153C22×C4×C47—22no—
154C22×C4×C47—44no—
155C22×C4×C47—84no—
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D412yes—
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2411yescharacteristic
160C2×C22⋊D47—28no—
161C2×C22⋊D47—44no—
162C2×C22⋊D47—44no—
163C22×Dic47—21no—
164C22×Dic47—21no—
165C22×Dic47—21no—
166C2×D188—22no—
167C2×D188—22no—
168C2×D188—21no—
169C2×D188C22×C412yescomplemented
170C2×C4×D47—44no—
171C2×C4×D47—41no—
172C2×C4×D47—41no—
173C4⋊C4×C47—24no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—22no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47C2×D412yes—
178C4⋊C4×C47C2×D412yes—
179C22⋊C4×C47—24no—
180C22⋊C4×C47—24no—
181C22⋊C4×C47—22no—
182C22⋊C4×C47—22no—
183C22⋊C4×C47—22no—
184C22⋊C4×C47—24no—
185C42×C47—24no—
186C42×C47—22no—
187C42×C47—22no—
188C42×C47—22no—
189C42×C47C2×D412yes—
190C42×C47C2×D412yes—
191C22⋊Dic47—44no—
192#10625336—28no—
193#10625334—28no—
194(C2×D47)⋊C4—44no—
195(C2×D47)⋊C4—42no—
196C4⋊Dic47—22no—
197Dic47⋊C4—44no—
198C4×Dic47—24no—
199C4×Dic47—21no—
200C4×Dic47—22no—
201C4×Dic47—42no—
202C4×Dic47C22×C412yescomplemented
203#10625329—22no—
204#10625329—22no—
205#10625329—22no—
206#10625329C22×C414yescomplemented
207C2×C47⋊C8—22no—
208C2×C47⋊C8—22no—
209C23×C47—28no—
210C23×C47—22no—
211C23×C47—22no—
212C23×C47—22no—
213C23×C47—42no—
214C23×C47—42no—
215C23×C47—82no—
216C23×C47—82no—
217C23×C47C4×D414yes—
218C23×C47C22×D411yescharacteristic
219C22×D47—22no—
220C22×D47—42no—
221C22×D47—42no—
222C22×D47—41no—
223C22×D47—41no—
224Q8×C47—22no—
225Q8×C47C4×D414yescomplemented
226D4×C47—216no—
227D4×C47—24no—
228D4×C47—24no—
229D4×C47—24no—
230D4×C47—22no—
231D4×C47—24no—
232D4×C47C4×D414yescomplemented
233C2×C4×C47—28no—
234C2×C4×C47—22no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—41no—
242C2×C4×C47—44no—
243C2×C4×C47—44no—
244C2×C4×C47—42no—
245C2×C4×C47—82no—
246C2×C4×C47—82no—
247C2×C4×C47—82no—
248C2×C4×C47—84no—
249C2×C4×C47—82no—
250C2×C4×C47#12012yes—
251C2×C4×C47#12011yescharacteristic
252C2×C4×C47#12011yescharacteristic
253C2×C4×C47C23×C411yescharacteristic
254C2×C4×C47C22×D412yes—
255C2×C4×C47C2×#4112yes—
256C2×C4×C47C2×#4111yescharacteristic
257C22⋊D47—416no—
258C2×Dic47—22no—
259C2×Dic47—22no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—42no—
265C2×Dic47—41no—
266D188—24no—
267D188—42no—
268C4×D47—82no—
269C4×D47—82no—
270C47⋊C8—24no—
271C47⋊C8—42no—
272C4×#453—471nomaximal, Sylow
273C22×C47—21no—
274C22×C47—21no—
275C22×C47—24no—
276C22×C47—24no—
277C22×C47—44no—
278C22×C47—42no—
279C22×C47—42no—
280C22×C47—42no—
281C22×C47—42no—
282C22×C47—82no—
283C22×C47—88no—
284C22×C47C2×C4×D411yescharacteristic
285C22×C47C2×#12011yescharacteristic
286C22×C47#52511yescharacteristic
287C2×D47—44no—
288C2×D47—42no—
289C2×D47—82no—
290C2×D47—81no—
291C2×D47—81no—
292C4×C47—21no—
293C4×C47—22no—
294C4×C47—24no—
295C4×C47—42no—
296C4×C47—42no—
297C4×C47—42no—
298C4×C47#45312yescomplemented
299C4×C47C2×C4×D412yes—
300C4×C47#52512yes—
301Dic47—24no—
302Dic47—42no—
303Dic47—81no—
304C4×H(5,𝔽2)D4711yescharacteristic, complemented
305C2×#453—471no—
306C4×AGL(1,ℤ/8)—472no—
307C4×#127—471no—
308#1887—471no—
309#1883—472no—
310#1859—472no—
311#1856—471no—
312#1792—471no—
313C4×#104—472no—
314C4×#100—471no—
315C2×C47—21no—
316C2×C47—21no—
317C2×C47—44no—
318C2×C47—44no—
319C2×C47C4×#12011yescharacteristic
320C2×C47C2×#45311yescharacteristic
321C2×C47#301811yescharacteristic
322D47—82no—
323C2×H(5,𝔽2)C2×D4711yescharacteristic
324C2×AGL(1,ℤ/8)—472no—
325C2×#127—471no—
326#520—22no—
327#520C2×D4712yes—
328#518—22no—
329#518—22no—
330#518C2×D4711yescharacteristic
331C4×#41—22no—
332C4×#41C2×D4712yes—
333C2×C4×D4—22no—
334C2×C4×D4—22no—
335C2×C4×D4—471no—
336C2×C4×D4—942no—
337C2×C4×D4C2×D4711yescharacteristic
338#453—478no—
339#442—942no—
340#440—942no—
341C4×SD16—942no—
342C4×D8—942no—
343#428—472no—
344C2×#104—472no—
345C2×#104—472no—
346#418—472no—
347#417—472no—
348C2×#100—471no—
349C2×#100—471no—
350C4×M16—472no—
351#390—471no—
352#390—471no—
353#390—942no—
354#384—471no—
355C43—471no—
356#343—472no—
357#339—472no—
358#339—472no—
359C47C4×#45311yescharacteristic, complemented, Sylow
360H(5,𝔽2)—24no—
361H(5,𝔽2)—22no—
362H(5,𝔽2)C4×D4714yescomplemented
363C2×#41—22no—
364C2×#41C22×D4712yes—
365C22×D4—22no—
366C22×D4—22no—
367C22×D4—471no—
368C22×D4—942no—
369C22×D4C22×D4711yescharacteristic
370C23×C4—22no—
371C23×C4—941no—
372C23×C4C22⋊D4712yes—
373AGL(1,ℤ/8)—478no—
374AGL(1,ℤ/8)—944no—
375C2×SD16—942no—
376C2×D8—942no—
377C2×M16—472no—
378C2×M16—472no—
379#127—474no—
380#127—942no—
381C4×Q8—22no—
382C4×Q8C22⋊D4712yescomplemented
383C4×Q8C22⋊D4712yescomplemented
384C4×D4—24no—
385C4×D4—24no—
386C4×D4—24no—
387C4×D4—24no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—24no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—942no—
397C4×D4—942no—
398C4×D4—1888no—
399C4×D4C22⋊D4712yescomplemented
400C4×D4C22⋊D4712yescomplemented
401#117—24no—
402#117—22no—
403#117—22no—
404#117C22×D4712yes—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4—942no—
409C2×C4⋊C4C22×D4711yescharacteristic
410C2×C22⋊C4—22no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—942no—
414C2×C22⋊C4—941no—
415C2×C22⋊C4—942no—
416C2×C22⋊C4C22⋊D4712yes—
417C2×C42—22no—
418C2×C42—22no—
419C2×C42—471no—
420C2×C42—471no—
421C2×C42—941no—
422C2×C42—941no—
423C2×C42C22×D4711yescharacteristic
424#107—942no—
425#106—942no—
426#104—478no—
427#104—944no—
428#103—942no—
429#102—942no—
430#102—942no—
431#102—942no—
432#100—474no—
433#100—942no—
434#97—942no—
435C4×C8—942no—
436C24—22no—
437C24—941no—
438C24C2×C22⋊D4712yes—
439#41—28no—
440#41—28no—
441#41—22no—
442#41—22no—
443#41C2×C4×D4714yes—
444#41#1062535014yes—
445C2×Q8C2×C22⋊D4712yes—
446C2×D4—24no—
447C2×D4—28no—
448C2×D4—28no—
449C2×D4—28no—
450C2×D4—22no—
451C2×D4—28no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—472no—
455C2×D4—948no—
456C2×D4—942no—
457C2×D4—942no—
458C2×D4—941no—
459C2×D4—1884no—
460C2×D4—1884no—
461C2×D4C2×C4×D4712yes—
462C2×D4#1062535012yes—
463C2×D4C2×C22⋊D4712yes—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—44no—
474C22×C4—84no—
475C22×C4—941no—
476C22×C4—941no—
477C22×C4—941no—
478C22×C4—1884no—
479C22×C4—1881no—
480C22×C4—1881no—
481C22×C4C2×C22⋊D4711yescharacteristic
482C22×C4C2×C22⋊D4712yes—
483C22×C4C2×C22⋊D4711yescharacteristic
484C22×C4C23×D4711yescharacteristic
485SD16—948no—
486D8—948no—
487M16—474no—
488M16—942no—
489M16—942no—
490M16—942no—
491C2×C8—942no—
492C2×C8—942no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—942no—
498C4⋊C4—1884no—
499C4⋊C4C2×C22⋊D4712yes—
500C4⋊C4C2×C22⋊D4712yes—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—24no—
507C22⋊C4—1884no—
508C22⋊C4—1884no—
509C22⋊C4—1882no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—472no—
515C42—944no—
516C42—941no—
517C42—942no—
518C42—1882no—
519C42C2×C22⋊D4712yes—
520C42C2×C22⋊D4712yes—
521C23—28no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—42no—
526C23—42no—
527C23—82no—
528C23—82no—
529C23—942no—
530C23—1882no—
531C23—1882no—
532C23—1881no—
533C23—1881no—
534C23C4×C22⋊D4714yes—
535C23C22×C22⋊D4711yescharacteristic
536Q8—22no—
537Q8C4×C22⋊D4714yescomplemented
538D4—216no—
539D4—24no—
540D4—24no—
541D4—24no—
542D4—24no—
543D4—22no—
544D4—944no—
545D4—18816no—
546D4—1882no—
547D4C4×C22⋊D4714yescomplemented
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—84no—
564C2×C4—82no—
565C2×C4—942no—
566C2×C4—942no—
567C2×C4—942no—
568C2×C4—942no—
569C2×C4—3762no—
570C2×C4—3762no—
571C2×C4—1882no—
572C2×C4—1881no—
573C2×C4—1881no—
574C2×C4—1881no—
575C2×C4#1309399312yes—
576C2×C4#1309399311yescharacteristic
577C2×C4#1309399311yescharacteristic
578C2×C4C22×C4×D4711yescharacteristic
579C2×C4C22×C22⋊D4712yes—
580C2×C4C2×#1062535012yes—
581C2×C4C2×#1062535011yescharacteristic
582C8—944no—
583C8—1882no—
584C22—21no—
585C22—21no—
586C22—24no—
587C22—24no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—82no—
594C22—88no—
595C22—3762no—
596C22—3761no—
597C22—3761no—
598C22—1884no—
599C22—1882no—
600C22C2×C4×C22⋊D4711yescharacteristic
601C22C2×#1309399311yescharacteristic
602C22#1568963411yescharacteristic
603C4—21no—
604C4—22no—
605C4—24no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—944no—
610C4—3761no—
611C4—1882no—
612C4#1568929112yesdirect product factor
613C4C2×C4×C22⋊D4712yes—
614C4#1568963412yes—
615C2—21no—
616C2—21no—
617C2—44no—
618C2—44no—
619C2—3762no—
620C2C4×#1309399311yescharacteristic
621C2C2×#1568929111yescharacteristic
622C2#2025273711yescharacteristic
subgroupG/H
centerC2×C4#13093993
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×H(5,𝔽2)×C47C2
socleC22×C47C2×#120
prime radicalC2×#15689291C2
cosocleC22×C4×C47C24
hypercenterC4×H(5,𝔽2)D47
hypocenterC47C4×#453
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C4×#13094049C2semidirect
3. C4×#13094020C2semidirect
4. #20243786C2semidirect
5. #20243764C2semidirect
6. #20243614C2semidirect
7. #20243598C2semidirect
8. #20243342C2semidirect
9. C4×#13093951C2semidirect
10. C4×#13093941C2semidirect
11. C4×H(5,𝔽2)×C47C2semidirect
14. C2×C4×D188C22semidirect
16. #15689291C4direct (N × H)
21. #15689166C22semidirect
24. #15689127C22semidirect
28. C4×#10625329C22semidirect
30. #15689024C22semidirect
32. #15688992C22semidirect
33. C42×Dic47C22semidirect
34. #15688892C22semidirect
35. #15688882C22semidirect
36. #15688882C22semidirect
52. #13094049C2×C4semidirect
58. #13094020C2×C4semidirect
74. #13093951C2×C4semidirect
80. #13093941C2×C4semidirect
85. H(5,𝔽2)×C47C2×C4semidirect
94. C4×Q8×C47D4semidirect
95. C4×Q8×C47D4semidirect
108. C4×D4×C47D4semidirect
109. C4×D4×C47D4semidirect
169. C2×D188C22×C4semidirect
202. C4×Dic47C22×C4semidirect
206. #10625329C22×C4semidirect
225. Q8×C47C4×D4semidirect
232. D4×C47C4×D4semidirect
298. C4×C47#453semidirect
304. C4×H(5,𝔽2)D47semidirect
359. C47C4×#453semidirect
362. H(5,𝔽2)C4×D47semidirect
382. C4×Q8C22⋊D47semidirect
383. C4×Q8C22⋊D47semidirect
399. C4×D4C22⋊D47semidirect
400. C4×D4C22⋊D47semidirect
537. Q8C4×C22⋊D47semidirect
547. D4C4×C22⋊D47semidirect
612. C4#15689291direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156892912
C4×#130940492
C4×#130940202
#202437862
#202437642
#202436142
#202435982
#202433422
C4×#130939512
C4×#130939412
C4×H(5,𝔽2)×C472
C4×#45347