Finite Group Databasebeta

← C4×#16810791 #27389966

Subgroups — C4×#16810791 #27389966

Order 13568.

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×#16810791C211yesmaximal, characteristic
2C4×#13133242C212yesmaximal, complemented
3C4×#13133213C211yesmaximal, characteristic, complemented
4#20275113C211yesmaximal, characteristic, complemented
5#20275091C212yesmaximal, complemented
6#20274941C212yesmaximal, complemented
7#20274925C211yesmaximal, characteristic, complemented
8#20274669C211yesmaximal, characteristic, complemented
9C4×#13133144C212yesmaximal, complemented
10C4×#13133134C211yesmaximal, characteristic, complemented
11C4×H(5,𝔽2)×C53C211yesmaximal, characteristic, complemented
12C2×#13133242C2212yes—
13C2×#13133213C2211yescharacteristic
14C2×C4×D212C2211yescharacteristic, complemented
15C2×C4×C22⋊D53—22no—
16#16810791C418yesdirect product factor
17#16810733—22no—
18#16810721—22no—
19C4×#11719924—22no—
20C4×#11719922—22no—
21#16810666C2212yescomplemented
22C2×#13133144C2212yes—
23C2×#13133144C2212yes—
24#16810627C2212yescomplemented
25#16810624C2212yes—
26C2×#13133134C2211yescharacteristic
27C2×#13133134C2211yescharacteristic
28C4×#11719917C2212yescomplemented
29#16810526C2211yescharacteristic
30#16810524C2211yescharacteristic, complemented
31#16810523—22no—
32#16810492C2211yescharacteristic, complemented
33C42×Dic53C2211yescharacteristic, complemented
34#16810392C2212yescomplemented
35#16810382C2212yescomplemented
36#16810382C2212yescomplemented
37C2×H(5,𝔽2)×C53C2211yescharacteristic
38C53×#520—22no—
39C53×#520C2212yes—
40C53×#518—22no—
41C53×#518—22no—
42C53×#518C2211yescharacteristic
43C4×#41×C53—22no—
44C4×#41×C53C2212yes—
45C2×C4×D4×C53—22no—
46C2×C4×D4×C53—22no—
47C2×C4×D4×C53C2211yescharacteristic
48C22×D212C2311yescharacteristic
49C22×C22⋊D53—22no—
50C22×C4×D53—21no—
51#13133242—24no—
52#13133242C2×C418yescomplemented
53C2×#11719924—22no—
54C2×#11719922—22no—
55C2×#11719917C2312yes—
56C2×#11719917C2312yes—
57#13133213—22no—
58#13133213C2×C414yescomplemented
59C4×D212—22no—
60C4×D212—22no—
61C4×C22⋊D53—48no—
62C2×C4⋊Dic53C2311yescharacteristic
63C2×#11719919—22no—
64C2×C22⋊Dic53—22no—
65C2×#11719921—21no—
66C2×#11719921—22no—
67C2×C4×Dic53—21no—
68C2×C4×Dic53—21no—
69C2×C4×Dic53C2311yescharacteristic
70C2×C4×Dic53C2311yescharacteristic
71#13133150—22no—
72#13133148—22no—
73#13133144—24no—
74#13133144C2×C418yescomplemented
75#13133141—22no—
76#13133138—22no—
77#13133137—22no—
78#13133137—22no—
79#13133134—22no—
80#13133134C2×C414yescomplemented
81#13133128—22no—
82C4×#88322—22no—
83H(5,𝔽2)×C53—24no—
84H(5,𝔽2)×C53—22no—
85H(5,𝔽2)×C53C2×C414yescomplemented
86C2×#41×C53—22no—
87C2×#41×C53C2312yes—
88C22×D4×C53—22no—
89C22×D4×C53—22no—
90C22×D4×C53C2311yescharacteristic
91C23×C4×C53—22no—
92C23×C4×C53D412yes—
93C4×Q8×C53—22no—
94C4×Q8×C53D412yescomplemented
95C4×Q8×C53D412yescomplemented
96C4×D4×C53—24no—
97C4×D4×C53—24no—
98C4×D4×C53—24no—
99C4×D4×C53—24no—
100C4×D4×C53—22no—
101C4×D4×C53—22no—
102C4×D4×C53—22no—
103C4×D4×C53—22no—
104C4×D4×C53—22no—
105C4×D4×C53—22no—
106C4×D4×C53—24no—
107C4×D4×C53—22no—
108C4×D4×C53D412yescomplemented
109C4×D4×C53D412yescomplemented
110#117×C53—24no—
111#117×C53—22no—
112#117×C53—22no—
113#117×C53C2312yes—
114C2×C4⋊C4×C53—22no—
115C2×C4⋊C4×C53—22no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—22no—
118C2×C22⋊C4×C53—24no—
119C2×C22⋊C4×C53—22no—
120C2×C22⋊C4×C53D412yes—
121C2×C42×C53—22no—
122C2×C42×C53—22no—
123C2×C42×C53C2311yescharacteristic
124C24×C53—22no—
125C24×C53C2×D412yes—
126C23×D53—21no—
127#41×C53—28no—
128#41×C53—28no—
129#41×C53—22no—
130#41×C53—22no—
131#41×C53C22×C414yes—
132#41×C53#4114yes—
133C2×Q8×C53C2×D412yes—
134C2×D4×C53—24no—
135C2×D4×C53—28no—
136C2×D4×C53—28no—
137C2×D4×C53—28no—
138C2×D4×C53—22no—
139C2×D4×C53—28no—
140C2×D4×C53—22no—
141C2×D4×C53—22no—
142C2×D4×C53C22×C412yes—
143C2×D4×C53C2×D412yes—
144C2×D4×C53#4112yes—
145C22×C4×C53—22no—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—22no—
149C22×C4×C53—24no—
150C22×C4×C53—22no—
151C22×C4×C53—22no—
152C22×C4×C53—22no—
153C22×C4×C53—22no—
154C22×C4×C53—44no—
155C22×C4×C53—84no—
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D412yes—
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2411yescharacteristic
160C2×C22⋊D53—28no—
161C2×C22⋊D53—44no—
162C2×C22⋊D53—44no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—21no—
166C2×D212—22no—
167C2×D212—22no—
168C2×D212—21no—
169C2×D212C22×C412yescomplemented
170C2×C4×D53—44no—
171C2×C4×D53—41no—
172C2×C4×D53—41no—
173C4⋊C4×C53—24no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53C2×D412yes—
178C4⋊C4×C53C2×D412yes—
179C22⋊C4×C53—24no—
180C22⋊C4×C53—24no—
181C22⋊C4×C53—22no—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—24no—
185C42×C53—24no—
186C42×C53—22no—
187C42×C53—22no—
188C42×C53—22no—
189C42×C53C2×D412yes—
190C42×C53C2×D412yes—
191C22⋊Dic53—44no—
192#11719924—28no—
193#11719922—28no—
194#11719921—44no—
195#11719921—42no—
196C4⋊Dic53—22no—
197#11719919—44no—
198C4×Dic53—24no—
199C4×Dic53—21no—
200C4×Dic53—22no—
201C4×Dic53—42no—
202C4×Dic53C22×C412yescomplemented
203#11719917—22no—
204#11719917—22no—
205#11719917—22no—
206#11719917C22×C414yescomplemented
207C2×#88322—22no—
208C2×#88322—22no—
209C23×C53—28no—
210C23×C53—22no—
211C23×C53—22no—
212C23×C53—22no—
213C23×C53—42no—
214C23×C53—42no—
215C23×C53—82no—
216C23×C53—82no—
217C23×C53C4×D414yes—
218C23×C53C22×D411yescharacteristic
219C22×D53—22no—
220C22×D53—42no—
221C22×D53—42no—
222C22×D53—41no—
223C22×D53—41no—
224Q8×C53—22no—
225Q8×C53C4×D414yescomplemented
226D4×C53—216no—
227D4×C53—24no—
228D4×C53—24no—
229D4×C53—24no—
230D4×C53—22no—
231D4×C53—24no—
232D4×C53C4×D414yescomplemented
233C2×C4×C53—28no—
234C2×C4×C53—22no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—41no—
242C2×C4×C53—44no—
243C2×C4×C53—44no—
244C2×C4×C53—42no—
245C2×C4×C53—82no—
246C2×C4×C53—82no—
247C2×C4×C53—82no—
248C2×C4×C53—84no—
249C2×C4×C53—82no—
250C2×C4×C53#12011yescharacteristic
251C2×C4×C53#12012yes—
252C2×C4×C53#12011yescharacteristic
253C2×C4×C53C23×C411yescharacteristic
254C2×C4×C53C22×D412yes—
255C2×C4×C53C2×#4112yes—
256C2×C4×C53C2×#4111yescharacteristic
257C22⋊D53—416no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—42no—
265C2×Dic53—41no—
266D212—24no—
267D212—42no—
268C4×D53—82no—
269C4×D53—82no—
270#88322—24no—
271#88322—42no—
272C4×#453—531nomaximal, Sylow
273C22×C53—21no—
274C22×C53—21no—
275C22×C53—24no—
276C22×C53—24no—
277C22×C53—44no—
278C22×C53—42no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—82no—
283C22×C53—88no—
284C22×C53C2×C4×D411yescharacteristic
285C22×C53C2×#12011yescharacteristic
286C22×C53#52511yescharacteristic
287C2×D53—44no—
288C2×D53—42no—
289C2×D53—82no—
290C2×D53—81no—
291C2×D53—81no—
292C4×C53—21no—
293C4×C53—22no—
294C4×C53—24no—
295C4×C53—42no—
296C4×C53—42no—
297C4×C53—42no—
298C4×C53#45312yescomplemented
299C4×C53C2×C4×D412yes—
300C4×C53#52512yes—
301Dic53—24no—
302Dic53—42no—
303Dic53—81no—
304C4×H(5,𝔽2)D5311yescharacteristic, complemented
305C2×#453—531no—
306C4×AGL(1,ℤ/8)—532no—
307C4×#127—531no—
308#1887—531no—
309#1883—532no—
310#1859—532no—
311#1856—531no—
312#1792—531no—
313C4×#104—532no—
314C4×#100—531no—
315C2×C53—21no—
316C2×C53—21no—
317C2×C53—44no—
318C2×C53—44no—
319C2×C53C4×#12011yescharacteristic
320C2×C53C2×#45311yescharacteristic
321C2×C53#301811yescharacteristic
322D53—82no—
323C2×H(5,𝔽2)C2×D5311yescharacteristic
324C2×AGL(1,ℤ/8)—532no—
325C2×#127—531no—
326#520—22no—
327#520C2×D5312yes—
328#518—22no—
329#518—22no—
330#518C2×D5311yescharacteristic
331C4×#41—22no—
332C4×#41C2×D5312yes—
333C2×C4×D4—22no—
334C2×C4×D4—22no—
335C2×C4×D4—531no—
336C2×C4×D4—1062no—
337C2×C4×D4C2×D5311yescharacteristic
338#453—538no—
339#442—1062no—
340#440—1062no—
341C4×SD16—1062no—
342C4×D8—1062no—
343#428—532no—
344C2×#104—532no—
345C2×#104—532no—
346#418—532no—
347#417—532no—
348C2×#100—531no—
349C2×#100—531no—
350C4×M16—532no—
351#390—531no—
352#390—531no—
353#390—1062no—
354#384—531no—
355C43—531no—
356#343—532no—
357#339—532no—
358#339—532no—
359C53C4×#45311yescharacteristic, complemented, Sylow
360H(5,𝔽2)—24no—
361H(5,𝔽2)—22no—
362H(5,𝔽2)C4×D5314yescomplemented
363C2×#41—22no—
364C2×#41C22×D5312yes—
365C22×D4—22no—
366C22×D4—22no—
367C22×D4—531no—
368C22×D4—1062no—
369C22×D4C22×D5311yescharacteristic
370C23×C4—22no—
371C23×C4—1061no—
372C23×C4C22⋊D5312yes—
373AGL(1,ℤ/8)—538no—
374AGL(1,ℤ/8)—1064no—
375C2×SD16—1062no—
376C2×D8—1062no—
377C2×M16—532no—
378C2×M16—532no—
379#127—534no—
380#127—1062no—
381C4×Q8—22no—
382C4×Q8C22⋊D5312yescomplemented
383C4×Q8C22⋊D5312yescomplemented
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—1062no—
397C4×D4—1062no—
398C4×D4—2128no—
399C4×D4C22⋊D5312yescomplemented
400C4×D4C22⋊D5312yescomplemented
401#117—24no—
402#117—22no—
403#117—22no—
404#117C22×D5312yes—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—1062no—
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C22⋊C4—22no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—1062no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1062no—
416C2×C22⋊C4C22⋊D5312yes—
417C2×C42—22no—
418C2×C42—22no—
419C2×C42—531no—
420C2×C42—531no—
421C2×C42—1061no—
422C2×C42—1061no—
423C2×C42C22×D5311yescharacteristic
424#107—1062no—
425#106—1062no—
426#104—538no—
427#104—1064no—
428#103—1062no—
429#102—1062no—
430#102—1062no—
431#102—1062no—
432#100—534no—
433#100—1062no—
434#97—1062no—
435C4×C8—1062no—
436C24—22no—
437C24—1061no—
438C24C2×C22⋊D5312yes—
439#41—28no—
440#41—28no—
441#41—22no—
442#41—22no—
443#41C2×C4×D5314yes—
444#41#1171994514yes—
445C2×Q8C2×C22⋊D5312yes—
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—532no—
455C2×D4—1068no—
456C2×D4—1062no—
457C2×D4—1062no—
458C2×D4—1061no—
459C2×D4—2124no—
460C2×D4—2124no—
461C2×D4C2×C4×D5312yes—
462C2×D4#1171994512yes—
463C2×D4C2×C22⋊D5312yes—
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—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—2124no—
479C22×C4—2121no—
480C22×C4—2121no—
481C22×C4C2×C22⋊D5311yescharacteristic
482C22×C4C2×C22⋊D5312yes—
483C22×C4C2×C22⋊D5311yescharacteristic
484C22×C4C23×D5311yescharacteristic
485SD16—1068no—
486D8—1068no—
487M16—534no—
488M16—1062no—
489M16—1062no—
490M16—1062no—
491C2×C8—1062no—
492C2×C8—1062no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—1062no—
498C4⋊C4—2124no—
499C4⋊C4C2×C22⋊D5312yes—
500C4⋊C4C2×C22⋊D5312yes—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—24no—
507C22⋊C4—2124no—
508C22⋊C4—2124no—
509C22⋊C4—2122no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—532no—
515C42—1064no—
516C42—1061no—
517C42—1062no—
518C42—2122no—
519C42C2×C22⋊D5312yes—
520C42C2×C22⋊D5312yes—
521C23—28no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—42no—
526C23—42no—
527C23—82no—
528C23—82no—
529C23—1062no—
530C23—2122no—
531C23—2122no—
532C23—2121no—
533C23—2121no—
534C23C4×C22⋊D5314yes—
535C23C22×C22⋊D5311yescharacteristic
536Q8—22no—
537Q8C4×C22⋊D5314yescomplemented
538D4—216no—
539D4—24no—
540D4—24no—
541D4—24no—
542D4—24no—
543D4—22no—
544D4—1064no—
545D4—21216no—
546D4—2122no—
547D4C4×C22⋊D5314yescomplemented
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—1062no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—4242no—
570C2×C4—4242no—
571C2×C4—2122no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4#1313318612yes—
576C2×C4#1313318611yescharacteristic
577C2×C4#1313318611yescharacteristic
578C2×C4C22×C4×D5311yescharacteristic
579C2×C4C22×C22⋊D5312yes—
580C2×C4C2×#1171994512yes—
581C2×C4C2×#1171994511yescharacteristic
582C8—1064no—
583C8—2122no—
584C22—21no—
585C22—21no—
586C22—24no—
587C22—24no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—82no—
594C22—88no—
595C22—4242no—
596C22—4241no—
597C22—4241no—
598C22—2124no—
599C22—2122no—
600C22C2×C4×C22⋊D5311yescharacteristic
601C22C2×#1313318611yescharacteristic
602C22#1681113411yescharacteristic
603C4—21no—
604C4—22no—
605C4—24no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—1064no—
610C4—4241no—
611C4—2122no—
612C4#1681079112yesdirect product factor
613C4C2×C4×C22⋊D5312yes—
614C4#1681113412yes—
615C2—21no—
616C2—21no—
617C2—44no—
618C2—44no—
619C2—4242no—
620C2C4×#1313318611yescharacteristic
621C2C2×#1681079111yescharacteristic
622C2#2028406411yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×H(5,𝔽2)×C53C2
socleC22×C53C2×#120
prime radicalC2×#16810791C2
cosocleC22×C4×C53C24
hypercenterC4×H(5,𝔽2)D53
hypocenterC53C4×#453
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
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×#13133242C2semidirect
3. C4×#13133213C2semidirect
4. #20275113C2semidirect
5. #20275091C2semidirect
6. #20274941C2semidirect
7. #20274925C2semidirect
8. #20274669C2semidirect
9. C4×#13133144C2semidirect
10. C4×#13133134C2semidirect
11. C4×H(5,𝔽2)×C53C2semidirect
14. C2×C4×D212C22semidirect
16. #16810791C4direct (N × H)
21. #16810666C22semidirect
24. #16810627C22semidirect
28. C4×#11719917C22semidirect
30. #16810524C22semidirect
32. #16810492C22semidirect
33. C42×Dic53C22semidirect
34. #16810392C22semidirect
35. #16810382C22semidirect
36. #16810382C22semidirect
52. #13133242C2×C4semidirect
58. #13133213C2×C4semidirect
74. #13133144C2×C4semidirect
80. #13133134C2×C4semidirect
85. H(5,𝔽2)×C53C2×C4semidirect
94. C4×Q8×C53D4semidirect
95. C4×Q8×C53D4semidirect
108. C4×D4×C53D4semidirect
109. C4×D4×C53D4semidirect
169. C2×D212C22×C4semidirect
202. C4×Dic53C22×C4semidirect
206. #11719917C22×C4semidirect
225. Q8×C53C4×D4semidirect
232. D4×C53C4×D4semidirect
298. C4×C53#453semidirect
304. C4×H(5,𝔽2)D53semidirect
359. C53C4×#453semidirect
362. H(5,𝔽2)C4×D53semidirect
382. C4×Q8C22⋊D53semidirect
383. C4×Q8C22⋊D53semidirect
399. C4×D4C22⋊D53semidirect
400. C4×D4C22⋊D53semidirect
537. Q8C4×C22⋊D53semidirect
547. D4C4×C22⋊D53semidirect
612. C4#16810791direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168107912
C4×#131332422
C4×#131332132
#202751132
#202750912
#202749412
#202749252
#202746692
C4×#131331442
C4×#131331342
C4×H(5,𝔽2)×C532
C4×#45353