Finite Group Databasebeta

← C4×#16810762 #27389683

Subgroups — C4×#16810762 #27389683

Order 13568.

1532 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810762C211yesmaximal, characteristic
2C2×C4×#11719924C212yesmaximal, complemented
3C4×#13133191C211yesmaximal, characteristic
4#20275070C212yesmaximal, complemented
5#20275032C211yesmaximal, characteristic
6#20274983C212yesmaximal, complemented
7#20274908C211yesmaximal, characteristic, complemented
8#20274650C211yesmaximal, characteristic, complemented
9C4×#13133141C212yesmaximal, complemented
10C4×C22⋊#88322C211yesmaximal, characteristic, complemented
11C22×C4×Q8×C53C211yesmaximal, characteristic, complemented
12C22×#11719924C2212yes—
13C2×#13133191C2211yescharacteristic
14C2×C4×D212C2211yescharacteristic
15C2×C4×C22⋊D53—21no—
16#16810762C418yesdirect product factor
17C4×#11719924—28no—
18C2×#13133148C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133141C2212yes—
22C2×#13133137C2212yes—
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×C4×#88322C2212yes—
26#16810525C2211yescharacteristic
27#16810521C2211yescharacteristic
28#16810514—21no—
29#16810502—21no—
30#16810483C2211yescharacteristic
31C4×C4⋊Dic53C2211yescharacteristic
32C4×#11719921—21no—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C23×Q8×C53C2211yescharacteristic
37C2×C4×Q8×C53—28no—
38C2×C4×Q8×C53—21no—
39C2×C4×Q8×C53—21no—
40C2×C4×Q8×C53C2212yes—
41C2×C4×Q8×C53C2212yes—
42C22×C4⋊C4×C53—21no—
43C22×C4⋊C4×C53C2211yescharacteristic
44C22×C42×C53—21no—
45C22×C42×C53C2211yescharacteristic
46C22×D212C2311yescharacteristic
47C22×C22⋊D53—21no—
48C22×C4×D53—21no—
49C2×#11719924—24no—
50C2×#11719924—24no—
51C2×#11719924C2×C418yescomplemented
52C22×#88322C2312yes—
53C22×#88322C2312yes—
54#13133191—22no—
55#13133191C2×C414yes—
56C4×D212—24no—
57C4×C22⋊D53—44no—
58C2×C4⋊Dic53C2311yescharacteristic
59C2×C4⋊Dic53C2311yescharacteristic
60C2×C4⋊Dic53C2311yescharacteristic
61C2×#11719919—21no—
62C2×C22⋊Dic53—21no—
63C2×#11719921—21no—
64C2×#11719921—21no—
65C2×#11719921—21no—
66C2×#11719921—21no—
67C2×C4×Dic53—21no—
68#13133148—24no—
69#13133141—24no—
70#13133141—24no—
71#13133141C2×C418yescomplemented
72#13133137—24no—
73C22⋊#88322—22no—
74C22⋊#88322C2×C414yescomplemented
75C4×#88322—24no—
76#13133124—21no—
77#13133124—21no—
78#13133124—21no—
79C22×Q8×C53—24no—
80C22×Q8×C53—24no—
81C22×Q8×C53—21no—
82C22×Q8×C53—22no—
83C22×Q8×C53C2×C414yescomplemented
84C22×Q8×C53C2312yes—
85C23×C4×C53—21no—
86C23×C4×C53—21no—
87C23×C4×C53D411yescharacteristic
88C23×C4×C53D411yescharacteristic
89C23×C4×C53C2311yescharacteristic
90C4×Q8×C53—28no—
91C4×Q8×C53—28no—
92C4×Q8×C53—24no—
93C4×Q8×C53—24no—
94C4×Q8×C53—24no—
95C4×Q8×C53D418yescomplemented
96C2×C4⋊C4×C53—28no—
97C2×C4⋊C4×C53—24no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—22no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53C2312yes—
102C2×C42×C53—28no—
103C2×C42×C53—24no—
104C2×C42×C53—22no—
105C2×C42×C53—22no—
106C2×C42×C53D411yescharacteristic
107C2×C42×C53D411yescharacteristic
108C2×C42×C53C2311yescharacteristic
109C2×C42×C53C2311yescharacteristic
110C24×C53C2×D411yescharacteristic
111C23×D53—21no—
112C2×Q8×C53—24no—
113C2×Q8×C53—216no—
114C2×Q8×C53—24no—
115C2×Q8×C53—216no—
116C2×Q8×C53—24no—
117C2×Q8×C53—22no—
118C2×Q8×C53—22no—
119C2×Q8×C53—22no—
120C2×Q8×C53C22×C414yes—
121C2×Q8×C53C2×D414yes—
122C2×Q8×C53#4114yes—
123C22×C4×C53—24no—
124C22×C4×C53—24no—
125C22×C4×C53—24no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—24no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—41no—
136C22×C4×C53—42no—
137C22×C4×C53—42no—
138C22×C4×C53SD1614yescomplemented
139C22×C4×C53C22×C412yes—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53#4112yes—
146C22×C4×C53C2411yescharacteristic
147C2×C22⋊D53—24no—
148C2×C22⋊D53—42no—
149C2×C22⋊D53—42no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C2×D212—22no—
154C2×D212—22no—
155C2×D212—21no—
156C2×D212C22×C412yes—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C4⋊C4×C53—28no—
161C4⋊C4×C53—24no—
162C4⋊C4×C53—28no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53C2×D414yes—
165C42×C53—28no—
166C42×C53—28no—
167C42×C53—22no—
168C42×C53—22no—
169C42×C53—22no—
170C42×C53C2×D414yes—
171C22⋊Dic53—42no—
172#11719924—216no—
173#11719921—24no—
174#11719921—42no—
175#11719921—42no—
176#11719921—42no—
177C4⋊Dic53—21no—
178C4⋊Dic53—22no—
179C4⋊Dic53—22no—
180C4⋊Dic53C22×C412yes—
181#11719919—42no—
182C4×Dic53—42no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322C22×C414yes—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53C4×D412yes—
194C23×C53C2×SD1612yes—
195C23×C53#13512yes—
196C23×C53C22×D411yescharacteristic
197C22×D53—22no—
198C22×D53—41no—
199C22×D53—41no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53—41no—
203C22×D53—41no—
204Q8×C53—216no—
205Q8×C53—24no—
206Q8×C53—24no—
207Q8×C53—24no—
208Q8×C53C4×D418yescomplemented
209C2×C4×C53—216no—
210C2×C4×C53—24no—
211C2×C4×C53—24no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—24no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—28no—
222C2×C4×C53—24no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—44no—
227C2×C4×C53—48no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53C4×D412yes—
231C2×C4×C53#12012yes—
232C2×C4×C53#12012yes—
233C2×C4×C53C2×SD1612yes—
234C2×C4×C53C2×SD1612yes—
235C2×C4×C53#13712yes—
236C2×C4×C53C23×C411yescharacteristic
237C2×C4×C53C22×D412yes—
238C2×C4×C53C2×#4112yes—
239C2×C4×C53C2×#4111yescharacteristic
240C22⋊D53—48no—
241C2×Dic53—22no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250D212—24no—
251D212—42no—
252C4×D53—84no—
253#88322—24no—
254#88322—42no—
255C4×#448—531nomaximal, Sylow
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53C4×SD1614yescomplemented
263C22×C53C2×C4×D411yescharacteristic
264C22×C53C2×#12011yescharacteristic
265C22×C53#52511yescharacteristic
266C22×C53C22×SD1611yescharacteristic
267C22×C53C2×#13511yescharacteristic
268C22×C53C2×#13711yescharacteristic
269C22×C53#57511yescharacteristic
270C2×D53—42no—
271C2×D53—42no—
272C2×D53—42no—
273C2×D53—81no—
274C2×D53—81no—
275C2×D53—81no—
276C2×D53—81no—
277C4×C53—22no—
278C4×C53—22no—
279C4×C53—28no—
280C4×C53—22no—
281C4×C53—41no—
282C4×C53—44no—
283C4×C53#44814yescomplemented
284C4×C53C2×C4×D412yes—
285C4×C53#52512yes—
286Dic53—42no—
287Dic53—81no—
288Dic53—81no—
289C22×C4×Q8D5311yescharacteristic, complemented
290C2×#448—531no—
291C2×C4×SD16—532no—
292C4×#121—531no—
293#1880—532no—
294#1874—531no—
295#1865—532no—
296#1853—531no—
297#1787—531no—
298C4×#103—532no—
299C4×C22⋊C8—531no—
300C2×C53—24no—
301C2×C53C4×#12011yescharacteristic
302C2×C53C2×C4×SD1611yescharacteristic
303C2×C53C4×#13711yescharacteristic
304C2×C53C2×#44811yescharacteristic
305C2×C53#300211yescharacteristic
306C2×C53#310311yescharacteristic
307C2×C53#310811yescharacteristic
308D53—82no—
309C23×Q8C2×D5311yescharacteristic
310C22×SD16—532no—
311C2×#121—531no—
312C2×C4×Q8—28no—
313C2×C4×Q8—21no—
314C2×C4×Q8—21no—
315C2×C4×Q8C2×D5312yes—
316C2×C4×Q8C2×D5312yes—
317C2×C4×D4—531no—
318C2×C4×D4—1061no—
319C22×C4⋊C4—21no—
320C22×C4⋊C4C2×D5311yescharacteristic
321C22×C42—21no—
322C22×C42C2×D5311yescharacteristic
323#448—538no—
324C4×SD16—1068no—
325C2×#106—532no—
326C2×#103—532no—
327C2×#103—532no—
328C2×#103—532no—
329C2×#102—532no—
330C2×C22⋊C8—531no—
331C2×C22⋊C8—531no—
332C2×C4×C8—532no—
333#390—531no—
334#389—531no—
335#388—1061no—
336#386—1061no—
337#380—531no—
338C4×C4⋊C4—531no—
339C4×C22⋊C4—1061no—
340#340—532no—
341#340—532no—
342#336—532no—
343C53C4×#44811yescharacteristic, complemented, Sylow
344C22×Q8—24no—
345C22×Q8—24no—
346C22×Q8—22no—
347C22×Q8—21no—
348C22×Q8C4×D5314yescomplemented
349C22×Q8C22×D5312yes—
350C22×D4—531no—
351C22×D4—1061no—
352C23×C4—21no—
353C23×C4—21no—
354C23×C4—1061no—
355C23×C4C22⋊D5311yescharacteristic
356C23×C4C22⋊D5311yescharacteristic
357C23×C4C22×D5311yescharacteristic
358C2×SD16—538no—
359C2×SD16—1064no—
360C2×SD16—1064no—
361C22×C8—532no—
362C22×C8—532no—
363#121—534no—
364#121—1062no—
365C4×Q8—28no—
366C4×Q8—28no—
367C4×Q8—24no—
368C4×Q8—24no—
369C4×Q8—24no—
370C4×Q8C22⋊D5318yescomplemented
371C4×D4—1064no—
372C4×D4—2124no—
373C2×C4⋊C4—28no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4C22×D5312yes—
383C2×C22⋊C4—1061no—
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C42—28no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—24no—
392C2×C42—1061no—
393C2×C42C22⋊D5311yescharacteristic
394C2×C42C22⋊D5311yescharacteristic
395C2×C42C22×D5311yescharacteristic
396C2×C42C22×D5311yescharacteristic
397#106—1064no—
398#103—538no—
399#103—1064no—
400#103—1064no—
401#102—1064no—
402C22⋊C8—534no—
403C22⋊C8—1062no—
404C4×C8—1064no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408C24—1061no—
409C24C2×C22⋊D5311yescharacteristic
410C2×Q8—24no—
411C2×Q8—216no—
412C2×Q8—24no—
413C2×Q8—216no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8C2×C4×D5314yes—
419C2×Q8#1171994514yes—
420C2×Q8C2×C22⋊D5314yes—
421C2×D4—532no—
422C2×D4—1064no—
423C2×D4—1062no—
424C2×D4—1062no—
425C2×D4—1061no—
426C2×D4—2122no—
427C2×D4—2122no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—2122no—
447C22×C4—2122no—
448C22×C4—2122no—
449C22×C4#1171992414yescomplemented
450C22×C4C2×C4×D5312yes—
451C22×C4#1171994512yes—
452C22×C4C2×C22⋊D5311yescharacteristic
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C23×D5311yescharacteristic
458SD16—10616no—
459C2×C8—534no—
460C2×C8—1062no—
461C2×C8—1062no—
462C2×C8—1062no—
463C2×C8—1062no—
464C2×C8—1062no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—28no—
468C4⋊C4—22no—
469C4⋊C4—532no—
470C4⋊C4—1061no—
471C4⋊C4—1062no—
472C4⋊C4—1062no—
473C4⋊C4—2122no—
474C4⋊C4C2×C22⋊D5314yes—
475C22⋊C4—1064no—
476C22⋊C4—2122no—
477C22⋊C4—2122no—
478C22⋊C4—2122no—
479C22⋊C4—2122no—
480C42—28no—
481C42—28no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—2122no—
486C42C2×C22⋊D5314yes—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—1062no—
492C23—2121no—
493C23—2121no—
494C23—2121no—
495C23—2121no—
496C23—2121no—
497C23—2121no—
498C23C4×C22⋊D5312yes—
499C23C2×#1171992412yes—
500C23#1313323712yes—
501C23C22×C22⋊D5311yescharacteristic
502Q8—216no—
503Q8—24no—
504Q8—24no—
505Q8—24no—
506Q8C4×C22⋊D5318yescomplemented
507D4—1064no—
508D4—2128no—
509D4—2122no—
510C2×C4—216no—
511C2×C4—24no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—21no—
524C2×C4—24no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—44no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—1062no—
532C2×C4—4244no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4C4×C22⋊D5312yes—
542C2×C4#1313318612yes—
543C2×C4#1313318612yes—
544C2×C4C2×#1171992412yes—
545C2×C4C2×#1171992412yes—
546C2×C4#1313324812yes—
547C2×C4C22×C4×D5311yescharacteristic
548C2×C4C22×C22⋊D5312yes—
549C2×C4C2×#1171994512yes—
550C2×C4C2×#1171994511yescharacteristic
551C8—1064no—
552C8—2122no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—4241no—
560C22—4241no—
561C22—4241no—
562C22—4241no—
563C22—2122no—
564C22—2122no—
565C22—2122no—
566C22C4×#1171992414yescomplemented
567C22C2×C4×C22⋊D5311yescharacteristic
568C22C2×#1313318611yescharacteristic
569C22#1681113411yescharacteristic
570C22C22×#1171992411yescharacteristic
571C22C2×#1313323711yescharacteristic
572C22C2×#1313324811yescharacteristic
573C22#1681142111yescharacteristic
574C4—22no—
575C4—28no—
576C4—22no—
577C4—22no—
578C4—41no—
579C4—44no—
580C4—4241no—
581C4—4241no—
582C4—2122no—
583C4#1681076214yesdirect product factor
584C4C2×C4×C22⋊D5312yes—
585C4#1681113412yes—
586C2—24no—
587C2—4242no—
588C2C4×#1313318611yescharacteristic
589C2C2×C4×#1171992411yescharacteristic
590C2C4×#1313324811yescharacteristic
591C2C2×#1681076211yescharacteristic
592C2#2028393311yescharacteristic
593C2#2028474611yescharacteristic
594C2#2028479511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC22×C4×Q8×C53C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
cosocleC22×C4×C53C24
hypercenterC22×C4×Q8D53
hypocenterC53C4×#448
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11719924C2semidirect
4. #20275070C2semidirect
6. #20274983C2semidirect
7. #20274908C2semidirect
8. #20274650C2semidirect
9. C4×#13133141C2semidirect
10. C4×C22⋊#88322C2semidirect
11. C22×C4×Q8×C53C2semidirect
16. #16810762C4direct (N × H)
51. C2×#11719924C2×C4semidirect
71. #13133141C2×C4semidirect
74. C22⋊#88322C2×C4semidirect
83. C22×Q8×C53C2×C4semidirect
95. C4×Q8×C53D4semidirect
138. C22×C4×C53SD16semidirect
208. Q8×C53C4×D4semidirect
262. C22×C53C4×SD16semidirect
283. C4×C53#448semidirect
289. C22×C4×Q8D53semidirect
343. C53C4×#448semidirect
348. C22×Q8C4×D53semidirect
370. C4×Q8C22⋊D53semidirect
449. C22×C4#11719924semidirect
506. Q8C4×C22⋊D53semidirect
566. C22C4×#11719924semidirect
583. C4#16810762direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168107622
C2×C4×#117199242
C4×#131331912
#202750702
#202750322
#202749832
#202749082
#202746502
C4×#131331412
C4×C22⋊#883222
C22×C4×Q8×C532
C4×#44853