Finite Group Databasebeta

← #27289684

Subgroups — #27289684

Order 13568.

1164 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×#16811389C211yesmaximal, characteristic
2#20283235C211yesmaximal, characteristic
3#20283232C211yesmaximal, characteristic
4C2×#16810395C211yesmaximal, characteristic, complemented
5C2×#16810381C211yesmaximal, characteristic, complemented
6C2×#16810379C211yesmaximal, characteristic, complemented
7C2×C53×#344C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9#16811389—22no—
10#16811389—22no—
11#16811389—22no—
12#16811389—22no—
13C2×#13133225C414yescomplemented
14C2×#13133223—22no—
15C22×#11719917C2211yescharacteristic
16C22×#11719917C2211yescharacteristic
17C2×M16×D53—22no—
18C2×#13133176C2211yescharacteristic
19#16810633—21no—
20#16810631—21no—
21#16810629—21no—
22#16810627—21no—
23#16810625—22no—
24#16810622—22no—
25#16810620C414yescomplemented
26#16810617C414yescomplemented
27#16810395—22no—
28#16810381—22no—
29#16810379—22no—
30C22×M16×C53C2211yescharacteristic
31C22×M16×C53C2211yescharacteristic
32C2×#117×C53C2211yescharacteristic
33C53×#344—22no—
34C2×#11719945—21no—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945C2×C412yes—
41C2×#11719945C2×C412yes—
42C2×#11719945C2×C412yes—
43C2×#11719945D412yes—
44C2×#11719945D412yes—
45C2×#11719945Q812yes—
46C22×Dic106D411yescharacteristic
47C22×D212D411yescharacteristic
48C22×C22⋊D53—21no—
49C22×C4×D53—21no—
50#13133225—28no—
51#13133225—28no—
52#13133223—28no—
53#13133223—28no—
54C2×#11719917—21no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57C2×#11719917—21no—
58C2×#11719917—21no—
59C2×#11719917—21no—
60C2×#11719917—21no—
61C2×#11719917—21no—
62C2×#11719917—21no—
63C2×#11719917—21no—
64C2×#11719917D412yes—
65C2×#11719917D412yes—
66C2×#11719912—24no—
67M16×D53—28no—
68M16×D53—28no—
69C22×#88322C2×C412yes—
70C22×#88322C2×C412yes—
71C2×C8×D53—24no—
72#13133176—21no—
73#13133176—21no—
74#13133176—21no—
75#13133176C2×C412yes—
76C2×C4⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C4×Dic53—21no—
79#13133142—24no—
80#13133140—24no—
81#13133139—24no—
82#13133137—24no—
83C23×C4×C53C2311yescharacteristic
84C2×M16×C53—21no—
85C2×M16×C53—21no—
86C2×M16×C53—21no—
87C2×M16×C53—21no—
88C2×M16×C53—21no—
89C2×M16×C53—21no—
90C2×M16×C53—21no—
91C2×M16×C53—21no—
92C2×M16×C53—21no—
93C2×M16×C53C2×C412yes—
94C2×M16×C53D412yes—
95C2×M16×C53D412yes—
96C22×C8×C53—21no—
97C22×C8×C53C2×C412yes—
98#117×C53—21no—
99#117×C53—21no—
100#117×C53—21no—
101#117×C53C2×C412yes—
102C2×C4⋊C4×C53—21no—
103C2×C22⋊C4×C53—21no—
104C2×C42×C53—21no—
105C24×C53#4111yescharacteristic
106C23×D53—21no—
107C22×C4×C53—41no—
108C22×C4×C53C22×C411yescharacteristic
109C22×C4×C53C22×C411yescharacteristic
110C22×C4×C53C22×C411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×Q811yescharacteristic
119C22×C4×C53#4111yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C22×C4×C53#4111yescharacteristic
122C2×C22⋊D53—22no—
123C2×C22⋊D53—22no—
124C2×C22⋊D53—22no—
125C2×C22⋊D53—22no—
126C2×C22⋊D53—22no—
127C2×C22⋊D53—22no—
128C22×Dic53—21no—
129C22×Dic53—41no—
130#11719945—22no—
131#11719945—22no—
132#11719945—22no—
133#11719945—22no—
134#11719945—22no—
135#11719945—22no—
136#11719945—22no—
137#11719945—22no—
138#11719945—22no—
139#11719945—22no—
140#11719945—22no—
141#11719945—22no—
142#11719945C4214yes—
143#11719945C22⋊C414yes—
144#11719945C4⋊C414yes—
145#11719945C4⋊C414yes—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212—21no—
149C2×D212C22⋊C412yes—
150C2×D212C22⋊C412yes—
151C2×D212C4⋊C412yes—
152C2×C4×D53—22no—
153C2×C4×D53—22no—
154C2×C4×D53—22no—
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106C22⋊C412yes—
162C2×Dic106C22⋊C412yes—
163C2×Dic106C4⋊C412yes—
164M16×C53—22no—
165M16×C53—22no—
166M16×C53—22no—
167M16×C53—41no—
168M16×C53—41no—
169M16×C53—41no—
170M16×C53—41no—
171M16×C53C22⋊C414yes—
172C2×C8×C53—22no—
173C2×C8×C53—22no—
174C2×C8×C53—22no—
175C2×C8×C53—22no—
176C2×C8×C53—22no—
177C2×C8×C53—22no—
178C4⋊C4×C53—22no—
179C4⋊C4×C53—41no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—41no—
182C42×C53—22no—
183C42×C53—41no—
184C22⋊Dic53—22no—
185C22⋊Dic53—41no—
186C4⋊Dic53—22no—
187C4⋊Dic53—41no—
188C4×Dic53—22no—
189C4×Dic53—41no—
190#11719917—22no—
191#11719917—22no—
192#11719917—22no—
193#11719917—22no—
194#11719917—41no—
195#11719917—41no—
196#11719917—41no—
197#11719917—41no—
198C2×#88322—22no—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—22no—
203C2×#88322C22⋊C414yes—
204#11719912—216no—
205C8×D53—216no—
206C23×C53—81no—
207C23×C53#11711yescharacteristic
208C23×C53C4×D411yescharacteristic
209C23×C53C4×D411yescharacteristic
210C23×C53#12111yescharacteristic
211C23×C53C22⋊Q811yescharacteristic
212C23×C53#12311yescharacteristic
213C23×C53#12311yescharacteristic
214C22×D53—22no—
215C22×D53—22no—
216C22×D53—22no—
217C22×D53—82no—
218C2×C4×C53—42no—
219C2×C4×C53—82no—
220C2×C4×C53C2×C4211yescharacteristic
221C2×C4×C53C2×C22⋊C411yescharacteristic
222C2×C4×C53C2×C22⋊C411yescharacteristic
223C2×C4×C53C2×C22⋊C411yescharacteristic
224C2×C4×C53C2×C22⋊C411yescharacteristic
225C2×C4×C53C2×C22⋊C411yescharacteristic
226C2×C4×C53C2×C4⋊C411yescharacteristic
227C2×C4×C53C2×C4⋊C411yescharacteristic
228C2×C4×C53C2×C4⋊C411yescharacteristic
229C2×C4×C53#11711yescharacteristic
230C2×C4×C53C4×D411yescharacteristic
231C2×C4×C53C4×D411yescharacteristic
232C2×C4×C53C4×D411yescharacteristic
233C2×C4×C53C4×D411yescharacteristic
234C2×C4×C53C4×D411yescharacteristic
235C2×C4×C53C4×D411yescharacteristic
236C2×C4×C53#12011yescharacteristic
237C2×C4×C53#12011yescharacteristic
238C2×C4×C53#12011yescharacteristic
239C2×C4×C53#12011yescharacteristic
240C2×C4×C53#12111yescharacteristic
241C2×C4×C53#12111yescharacteristic
242C2×C4×C53#12111yescharacteristic
243C2×C4×C53C22⋊Q811yescharacteristic
244C2×C4×C53C22⋊Q811yescharacteristic
245C2×C4×C53C22⋊Q811yescharacteristic
246C2×C4×C53#12311yescharacteristic
247C2×C4×C53#12311yescharacteristic
248C22⋊D53—24no—
249C22⋊D53—24no—
250C22⋊D53—24no—
251C22⋊D53—24no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—42no—
256C2×Dic53—82no—
257D212—22no—
258D212—22no—
259D212—22no—
260D212#9514yes—
261C4×D53—24no—
262C4×D53—24no—
263C4×D53—24no—
264C4×D53—24no—
265Dic106—22no—
266Dic106—22no—
267Dic106—22no—
268Dic106#9514yes—
269C8×C53—24no—
270C8×C53—42no—
271#88322—24no—
272#88322—42no—
273#10793—531nomaximal, Sylow
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53C4×C22⋊C411yescharacteristic
282C22×C53#38011yescharacteristic
283C22×C53#38111yescharacteristic
284C22×C53#38511yescharacteristic
285C22×C53#38511yescharacteristic
286C22×C53#38611yescharacteristic
287C22×C53#38611yescharacteristic
288C2×D53—24no—
289C2×D53—44no—
290C2×D53—82no—
291C2×D53—82no—
292C4×C53—82no—
293C4×C53C2×#9511yescharacteristic
294C4×C53C4×C22⋊C411yescharacteristic
295C4×C53#37911yescharacteristic
296C4×C53#38011yescharacteristic
297C4×C53#38511yescharacteristic
298C4×C53#38511yescharacteristic
299C4×C53#38611yescharacteristic
300C4×C53#38611yescharacteristic
301Dic53—24no—
302Dic53—82no—
303C2×#568—531no—
304#2896—531no—
305#2896—531no—
306C2×#344—531no—
307C2×#344D5311yescharacteristic, complemented
308C2×#338—531no—
309C2×H(3,ℤ/4)—531no—
310C2×C53—21no—
311C2×C53—21no—
312C2×C53—21no—
313C2×C53—21no—
314C2×C53—21no—
315C2×C53—21no—
316C2×C53—21no—
317C2×C53C22⋊#9511yescharacteristic
318D53—44no—
319C22×#41—531no—
320#568—1062no—
321#568—1062no—
322#568—1062no—
323#568—1062no—
324C2×#131—534no—
325C2×#131—1062no—
326C22×M16—531no—
327C22×M16—531no—
328C22×M16—1062no—
329C22×M16C2×D5311yescharacteristic
330C22×M16C2×D5311yescharacteristic
331C2×#117—531no—
332C2×#117C2×D5311yescharacteristic
333#419—1061no—
334#419—1061no—
335#418—1061no—
336#418—1061no—
337#417—1062no—
338#417—1062no—
339#416—534no—
340#416—534no—
341#344—22no—
342#344—1062no—
343#338—1062no—
344H(3,ℤ/4)—1062no—
345C53#1079311yescharacteristic, complemented, Sylow
346C2×#41—532no—
347C2×#41—532no—
348C2×#41—532no—
349C2×#41—532no—
350C2×#41—532no—
351C2×#41—532no—
352C2×#41—1061no—
353C2×#41—1061no—
354C2×#41—1061no—
355C2×#41—1061no—
356C2×#41—1061no—
357C2×#41—1061no—
358C22×Q8—531no—
359C22×D4—531no—
360C22×D4—1061no—
361C23×C4—1061no—
362C23×C4C22×D5311yescharacteristic
363#131—1068no—
364#131—1068no—
365#131—1068no—
366#131—1068no—
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—21no—
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—21no—
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—532no—
377C2×M16—532no—
378C2×M16—1064no—
379C2×M16—1068no—
380C2×M16—1068no—
381C2×M16—1061no—
382C2×M16—1061no—
383C2×M16—1061no—
384C2×M16—1061no—
385C2×M16—1061no—
386C2×M16—1061no—
387C2×M16—1061no—
388C2×M16—1061no—
389C2×M16—1061no—
390C2×M16—1061no—
391C2×M16C4×D5312yes—
392C2×M16D21212yes—
393C2×M16C22⋊D5312yes—
394C22×C8—21no—
395C22×C8—532no—
396C22×C8—532no—
397C22×C8—1064no—
398C22×C8C4×D5312yes—
399#117—21no—
400#117—21no—
401#117—21no—
402#117—532no—
403#117—1061no—
404#117—1061no—
405#117—1061no—
406#117C4×D5312yes—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—1061no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1061no—
411C2×C42—21no—
412C2×C42—1061no—
413#103—1064no—
414#103—1064no—
415#102—1064no—
416#102—1064no—
417C24—1061no—
418C24#1171994511yescharacteristic
419#41—534no—
420#41—534no—
421#41—534no—
422#41—534no—
423#41—1062no—
424#41—1062no—
425#41—1062no—
426#41—1062no—
427#41—1062no—
428#41—1062no—
429#41—1062no—
430#41—1062no—
431#41—1062no—
432#41—1062no—
433#41—1062no—
434#41—1062no—
435C2×Q8—532no—
436C2×Q8—532no—
437C2×Q8—532no—
438C2×Q8—1061no—
439C2×Q8—1061no—
440C2×Q8—1061no—
441C2×D4—532no—
442C2×D4—532no—
443C2×D4—532no—
444C2×D4—1062no—
445C2×D4—1062no—
446C2×D4—1062no—
447C2×D4—1062no—
448C2×D4—1062no—
449C2×D4—1062no—
450C2×D4—1061no—
451C2×D4—1061no—
452C2×D4—1061no—
453C22×C4—41no—
454C22×C4—1061no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—2121no—
462C22×C4C2×C4×D5311yescharacteristic
463C22×C4C2×C4×D5311yescharacteristic
464C22×C4C2×C4×D5311yescharacteristic
465C22×C4C2×D21211yescharacteristic
466C22×C4#1171994511yescharacteristic
467C22×C4D4×D5311yescharacteristic
468C22×C4D4×D5311yescharacteristic
469C22×C4D4×D5311yescharacteristic
470C22×C4D4×D5311yescharacteristic
471C22×C4D4×D5311yescharacteristic
472C22×C4#1171994711yescharacteristic
473C22×C4Q8×D5311yescharacteristic
474C22×C4#1171994911yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476M16—22no—
477M16—22no—
478M16—22no—
479M16—41no—
480M16—41no—
481M16—41no—
482M16—41no—
483M16—10616no—
484M16—1062no—
485M16—1062no—
486M16—1062no—
487M16—1062no—
488M16—2121no—
489M16—2121no—
490M16—2121no—
491M16—2121no—
492M16#1171992114yes—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—534no—
500C2×C8—10616no—
501C2×C8—1062no—
502C2×C8—1062no—
503C2×C8—1062no—
504C2×C8—1062no—
505C2×C8—1062no—
506C4⋊C4—22no—
507C4⋊C4—41no—
508C4⋊C4—1062no—
509C4⋊C4—2121no—
510C22⋊C4—22no—
511C22⋊C4—41no—
512C22⋊C4—1062no—
513C22⋊C4—2121no—
514C42—22no—
515C42—41no—
516C42—1062no—
517C42—2121no—
518C23—81no—
519C23—1062no—
520C23—1062no—
521C23—1062no—
522C23—4242no—
523C23#1313317411yescharacteristic
524C23C4×C22⋊D5311yescharacteristic
525C23C4×D21211yescharacteristic
526C23#1313319011yescharacteristic
527C23#1313319511yescharacteristic
528C23#1313319911yescharacteristic
529C23#1313320211yescharacteristic
530Q8—534no—
531Q8—1062no—
532Q8—1062no—
533Q8—1062no—
534D4—534no—
535D4—1064no—
536D4—1064no—
537D4—1064no—
538D4—1064no—
539D4—1062no—
540D4—1062no—
541D4—1062no—
542C2×C4—42no—
543C2×C4—82no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1062no—
547C2×C4—1064no—
548C2×C4—1064no—
549C2×C4—1064no—
550C2×C4—1064no—
551C2×C4—4242no—
552C2×C4—2122no—
553C2×C4C42×D5311yescharacteristic
554C2×C4C22⋊C4×D5311yescharacteristic
555C2×C4C22⋊C4×D5311yescharacteristic
556C2×C4C22⋊C4×D5311yescharacteristic
557C2×C4C22⋊C4×D5311yescharacteristic
558C2×C4C2×#1171992111yescharacteristic
559C2×C4C4⋊C4×D5311yescharacteristic
560C2×C4C4⋊C4×D5311yescharacteristic
561C2×C4C4⋊C4×D5311yescharacteristic
562C2×C4#1313317511yescharacteristic
563C2×C4#1313317711yescharacteristic
564C2×C4#1313317711yescharacteristic
565C2×C4C4×C22⋊D5311yescharacteristic
566C2×C4#1313317911yescharacteristic
567C2×C4#1313317911yescharacteristic
568C2×C4C4×D21211yescharacteristic
569C2×C4#1313318511yescharacteristic
570C2×C4#1313318511yescharacteristic
571C2×C4#1313318711yescharacteristic
572C2×C4#1313318711yescharacteristic
573C2×C4#1313318811yescharacteristic
574C2×C4#1313319011yescharacteristic
575C2×C4#1313319211yescharacteristic
576C2×C4#1313319311yescharacteristic
577C2×C4#1313319511yescharacteristic
578C2×C4#1313319711yescharacteristic
579C2×C4#1313319911yescharacteristic
580C2×C4#1313320211yescharacteristic
581C8—24no—
582C8—42no—
583C8—1064no—
584C8—2122no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—1064no—
593C22—4242no—
594C22—4242no—
595C22—2124no—
596C22C4×#1171992111yescharacteristic
597C22#1681048211yescharacteristic
598C22#1681048511yescharacteristic
599C22#1681049411yescharacteristic
600C22#1681049711yescharacteristic
601C22#1681049911yescharacteristic
602C22#1681050111yescharacteristic
603C4—82no—
604C4—1064no—
605C4—4242no—
606C4#95×D5311yescharacteristic
607C4#1681047011yescharacteristic
608C4#1681047811yescharacteristic
609C4#1681048011yescharacteristic
610C4#1681049411yescharacteristic
611C4#1681049711yescharacteristic
612C4#1681049911yescharacteristic
613C4#1681050111yescharacteristic
614C2—21no—
615C2—21no—
616C2—21no—
617C2—21no—
618C2—21no—
619C2—21no—
620C2—21no—
621C2—2124no—
622C2#2027293611yescharacteristic
subgroupG/H
centerC4#16810470
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC2×C53×#344C2
socleC2×C53C22⋊#95
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenterC2×#344D53
hypocenterC53#10793
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
4. C2×#16810395C2semidirect
5. C2×#16810381C2semidirect
6. C2×#16810379C2semidirect
7. C2×C53×#344C2semidirect
13. C2×#13133225C4semidirect
25. #16810620C4semidirect
26. #16810617C4semidirect
307. C2×#344D53semidirect
345. C53#10793semidirect

Maximal subgroups

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

subgroupindex
C2×#168113892
#202832352
#202832322
C2×#168103952
C2×#168103812
C2×#168103792
C2×C53×#3442
#1079353