Finite Group Databasebeta

← #27290456

Subgroups — #27290456

Order 13568.

930 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810639C211yesmaximal, characteristic
2C2×#16810617C211yesmaximal, characteristic
3C2×#16810596C211yesmaximal, characteristic
4C2×#16810581C211yesmaximal, characteristic, complemented
5#20274448C211yesmaximal, characteristic, complemented
6#20274431C211yesmaximal, characteristic, complemented
7C53×#1733C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C23×#88322C2211yescharacteristic
10C2×#13133176C2211yescharacteristic
11C2×#13133176C2211yescharacteristic
12#16810639—22no—
13#16810639C414yescomplemented
14C2×#13133143—22no—
15#16810617—21no—
16#16810617—21no—
17#16810617—22no—
18C2×#13133140C412yescomplemented
19C2×#13133137C412yescomplemented
20#16810596—22no—
21#16810596C414yescomplemented
22C2×#13133131—22no—
23#16810581—22no—
24#16810581—22no—
25C2×#13133127—21no—
26C2×C8×Dic53—21no—
27#16810390—22no—
28#16810385—22no—
29C22×M16×C53C2211yescharacteristic
30C2×#117×C53C2211yescharacteristic
31C2×#117×C53C2211yescharacteristic
32C53×#339—22no—
33C2×#11719945—21no—
34C2×#11719945—21no—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945C2×C412yes—
44C2×#11719945C2×C412yes—
45C22×Dic106C2×C411yescharacteristic
46C22×D212C2×C411yescharacteristic
47C22×C22⋊D53—21no—
48C22×C4×D53—21no—
49C22×#88322—21no—
50C22×#88322—21no—
51C22×#88322—21no—
52C22×#88322—21no—
53C22×#88322C2×C412yes—
54C22×#88322D411yescharacteristic
55C22×#88322D411yescharacteristic
56C22×#88322D412yes—
57#13133176—21no—
58#13133176—21no—
59#13133176—21no—
60#13133176—21no—
61#13133176—21no—
62#13133176C2×C412yes—
63#13133176C2×C412yes—
64#13133176D412yes—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C22⋊Dic53—21no—
68C2×C22⋊Dic53D412yes—
69C2×C4×Dic53—21no—
70C2×C4×Dic53—21no—
71#13133143—28no—
72#13133140—22no—
73#13133140—22no—
74#13133137—22no—
75#13133137—22no—
76#13133131—28no—
77#13133127—24no—
78C8×Dic53—24no—
79C23×C4×C53C2311yescharacteristic
80C2×M16×C53—21no—
81C2×M16×C53—21no—
82C2×M16×C53—21no—
83C2×M16×C53C2×C412yes—
84C2×M16×C53D412yes—
85C2×M16×C53D412yes—
86C22×C8×C53—21no—
87#117×C53—21no—
88#117×C53—21no—
89#117×C53—21no—
90#117×C53—21no—
91#117×C53—21no—
92#117×C53—21no—
93#117×C53—21no—
94#117×C53C2×C412yes—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53C2×C412yes—
97C2×C22⋊C4×C53—21no—
98C2×C22⋊C4×C53—21no—
99C2×C42×C53—21no—
100C2×C42×C53—21no—
101C24×C53C2×D411yescharacteristic
102C23×D53—21no—
103C22×C4×C53—22no—
104C22×C4×C53—41no—
105C22×C4×C53C22×C411yescharacteristic
106C22×C4×C53C22×C411yescharacteristic
107C22×C4×C53C22×C411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C22×C4×C53#4111yescharacteristic
118C22×C4×C53#4111yescharacteristic
119C2×C22⋊D53—22no—
120C2×C22⋊D53—22no—
121C2×C22⋊D53—41no—
122C2×C22⋊D53—41no—
123C2×C22⋊D53—41no—
124C2×C22⋊D53—41no—
125C22×Dic53—21no—
126C22×Dic53—22no—
127C22×Dic53—41no—
128#11719945—22no—
129#11719945—22no—
130#11719945—22no—
131#11719945—22no—
132#11719945—22no—
133#11719945—22no—
134#11719945—22no—
135#11719945—22no—
136#11719945—41no—
137#11719945—41no—
138#11719945—41no—
139#11719945—41no—
140#11719945—41no—
141#11719945—41no—
142#11719945—41no—
143#11719945—41no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212—21no—
149C2×D212C4212yes—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—41no—
153C2×C4×D53—41no—
154C2×C4×D53—41no—
155C2×C4×D53—41no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106C4212yes—
162M16×C53—22no—
163M16×C53—22no—
164M16×C53—22no—
165M16×C53C22⋊C414yes—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—41no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—41no—
175C22⋊C4×C53—41no—
176C22⋊C4×C53—41no—
177C42×C53—22no—
178C42×C53—41no—
179C42×C53—41no—
180C42×C53—41no—
181C22⋊Dic53—22no—
182C22⋊Dic53—22no—
183C22⋊Dic53—41no—
184C22⋊Dic53C22⋊C414yes—
185C4⋊Dic53—22no—
186C4⋊Dic53—22no—
187C4⋊Dic53—22no—
188C4⋊Dic53—41no—
189C4×Dic53—22no—
190C4×Dic53—22no—
191C4×Dic53—22no—
192C4×Dic53—41no—
193C2×#88322—22no—
194C2×#88322—22no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322C22⋊C412yes—
202C2×#88322C22⋊C412yes—
203C23×C53—42no—
204C23×C53C2×C22⋊C411yescharacteristic
205C23×C53C4×D411yescharacteristic
206C23×C53C4×D411yescharacteristic
207C23×C53#12111yescharacteristic
208C23×C53#12111yescharacteristic
209C23×C53#12411yescharacteristic
210C23×C53#12711yescharacteristic
211C22×D53—22no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—82no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—82no—
218C2×C4×C53—82no—
219C2×C4×C53C2×C4211yescharacteristic
220C2×C4×C53C2×C22⋊C411yescharacteristic
221C2×C4×C53C2×C22⋊C411yescharacteristic
222C2×C4×C53#11711yescharacteristic
223C2×C4×C53#11711yescharacteristic
224C2×C4×C53#11711yescharacteristic
225C2×C4×C53C4×D411yescharacteristic
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53C4×D411yescharacteristic
228C2×C4×C53C4×D411yescharacteristic
229C2×C4×C53C4×D411yescharacteristic
230C2×C4×C53C4×D411yescharacteristic
231C2×C4×C53C4×D411yescharacteristic
232C2×C4×C53C4×D411yescharacteristic
233C2×C4×C53C4×D411yescharacteristic
234C2×C4×C53C4×D411yescharacteristic
235C2×C4×C53#12011yescharacteristic
236C2×C4×C53#12111yescharacteristic
237C2×C4×C53#12111yescharacteristic
238C2×C4×C53#12111yescharacteristic
239C2×C4×C53#12111yescharacteristic
240C2×C4×C53#12311yescharacteristic
241C2×C4×C53#12311yescharacteristic
242C2×C4×C53#12311yescharacteristic
243C2×C4×C53#12411yescharacteristic
244C2×C4×C53#12411yescharacteristic
245C2×C4×C53#12611yescharacteristic
246C2×C4×C53#12611yescharacteristic
247C22⋊D53—42no—
248C22⋊D53—42no—
249C22⋊D53—42no—
250C22⋊D53—42no—
251C2×Dic53—22no—
252C2×Dic53—24no—
253C2×Dic53—44no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—42no—
257C2×Dic53—82no—
258D212—22no—
259D212—22no—
260D212—41no—
261D212—41no—
262C4×D53—42no—
263C4×D53—42no—
264C4×D53—42no—
265C4×D53—42no—
266Dic106—22no—
267Dic106—22no—
268Dic106—41no—
269Dic106—41no—
270C8×C53—24no—
271#88322—22no—
272#88322—22no—
273#10944—531nomaximal, Sylow
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53C4×C22⋊C411yescharacteristic
282C22×C53#38611yescharacteristic
283C22×C53#38811yescharacteristic
284C22×C53#38811yescharacteristic
285C22×C53#39011yescharacteristic
286C22×C53#39011yescharacteristic
287C22×C53#39011yescharacteristic
288C2×D53—42no—
289C2×D53—82no—
290C2×D53—82no—
291C2×D53—82no—
292C4×C53—82no—
293C4×C53—82no—
294C4×C53C4×C22⋊C411yescharacteristic
295C4×C53C4×C22⋊C411yescharacteristic
296C4×C53#38611yescharacteristic
297C4×C53#38611yescharacteristic
298C4×C53#38811yescharacteristic
299C4×C53#38811yescharacteristic
300C4×C53#38811yescharacteristic
301C4×C53#38811yescharacteristic
302Dic53—44no—
303Dic53—42no—
304Dic53—82no—
305C2×#421—531no—
306C2×#416—531no—
307C2×#410—531no—
308C2×#405—531no—
309#1740—531no—
310#1735—531no—
311#1733D5311yescharacteristic, complemented
312C2×C53—21no—
313C2×C53—21no—
314C2×C53—21no—
315C2×C53—21no—
316C2×C53—21no—
317C2×C53—21no—
318C2×C53#143711yescharacteristic
319C2×C53#197612yes—
320D53—82no—
321C22×#41—531no—
322C22×M16C2×D5311yescharacteristic
323C23×C8—531no—
324C2×#117—531no—
325C2×#117—531no—
326C2×#117C2×D5311yescharacteristic
327C2×#117C2×D5311yescharacteristic
328#421—534no—
329#421—1062no—
330C2×#104—1062no—
331#416—1062no—
332#416—1061no—
333#416—1061no—
334C2×#103—532no—
335C2×#102—532no—
336#410—534no—
337#410—1062no—
338C2×#99—1062no—
339#405—1062no—
340#405—1062no—
341C2×#97—1061no—
342C2×C4×C8—1061no—
343#342—1062no—
344#340—1062no—
345#339—22no—
346C53#1094411yescharacteristic, complemented, Sylow
347C2×#41—532no—
348C2×#41—532no—
349C2×#41—1061no—
350C2×#41—1061no—
351C2×#41—1061no—
352C2×#41—1061no—
353C2×#41—1061no—
354C2×#41—1061no—
355C2×#41—1061no—
356C2×#41—1061no—
357C2×#41—1061no—
358C2×#41—1061no—
359C22×Q8—531no—
360C22×D4—531no—
361C22×D4—1061no—
362C23×C4—1061no—
363C23×C4C22×D5311yescharacteristic
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16C4×D5312yes—
368C2×M16D21212yes—
369C2×M16C22⋊D5312yes—
370C22×C8—21no—
371C22×C8—532no—
372C22×C8—532no—
373C22×C8—531no—
374C22×C8—531no—
375C22×C8—1061no—
376C22×C8—1061no—
377C22×C8—1061no—
378C22×C8—1061no—
379#117—21no—
380#117—21no—
381#117—21no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117—532no—
387#117—532no—
388#117—532no—
389#117—1061no—
390#117—1061no—
391#117—1061no—
392#117—1061no—
393#117—1061no—
394#117C4×D5312yes—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4C4×D5312yes—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—532no—
402C2×C22⋊C4—1061no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—1061no—
406C2×C42—1061no—
407#104—1068no—
408#103—1062no—
409#103—1062no—
410#102—1062no—
411#102—1062no—
412#99—1068no—
413#97—1064no—
414C4×C8—1064no—
415C24—1061no—
416C24D4×D5311yescharacteristic
417#41—1062no—
418#41—1062no—
419#41—1062no—
420#41—1062no—
421#41—1062no—
422#41—1062no—
423#41—1062no—
424#41—1062no—
425#41—2121no—
426#41—2121no—
427#41—2121no—
428#41—2121no—
429#41—2121no—
430#41—2121no—
431#41—2121no—
432#41—2121no—
433C2×Q8—532no—
434C2×Q8—1061no—
435C2×Q8—1061no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8—1061no—
439C2×D4—532no—
440C2×D4—1062no—
441C2×D4—1062no—
442C2×D4—1061no—
443C2×D4—1061no—
444C2×D4—1061no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—2121no—
448C2×D4—2121no—
449C2×D4—2121no—
450C2×D4—2121no—
451C22×C4—22no—
452C22×C4—41no—
453C22×C4—1061no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—2121no—
458C22×C4—2121no—
459C22×C4—2121no—
460C22×C4—2121no—
461C22×C4—2121no—
462C22×C4C2×C4×D5311yescharacteristic
463C22×C4C2×C4×D5311yescharacteristic
464C22×C4C2×C4×D5311yescharacteristic
465C22×C4C2×D21211yescharacteristic
466C22×C4#1171994511yescharacteristic
467C22×C4#1171994511yescharacteristic
468C22×C4D4×D5311yescharacteristic
469C22×C4D4×D5311yescharacteristic
470C22×C4D4×D5311yescharacteristic
471C22×C4#1171994711yescharacteristic
472C22×C4#1171994711yescharacteristic
473C22×C4#1171994911yescharacteristic
474C22×C4#1171994911yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476M16—22no—
477M16—22no—
478M16—22no—
479M16#1171992114yes—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—532no—
484C2×C8—532no—
485C2×C8—1062no—
486C2×C8—1062no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1062no—
490C2×C8—1062no—
491C2×C8—1061no—
492C2×C8—1061no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—41no—
497C4⋊C4—1062no—
498C4⋊C4—1062no—
499C4⋊C4—1062no—
500C4⋊C4—2121no—
501C22⋊C4—22no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—534no—
506C22⋊C4—1062no—
507C22⋊C4—1062no—
508C22⋊C4—2121no—
509C42—22no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—1062no—
514C42—1062no—
515C42—1062no—
516C42—2121no—
517C23—42no—
518C23—1062no—
519C23—4242no—
520C23—2121no—
521C23—2121no—
522C23C22⋊C4×D5311yescharacteristic
523C23#1313317711yescharacteristic
524C23#1313317911yescharacteristic
525C23#1313319011yescharacteristic
526C23#1313319211yescharacteristic
527C23#1313320311yescharacteristic
528C23#1313321311yescharacteristic
529Q8—1062no—
530Q8—1062no—
531Q8—2121no—
532Q8—2121no—
533D4—1062no—
534D4—1062no—
535D4—2122no—
536D4—2122no—
537D4—2122no—
538D4—2122no—
539D4—2121no—
540D4—2121no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—1064no—
546C2×C4—1062no—
547C2×C4—4242no—
548C2×C4—2122no—
549C2×C4—2124no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2122no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4C42×D5311yescharacteristic
557C2×C4C22⋊C4×D5311yescharacteristic
558C2×C4C2×#1171992111yescharacteristic
559C2×C4#1313317411yescharacteristic
560C2×C4#1313317511yescharacteristic
561C2×C4#1313317511yescharacteristic
562C2×C4#1313317711yescharacteristic
563C2×C4#1313317711yescharacteristic
564C2×C4#1313317711yescharacteristic
565C2×C4C4×C22⋊D5311yescharacteristic
566C2×C4C4×C22⋊D5311yescharacteristic
567C2×C4#1313317911yescharacteristic
568C2×C4#1313317911yescharacteristic
569C2×C4#1313317911yescharacteristic
570C2×C4C4×D21211yescharacteristic
571C2×C4C4×D21211yescharacteristic
572C2×C4#1313318711yescharacteristic
573C2×C4#1313318911yescharacteristic
574C2×C4#1313318911yescharacteristic
575C2×C4#1313319011yescharacteristic
576C2×C4#1313319211yescharacteristic
577C2×C4#1313320111yescharacteristic
578C2×C4#1313320211yescharacteristic
579C2×C4#1313320211yescharacteristic
580C2×C4#1313320311yescharacteristic
581C2×C4#1313320611yescharacteristic
582C2×C4#1313321211yescharacteristic
583C2×C4#1313321211yescharacteristic
584C8—24no—
585C8—1062no—
586C8—1062no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—4242no—
595C22—4242no—
596C22—4242no—
597C22—2122no—
598C22#1681047011yescharacteristic
599C22#1681050111yescharacteristic
600C22#1681051211yescharacteristic
601C22#1681051511yescharacteristic
602C22#1681052311yescharacteristic
603C22#1681052311yescharacteristic
604C22#1681052411yescharacteristic
605C4—82no—
606C4—82no—
607C4—4242no—
608C4—2124no—
609C4—2122no—
610C4#1681047011yescharacteristic
611C4C4×#1171992111yescharacteristic
612C4#1681050011yescharacteristic
613C4#1681050111yescharacteristic
614C4#1681051211yescharacteristic
615C4#1681051311yescharacteristic
616C4#1681051311yescharacteristic
617C4#1681051511yescharacteristic
618C2—21no—
619C2—21no—
620C2—21no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—4242no—
625C2#2027295111yescharacteristic
626C2#2027561212yes—
subgroupG/H
centerC2×C4#13133189
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1733C2
socleC22×C53#390
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenter#1733D53
hypocenterC53#10944
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
4. C2×#16810581C2semidirect
5. #20274448C2semidirect
6. #20274431C2semidirect
7. C53×#1733C2semidirect
13. #16810639C4semidirect
18. C2×#13133140C4semidirect
19. C2×#13133137C4semidirect
21. #16810596C4semidirect
311. #1733D53semidirect
346. C53#10944semidirect

Maximal subgroups

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

subgroupindex
C2×#168106392
C2×#168106172
C2×#168105962
C2×#168105812
#202744482
#202744312
C53×#17332
#1094453