Finite Group Databasebeta

← #27355540

Subgroups — #27355540

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283731C211yesmaximal, characteristic, complemented
2#20283723C211yesmaximal, characteristic, complemented
3C4×#13133236C211yesmaximal, characteristic
4#20273608C211yesmaximal, characteristic, complemented
5#20273603C211yesmaximal, characteristic, complemented
6#20273567C211yesmaximal, characteristic, complemented
7#20273564C211yesmaximal, characteristic, complemented
8#20273545C211yesmaximal, characteristic, complemented
9#20273518C211yesmaximal, characteristic, complemented
10C8×#11719923C211yesmaximal, characteristic, complemented
11#20273477C211yesmaximal, characteristic, complemented
12#20273146C211yesmaximal, characteristic, complemented
13#20273146C211yesmaximal, characteristic, complemented
14#20272994C211yesmaximal, characteristic, complemented
15C53×#1581C211yesmaximal, characteristic, complemented
16C2×#13133236C411yescharacteristic
17C4×#11719947C2211yescharacteristic
18C4×#11719947C2211yescharacteristic
19#16810753C2211yescharacteristic
20#16810750C2211yescharacteristic
21C8×Dic106C2211yescharacteristic
22#16810747C2211yescharacteristic
23C4×Dic212C2211yescharacteristic
24#16810717C411yescharacteristic
25#16810714C411yescharacteristic
26#16810714C411yescharacteristic
27C4×#11719923C2211yescharacteristic
28C4×#11719923C2211yescharacteristic
29D8×Dic53C411yescharacteristic
30#16810701C2211yescharacteristic
31#16810699—21no—
32#16810698—21no—
33#16810694—21no—
34#16810691—21no—
35D4×#88322C2211yescharacteristic
36#16810679C2211yescharacteristic
37#16810679C2211yescharacteristic
38#16810677—21no—
39#16810677—21no—
40#16810674C2211yescharacteristic
41#16810660C411yescharacteristic
42#16810616C411yescharacteristic
43#16810616C411yescharacteristic
44M16×Dic53—21no—
45C4×#11719912C2211yescharacteristic
46C2×#13133127—21no—
47C4×C8×D53C2211yescharacteristic
48#16810376C2211yescharacteristic
49#16810373C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810355C2211yescharacteristic
52#16810355C2211yescharacteristic
53#16810353C2211yescharacteristic
54#16810353C2211yescharacteristic
55D4⋊#88322C2211yescharacteristic
56D4⋊#88322C2211yescharacteristic
57#16810345C2211yescharacteristic
58C8×#88322C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C4×D8×C53C2211yescharacteristic
61C53×#436C2211yescharacteristic
62C8×D4×C53C2211yescharacteristic
63C53×#335C2211yescharacteristic
64C53×D4⋊C8C2211yescharacteristic
65C53×D4⋊C8C2211yescharacteristic
66C53×#322C2211yescharacteristic
67C2×#11719947C2×C411yescharacteristic
68C2×#11719947C2×C411yescharacteristic
69#13133236—24no—
70C2×Dic212C2×C411yescharacteristic
71C2×#11719923C2×C411yescharacteristic
72C2×#11719923C2×C411yescharacteristic
73C2×#11719917—21no—
74C2×#11719912D411yescharacteristic
75C2×#11719912D411yescharacteristic
76C22×#88322—21no—
77C2×C8×D53C2×C411yescharacteristic
78C2×C8×D53C2×C411yescharacteristic
79C4×Dic106C2311yescharacteristic
80C4×Dic106C2311yescharacteristic
81#13133182C2×C411yescharacteristic
82#13133182C2×C411yescharacteristic
83D4×Dic53C2×C411yescharacteristic
84D4×Dic53C2×C411yescharacteristic
85C4×C22⋊D53—21no—
86C4×C22⋊D53—21no—
87#13133177—21no—
88#13133177—21no—
89#13133175C2×C411yescharacteristic
90#13133175C2×C411yescharacteristic
91#13133173—21no—
92#13133173—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C42×D53C2311yescharacteristic
96#13133151C2×C411yescharacteristic
97#13133148C2×C411yescharacteristic
98#13133148C2×C411yescharacteristic
99C4⋊#88322C2311yescharacteristic
100C4⋊#88322C2311yescharacteristic
101#13133146—21no—
102#13133146—21no—
103#13133146—21no—
104#13133146—21no—
105#13133146—21no—
106#13133142C2×C411yescharacteristic
107#13133142C2×C411yescharacteristic
108#13133140C2×C411yescharacteristic
109#13133140C2×C411yescharacteristic
110#13133138C2×C411yescharacteristic
111#13133138C2×C411yescharacteristic
112C22⋊#88322—21no—
113C22⋊#88322—21no—
114#13133129—21no—
115#13133129—21no—
116#13133129—21no—
117#13133127—21no—
118#13133127—21no—
119#13133127—21no—
120#13133127—21no—
121#13133127D411yescharacteristic
122#13133127D411yescharacteristic
123C4×#88322C2311yescharacteristic
124C4×#88322C2311yescharacteristic
125C4×#88322C2311yescharacteristic
126C8×Dic53—21no—
127C8×Dic53—21no—
128C8×Dic53C2×C411yescharacteristic
129C8×Dic53C2×C411yescharacteristic
130C2×D8×C53C2×C411yescharacteristic
131C2×M16×C53—21no—
132C22×C8×C53—21no—
133C4×D4×C53C2311yescharacteristic
134C4×D4×C53C2311yescharacteristic
135#107×C53C2×C411yescharacteristic
136C4⋊C8×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138#102×C53C2×C411yescharacteristic
139#102×C53C2×C411yescharacteristic
140C22⋊C8×C53—21no—
141C22⋊C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C4×C8×C53C2311yescharacteristic
145C2×D4×C53C22×C411yescharacteristic
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C2×C22⋊D53—21no—
150C2×C22⋊D53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155#11719947—22no—
156#11719947—22no—
157#11719947—41no—
158#11719947—41no—
159C2×C4×D53C22×C411yescharacteristic
160C2×C4×D53C2×D411yescharacteristic
161C2×C4×D53#4111yescharacteristic
162C2×Dic106C22×C411yescharacteristic
163C2×Dic106C22×C411yescharacteristic
164D8×C53—22no—
165M16×C53—42no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—42no—
170C2×C8×C53C22×C411yescharacteristic
171C2×C8×C53C22×C411yescharacteristic
172C2×C8×C53C2×D411yescharacteristic
173C2×C8×C53C2×D411yescharacteristic
174C4⋊C4×C53C22×C411yescharacteristic
175C4⋊C4×C53C22×C411yescharacteristic
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C42×C53C2411yescharacteristic
179C22⋊Dic53—21no—
180C22⋊Dic53—21no—
181#11719923—22no—
182#11719923M1614yescomplemented
183#11719921—21no—
184#11719921—21no—
185C4⋊Dic53C22×C411yescharacteristic
186C4⋊Dic53C22×C411yescharacteristic
187#11719919—21no—
188#11719919—21no—
189#11719919—21no—
190#11719919—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53C22×C411yescharacteristic
198C4×Dic53C2×D411yescharacteristic
199C4×Dic53#4111yescharacteristic
200#11719917—42no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—42no—
205C2×#88322C22×C411yescharacteristic
206C2×#88322C22×C411yescharacteristic
207C2×#88322C2×D411yescharacteristic
208C2×#88322C2×D411yescharacteristic
209Dic212—22no—
210#11719912—22no—
211#11719912—22no—
212C8×D53—22no—
213C8×D53—41no—
214C23×C53—21no—
215C23×C53—21no—
216C22×D53C4×D411yescharacteristic
217D4×C53—21no—
218D4×C53—21no—
219D4×C53—21no—
220D4×C53C2×M1612yes—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53C23×C411yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C22⋊D53—42no—
231C22⋊D53—42no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—81no—
239C2×Dic53—81no—
240C2×Dic53C4×D411yescharacteristic
241C2×Dic53C4×D411yescharacteristic
242C2×Dic53C4×D411yescharacteristic
243C4×D53—21no—
244C4×D53—21no—
245C4×D53—21no—
246C4×D53—21no—
247C4×D53—41no—
248Dic106—21no—
249Dic106—21no—
250Dic106—21no—
251Dic106C2×M1612yes—
252C8×C53—21no—
253C8×C53—21no—
254C8×C53—21no—
255C8×C53—41no—
256C8×C53—41no—
257C8×C53—41no—
258C8×C53#13512yescomplemented
259#88322—21no—
260#88322—21no—
261#88322—41no—
262#88322—41no—
263#88322—41no—
264#88322C2×M1612yes—
265#88322#13512yescomplemented
266#18037—531nomaximal, Sylow
267C22×C53—22no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53C2×C4×D411yescharacteristic
274C2×D53—21no—
275C2×D53—21no—
276C2×D53—21no—
277C4×C53—21no—
278C4×C53—41no—
279C4×C53—41no—
280C4×C53C22×M1611yescharacteristic
281C4×C53#56811yescharacteristic
282C4×C53C2×#13511yescharacteristic
283C4×C53C2×#13511yescharacteristic
284Dic53—22no—
285Dic53—21no—
286Dic53—21no—
287Dic53—21no—
288Dic53—41no—
289Dic53—41no—
290Dic53—41no—
291#2971—531no—
292#2970—531no—
293C4×#135—531no—
294#1590—531no—
295#1590—531no—
296#1584—531no—
297#1584—531no—
298#1581—531no—
299#1581D5311yescharacteristic, complemented
300C8×SD16—531no—
301C8×SD16—531no—
302#1568—531no—
303#1488—531no—
304#1488—531no—
305C8×M16—531no—
306C2×C53—42no—
307C2×C53—81no—
308C2×C53D4×M1611yescharacteristic
309C2×C53C4×#13511yescharacteristic
310C2×C53#295711yescharacteristic
311D53—41no—
312C2×#135—531no—
313C4×#41—531no—
314C4×#41—531no—
315#446—531no—
316#446—531no—
317C8×Q8—531no—
318C8×Q8—531no—
319C4×Q16—531no—
320C4×Q16—531no—
321C4×SD16—531no—
322C4×SD16—531no—
323C4×SD16—531no—
324C4×SD16—531no—
325C4×D8—531no—
326C4×D8C2×D5311yescharacteristic
327#436—531no—
328#436—1061no—
329#436—1061no—
330#436C2×D5311yescharacteristic
331#435—1061no—
332#435—1061no—
333C8×D4—531no—
334C8×D4C2×D5311yescharacteristic
335#432—531no—
336#432—531no—
337#432—1061no—
338#432—1061no—
339#431—531no—
340#427—531no—
341#416—531no—
342#416—531no—
343C4×M16—531no—
344C4×M16—1061no—
345C2×#97—1061no—
346C2×C4×C8—531no—
347#335—531no—
348#335C2×D5311yescharacteristic
349#334—531no—
350#334—531no—
351Q8⋊C8—531no—
352Q8⋊C8—531no—
353Q8⋊C8—531no—
354Q8⋊C8—531no—
355D4⋊C8—531no—
356D4⋊C8—531no—
357D4⋊C8C2×D5311yescharacteristic
358D4⋊C8C2×D5311yescharacteristic
359#322—531no—
360#322C2×D5311yescharacteristic
361C82—531no—
362C82—531no—
363C53#1803711yescharacteristic, complemented, Sylow
364C2×#41—531no—
365C2×#41—531no—
366#135—1064no—
367C2×Q16—531no—
368C2×SD16—531no—
369C2×SD16—531no—
370C2×D8C4×D5311yescharacteristic
371C2×M16—21no—
372C2×M16—531no—
373C2×M16—531no—
374C2×M16—1061no—
375C22×C8—21no—
376C22×C8—531no—
377C22×C8—531no—
378C22×C8—1061no—
379C4×Q8—531no—
380C4×Q8—531no—
381C4×Q8—531no—
382C4×Q8—531no—
383C4×D4—531no—
384C4×D4—531no—
385C4×D4—1061no—
386C4×D4—1061no—
387C4×D4—1061no—
388C4×D4—1061no—
389C4×D4C22×D5311yescharacteristic
390C4×D4C22×D5311yescharacteristic
391#117—531no—
392#117—531no—
393#117—1061no—
394#117—1061no—
395C2×C42—531no—
396C2×C42—1061no—
397C2×C42—1061no—
398#107—531no—
399#107C4×D5311yescharacteristic
400#106—531no—
401#106—531no—
402C4⋊C8—531no—
403C4⋊C8—531no—
404C4⋊C8—1061no—
405C4⋊C8—1061no—
406C4⋊C8—1061no—
407C4⋊C8—1061no—
408C4⋊C8—1061no—
409C4⋊C8C22×D5311yescharacteristic
410C4⋊C8C22×D5311yescharacteristic
411#103—531no—
412#103—531no—
413#103—531no—
414#103—531no—
415#102—531no—
416#102—531no—
417#102C4×D5311yescharacteristic
418#102C4×D5311yescharacteristic
419C22⋊C8—21no—
420C22⋊C8—21no—
421C22⋊C8—1061no—
422C22⋊C8—1061no—
423C22⋊C8—1061no—
424C22⋊C8—1061no—
425C22⋊C8—1061no—
426#97—531no—
427#97—531no—
428#97—1061no—
429#97—1061no—
430#97—1061no—
431#97—1061no—
432C4×C8—531no—
433C4×C8—531no—
434C4×C8—531no—
435C4×C8—531no—
436C4×C8—531no—
437C4×C8—1061no—
438C4×C8—1061no—
439C4×C8C22×D5311yescharacteristic
440C4×C8C22×D5311yescharacteristic
441C4×C8C22×D5311yescharacteristic
442#41—1062no—
443#41—1062no—
444#41—2121no—
445#41—2121no—
446C2×Q8—531no—
447C2×Q8—531no—
448C2×D4—1061no—
449C2×D4—1061no—
450C2×D4C2×C4×D5311yescharacteristic
451C2×D4C2×C4×D5311yescharacteristic
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461Q16—1062no—
462SD16—534no—
463SD16—1062no—
464D8—22no—
465M16—42no—
466M16—1062no—
467M16—1062no—
468M16—2122no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—42no—
473C2×C8—531no—
474C2×C8—531no—
475C2×C8—531no—
476C2×C8—531no—
477C2×C8—1062no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—2121no—
482C2×C8—2122no—
483C2×C8C2×C4×D5311yescharacteristic
484C2×C8C2×C4×D5311yescharacteristic
485C2×C8D4×D5311yescharacteristic
486C2×C8D4×D5311yescharacteristic
487C4⋊C4—531no—
488C4⋊C4—531no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4C2×C4×D5311yescharacteristic
494C4⋊C4C2×C4×D5311yescharacteristic
495C22⋊C4—21no—
496C22⋊C4—21no—
497C22⋊C4—1061no—
498C22⋊C4—1061no—
499C22⋊C4—1061no—
500C22⋊C4—1061no—
501C42—531no—
502C42—531no—
503C42—531no—
504C42—1061no—
505C42—1061no—
506C42—1061no—
507C42—1061no—
508C42—1061no—
509C42—1061no—
510C42C23×D5311yescharacteristic
511C23—21no—
512C23—21no—
513C23—531no—
514Q8—532no—
515Q8—1061no—
516Q8—1061no—
517Q8—1061no—
518D4—21no—
519D4—21no—
520D4—21no—
521D4—2122no—
522D4—2122no—
523D4M16×D5312yes—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—531no—
531C2×C4—531no—
532C2×C4—531no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—4241no—
542C2×C4—4241no—
543C2×C4—2121no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4C22×C4×D5311yescharacteristic
547C2×C4C2×D4×D5311yescharacteristic
548C2×C4#41×D5311yescharacteristic
549C8—21no—
550C8—21no—
551C8—21no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—532no—
556C8—532no—
557C8—1061no—
558C8—1061no—
559C8—2121no—
560C8—2121no—
561C8—2121no—
562C8#1313323612yescomplemented
563C22—22no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—1061no—
570C22—1061no—
571C22—1061no—
572C22C4×D4×D5311yescharacteristic
573C4—21no—
574C4—41no—
575C4—41no—
576C4—1061no—
577C4—1062no—
578C4—1061no—
579C4—1061no—
580C4—2121no—
581C4—2121no—
582C4—2121no—
583C4C2×M16×D5311yescharacteristic
584C4#1681138811yescharacteristic
585C4#135×D5311yescharacteristic
586C4C2×#1313323611yescharacteristic
587C2—42no—
588C2—81no—
589C2—2121no—
590C2#2028344811yescharacteristic
591C2C4×#1313323611yescharacteristic
592C2#2028362911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1581C2
socleC22×C53C2×C4×D4
prime radicalC2×#13133236C4
cosocleC42×C53C24
hypercenter#1581D53
hypocenterC53#18037
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1581

Complements

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

NH ≅ G/Ntype
1. #20283731C2semidirect
2. #20283723C2semidirect
4. #20273608C2semidirect
5. #20273603C2semidirect
6. #20273567C2semidirect
7. #20273564C2semidirect
8. #20273545C2semidirect
9. #20273518C2semidirect
10. C8×#11719923C2semidirect
11. #20273477C2semidirect
12. #20273146C2semidirect
13. #20273146C2semidirect
14. #20272994C2semidirect
15. C53×#1581C2semidirect
182. #11719923M16semidirect
258. C8×C53#135semidirect
265. #88322#135semidirect
299. #1581D53semidirect
363. C53#18037semidirect
562. C8#13133236semidirect

Maximal subgroups

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

subgroupindex
#202837312
#202837232
C4×#131332362
#202736082
#202736032
#202735672
#202735642
#202735452
#202735182
C8×#117199232
#202734772
#202731462
#202729942
C53×#15812
#1803753