Finite Group Databasebeta

← #27469480

Subgroups — #27469480

Order 13568.

930 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810944C211yesmaximal, characteristic, complemented
2C2×#16810936C211yesmaximal, characteristic, complemented
3C2×#16810874C211yesmaximal, characteristic
4C2×#16810663C211yesmaximal, characteristic, complemented
5#20279177C211yesmaximal, characteristic, complemented
6#20276243C211yesmaximal, characteristic, complemented
7#20276239C211yesmaximal, characteristic, complemented
8#20276239C211yesmaximal, characteristic, complemented
9#20276236C211yesmaximal, characteristic, complemented
10#20275590C211yesmaximal, characteristic, complemented
11#20275580C211yesmaximal, characteristic, complemented
12#20274884C211yesmaximal, characteristic, complemented
13#20274877C211yesmaximal, characteristic, complemented
14#20274499C211yesmaximal, characteristic, complemented
15C53×#2504C211yesmaximal, characteristic, complemented
16C22×#11719922C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×#13133191C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20#16810944—24no—
21#16810936—24no—
22#16810874—24no—
23#16810663—24no—
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28C2×#13133150C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31C2×#13133148C2211yescharacteristic
32C2×#13133148C2211yescharacteristic
33C2×#13133138C2211yescharacteristic
34C2×#13133138C2211yescharacteristic
35C2×#13133138C2211yescharacteristic
36C2×#13133137C2211yescharacteristic
37C2×#13133137C2211yescharacteristic
38C2×#13133137C2211yescharacteristic
39C2×C22⋊#88322C2211yescharacteristic
40C2×#13133128C2211yescharacteristic
41C2×C4×#88322C2211yescharacteristic
42#16810539—22no—
43#16810525C2211yescharacteristic
44#16810525C2211yescharacteristic
45#16810515—22no—
46#16810493C2211yescharacteristic
47#16810489C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#121×C53C2211yescharacteristic
51C22×C4⋊C4×C53C2211yescharacteristic
52C53×#397—22no—
53C53×#390C2211yescharacteristic
54C53×#390C2211yescharacteristic
55C53×#386—22no—
56C53×#382C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C2×#11719922—22no—
62C2×#11719922—22no—
63C2×#11719922—22no—
64C2×#11719917—22no—
65C2×#11719917—22no—
66C2×#11719917D414yes—
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71#13133191—22no—
72#13133191—22no—
73#13133176—22no—
74#13133176—22no—
75C2×C4⋊Dic53—22no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C22⋊Dic53—21no—
80C2×#11719921—21no—
81C2×#11719921—22no—
82C2×#11719921—22no—
83C2×C4×Dic53—21no—
84#13133150—22no—
85#13133150—22no—
86#13133150—22no—
87#13133150—22no—
88#13133150—22no—
89#13133148—22no—
90#13133148—22no—
91#13133148—22no—
92#13133148—22no—
93#13133138—22no—
94#13133138—22no—
95#13133138—22no—
96#13133137—22no—
97#13133137—22no—
98#13133137—22no—
99C22⋊#88322D414yes—
100#13133128—22no—
101C4×#88322—22no—
102#13133124—22no—
103C22×D4×C53—21no—
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53—21no—
106C23×C4×C53C2311yescharacteristic
107#121×C53—22no—
108#121×C53—22no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—22no—
111C2×C4⋊C4×C53—22no—
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—22no—
117C2×C22⋊C4×C53—22no—
118C2×C22⋊C4×C53D412yes—
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121#95×C53—22no—
122#95×C53—22no—
123#95×C53D412yes—
124C24×C53—21no—
125C24×C53C2×D411yescharacteristic
126C23×D53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—44no—
134C2×D4×C53—42no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—42no—
143C22×C4×C53—82no—
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D412yes—
148C22×C4×C53C2×D412yes—
149C22×C4×C53C2411yescharacteristic
150C2×C22⋊D53—42no—
151C2×C22⋊D53—82no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—42no—
168C22⋊C4×C53—42no—
169C22⋊C4×C53—42no—
170C22⋊C4×C53—82no—
171C22⋊C4×C53—82no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—21no—
176C22⋊Dic53—42no—
177#11719922—44no—
178#11719921—44no—
179#11719921—42no—
180#11719921—82no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—82no—
188C4×Dic53—42no—
189#11719917—28no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322C2×D412yes—
197C2×#88322C2×D412yes—
198C2×#88322C2×D412yes—
199C2×#88322C2×D412yes—
200C2×#88322#4112yes—
201C2×#88322#4112yes—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—42no—
210C23×C53—42no—
211C23×C53#12712yes—
212C23×C53C22×D411yescharacteristic
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216C22×D53—42no—
217C22×D53—42no—
218D4×C53—41no—
219D4×C53—41no—
220D4×C53—41no—
221D4×C53—41no—
222D4×C53—84no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—84no—
242C2×C4×C53#12712yes—
243C2×C4×C53C22×D411yescharacteristic
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C532−1+411yescharacteristic
248C2×C4×C532−1+411yescharacteristic
249C22⋊D53—84no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256D212—41no—
257D212—41no—
258D212—41no—
259D212—41no—
260#88322—22no—
261#88322—22no—
262#88322—22no—
263#88322#12314yes—
264#27942—531nomaximal, Sylow
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—82no—
271C22×C53—82no—
272C22×C53—81no—
273C22×C53—81no—
274C22×C53—81no—
275C22×C53C2×#12711yescharacteristic
276C22×C53#54711yescharacteristic
277C22×C53#54711yescharacteristic
278C22×C53#57512yes—
279C22×C53#57612yes—
280C2×D53—82no—
281C2×D53—82no—
282C2×D53—81no—
283C2×D53—81no—
284C2×D53—81no—
285C4×C53—22no—
286C4×C53—42no—
287C4×C53—42no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53—81no—
291C4×C53C2×#12311yescharacteristic
292C4×C53#53611yescharacteristic
293C4×C53#54711yescharacteristic
294C4×C53#54711yescharacteristic
295Dic53—81no—
296Dic53—81no—
297Dic53—81no—
298C2×#482—531no—
299C2×#480—531no—
300C2×#469—531no—
301C2×#428—531no—
302#2504D5311yescharacteristic, complemented
303#2500—531no—
304#2093—531no—
305#2092—531no—
306#2092—531no—
307#2092—531no—
308#1972—531no—
309#1970—531no—
310#1847—531no—
311#1846—531no—
312#1753—531no—
313C2×C53—42no—
314C2×C53—161no—
315C2×C53#250111yescharacteristic
316C2×C53#315311yescharacteristic
317C2×C53#315511yescharacteristic
318C2×C53#334712yes—
319C2×C53#335512yes—
320D53—161no—
321C22×D8—531no—
322C22×M16—531no—
323C2×#121—531no—
324C2×#121C2×D5311yescharacteristic
325C2×#117—531no—
326C22×C4⋊C4C2×D5311yescharacteristic
327#482—1064no—
328#480—1064no—
329#469—1064no—
330#428—1064no—
331C2×#107—531no—
332C2×#107—531no—
333C2×#107—531no—
334C2×#107—531no—
335C2×#107—531no—
336C2×#106—531no—
337C2×#106—531no—
338C2×#106—531no—
339C2×#106—531no—
340C2×#102—531no—
341C2×#102—531no—
342C2×#102—531no—
343C2×#102—531no—
344C2×#102—531no—
345C2×#102—531no—
346C2×C22⋊C8—531no—
347C2×#97—531no—
348C2×C4×C8—531no—
349#397—22no—
350#394—1062no—
351#390—531no—
352#390—531no—
353#390C2×D5311yescharacteristic
354#390C2×D5311yescharacteristic
355#388—1062no—
356#386—22no—
357#384—531no—
358#382—531no—
359#382C2×D5311yescharacteristic
360#382C2×D5311yescharacteristic
361C4×C22⋊C4C2×D5311yescharacteristic
362#336—531no—
363#336—531no—
364C53#2794211yescharacteristic, complemented, Sylow
365C22×D4—21no—
366C22×D4—531no—
367C22×D4—1061no—
368C22×D4C22×D5311yescharacteristic
369C23×C4—21no—
370C23×C4C22×D5311yescharacteristic
371C2×D8—1062no—
372C2×D8—1062no—
373C2×D8—1062no—
374C2×M16—534no—
375C2×M16—1062no—
376C2×M16—1062no—
377C22×C8—531no—
378C22×C8—531no—
379C22×C8—531no—
380C22×C8—531no—
381#121—22no—
382#121—22no—
383#121—1062no—
384#121—1062no—
385#117—1062no—
386#117—1062no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—1062no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1062no—
404C2×C22⋊C4C22⋊D5312yes—
405C2×C42—1061no—
406C2×C42C22×D5311yescharacteristic
407C2×C42C22×D5311yescharacteristic
408#107—1062no—
409#107—1062no—
410#107—1062no—
411#107—1062no—
412#107—1062no—
413#106—1062no—
414#106—1062no—
415#106—1062no—
416#106—1062no—
417#102—1062no—
418#102—1062no—
419#102—1062no—
420#102—1062no—
421#102—1062no—
422#102—1062no—
423C22⋊C8—534no—
424#97—1062no—
425C4×C8—1062no—
426#95—22no—
427#95—22no—
428#95—1062no—
429#95C22⋊D5312yes—
430C24—21no—
431C24—1061no—
432C24C2×C22⋊D5311yescharacteristic
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—44no—
440C2×D4—42no—
441C2×D4—1061no—
442C2×D4—1061no—
443C2×D4—1061no—
444C2×D4—1061no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—4242no—
448C2×D4—2122no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—42no—
457C22×C4—82no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4D4×D5312yes—
462C22×C4D4×D5312yes—
463C22×C4C2×C22⋊D5312yes—
464C22×C4C2×C22⋊D5312yes—
465C22×C4C2×C22⋊D5311yescharacteristic
466C22×C4C23×D5311yescharacteristic
467D8—2124no—
468M16—1068no—
469C2×C8—532no—
470C2×C8—532no—
471C2×C8—532no—
472C2×C8—532no—
473C2×C8—532no—
474C2×C8—532no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—4242no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—4242no—
500C22⋊C4—2124no—
501C22⋊C4—2122no—
502C22⋊C4—2122no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—2122no—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—2121no—
518C23—2121no—
519C23—2121no—
520C23—2122no—
521C23—2122no—
522C23#1313321312yes—
523C23C22×C22⋊D5311yescharacteristic
524D4—41no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—84no—
529D4—4244no—
530D4—2121no—
531D4—2121no—
532D4—2121no—
533D4—2121no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—84no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4#1313321312yes—
560C2×C4C2×D4×D5311yescharacteristic
561C2×C4C2×D4×D5311yescharacteristic
562C2×C4#41×D5311yescharacteristic
563C2×C4#41×D5311yescharacteristic
564C2×C4#1313326411yescharacteristic
565C2×C4#1313326411yescharacteristic
566C8—534no—
567C8—1062no—
568C8—1062no—
569C8—1062no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—82no—
577C22—81no—
578C22—82no—
579C22—81no—
580C22—4241no—
581C22—4242no—
582C22—4241no—
583C22—4242no—
584C22—4241no—
585C22C2×#1313321311yescharacteristic
586C22#1681127211yescharacteristic
587C22#1681127211yescharacteristic
588C22#1681142112yes—
589C22#1681142612yes—
590C4—22no—
591C4—42no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—4241no—
597C4—4241no—
598C4—4241no—
599C4#123×D5311yescharacteristic
600C4#1681119511yescharacteristic
601C4#1681127111yescharacteristic
602C4#1681127111yescharacteristic
603C2—42no—
604C2—161no—
605C2—8481no—
606C2#2027918411yescharacteristic
607C2#2028517911yescharacteristic
608C2#2028519811yescharacteristic
609C2#2028720912yes—
610C2#2028731512yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2504C2
socleC23×C53C22×D4
prime radicalC2×#16810874C2
hypercenter#2504D53
hypocenterC53#27942
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2504

Complements

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

NH ≅ G/Ntype
1. C2×#16810944C2semidirect
2. C2×#16810936C2semidirect
4. C2×#16810663C2semidirect
5. #20279177C2semidirect
6. #20276243C2semidirect
7. #20276239C2semidirect
8. #20276239C2semidirect
9. #20276236C2semidirect
10. #20275590C2semidirect
11. #20275580C2semidirect
12. #20274884C2semidirect
13. #20274877C2semidirect
14. #20274499C2semidirect
15. C53×#2504C2semidirect
302. #2504D53semidirect
364. C53#27942semidirect

Maximal subgroups

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

subgroupindex
C2×#168109442
C2×#168109362
C2×#168108742
C2×#168106632
#202791772
#202762432
#202762392
#202762362
#202755902
#202755802
#202748842
#202748772
#202744992
C53×#25042
#2794253