Finite Group Databasebeta

← #27469510

Subgroups — #27469510

Order 13568.

944 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×#16810956C211yesmaximal, characteristic, complemented
2C2×#16810947C211yesmaximal, characteristic
3C2×#16810871C211yesmaximal, characteristic
4C2×#16810666C211yesmaximal, characteristic, complemented
5#20279186C211yesmaximal, characteristic
6#20276258C211yesmaximal, characteristic, complemented
7#20276253C211yesmaximal, characteristic, complemented
8#20276244C211yesmaximal, characteristic, complemented
9#20276239C211yesmaximal, characteristic, complemented
10#20275599C211yesmaximal, characteristic, complemented
11#20275572C211yesmaximal, characteristic, complemented
12#20274884C211yesmaximal, characteristic, complemented
13#20274874C211yesmaximal, characteristic, complemented
14#20274499C211yesmaximal, characteristic, complemented
15C53×#2499C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×#13133201—22no—
19C2×#13133191C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21#16810956—24no—
22#16810947—24no—
23#16810871—24no—
24#16810666—24no—
25C2×#13133150C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28C2×#13133150C2211yescharacteristic
29C2×#13133150C2211yescharacteristic
30C2×#13133150C2211yescharacteristic
31C2×#13133148C2211yescharacteristic
32C2×#13133148C2211yescharacteristic
33C2×#13133148C2211yescharacteristic
34C2×#13133141C2211yescharacteristic
35C2×#13133141C2211yescharacteristic
36C2×#13133141C2211yescharacteristic
37C2×#13133137C2211yescharacteristic
38C2×#13133137C2211yescharacteristic
39C2×#13133137C2211yescharacteristic
40C2×C22⋊#88322C2211yescharacteristic
41C2×#13133128C2211yescharacteristic
42C2×C4×#88322C2211yescharacteristic
43#16810560—22no—
44#16810525C2211yescharacteristic
45#16810525C2211yescharacteristic
46#16810493C2211yescharacteristic
47#16810493C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×#117×C53C2211yescharacteristic
52C53×#397—22no—
53C53×#391C2211yescharacteristic
54C53×#391C2211yescharacteristic
55C53×#387—22no—
56C53×#384C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C23×Dic53—21no—
62C2×#11719924—22no—
63C2×#11719924—22no—
64C2×#11719924—22no—
65C2×#11719917—22no—
66C2×#11719917—22no—
67C2×#11719917D414yes—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72#13133201—48no—
73#13133191—22no—
74#13133191—22no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53—22no—
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719921—22no—
84C2×#11719921—22no—
85C2×#11719921—21no—
86#13133150—22no—
87#13133150—22no—
88#13133150—22no—
89#13133150—22no—
90#13133150—22no—
91#13133150—22no—
92#13133148—22no—
93#13133148—22no—
94#13133148—22no—
95#13133141—22no—
96#13133141—22no—
97#13133141—22no—
98#13133137—22no—
99#13133137—22no—
100#13133137—22no—
101C22⋊#88322D414yes—
102#13133128—22no—
103C4×#88322—22no—
104C22×Q8×C53C2311yescharacteristic
105C23×C4×C53C2311yescharacteristic
106C22⋊Q8×C53—22no—
107C22⋊Q8×C53—22no—
108#117×C53—22no—
109#117×C53—22no—
110C2×C4⋊C4×C53—21no—
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—21no—
117C2×C22⋊C4×C53D412yes—
118C2×C42×C53—21no—
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121#95×C53—22no—
122#95×C53—22no—
123#95×C53—22no—
124#95×C53D412yes—
125C24×C53C2×D411yescharacteristic
126C23×D53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
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×C53C2×D412yes—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D412yes—
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2411yescharacteristic
147C2×C22⋊D53—44no—
148C2×C22⋊D53—42no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—44no—
153C22×Dic53—42no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—42no—
167C4⋊C4×C53—44no—
168C22⋊C4×C53—24no—
169C22⋊C4×C53—42no—
170C22⋊C4×C53—42no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—42no—
176#11719924—44no—
177#11719921—44no—
178#11719921—42no—
179#11719921—44no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—44no—
187C4⋊Dic53—42no—
188C4⋊Dic53—44no—
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#12712yes—
207C23×C53C22×D411yescharacteristic
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—42no—
212C22×D53—42no—
213Q8×C53—41no—
214Q8×C53—41no—
215Q8×C53—41no—
216Q8×C53—41no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53#12712yes—
238C2×C4×C53AGL(1,ℤ/8)14yes—
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C532−1+411yescharacteristic
244C2×C4×C532−1+411yescharacteristic
245C22⋊D53—84no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—84no—
253D212—41no—
254D212—41no—
255D212—41no—
256D212—41no—
257#88322—22no—
258#88322—22no—
259#88322—22no—
260#88322#12314yescomplemented
261#27944—531nomaximal, Sylow
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—22no—
265C22×C53—42no—
266C22×C53—42no—
267C22×C53C2×#12711yescharacteristic
268C22×C53#54711yescharacteristic
269C22×C53#54711yescharacteristic
270C22×C53C2×AGL(1,ℤ/8)12yes—
271C22×C53#57812yes—
272C2×D53—82no—
273C2×D53—82no—
274C2×D53—81no—
275C2×D53—81no—
276C2×D53—81no—
277C4×C53—24no—
278C4×C53—22no—
279C4×C53—42no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53—81no—
283C4×C53—81no—
284C4×C53C2×#12311yescharacteristic
285C4×C53#53611yescharacteristic
286C4×C53#54711yescharacteristic
287C4×C53#54711yescharacteristic
288Dic53—81no—
289Dic53—81no—
290Dic53—81no—
291C2×#484—531no—
292C2×#482—531no—
293C2×#468—531no—
294C2×#428—531no—
295#2501—531no—
296#2499D5311yescharacteristic, complemented
297#2096—531no—
298#2095—531no—
299#2093—531no—
300#2092—531no—
301#1973—531no—
302#1968—531no—
303#1847—531no—
304#1845—531no—
305#1753—531no—
306C2×C53—42no—
307C2×C53#250111yescharacteristic
308C2×C53#315111yescharacteristic
309C2×C53#315611yescharacteristic
310C2×C53#334312yes—
311C2×C53#335312yes—
312D53—161no—
313C22×SD16—531no—
314C22×M16—531no—
315C2×#123—1062no—
316C2×C22⋊Q8C2×D5311yescharacteristic
317C2×#121—531no—
318C2×#117C2×D5311yescharacteristic
319C22×C4⋊C4—531no—
320#484—1064no—
321#482—1064no—
322#468—1064no—
323#428—1064no—
324C2×#107—531no—
325C2×#107—531no—
326C2×#107—531no—
327C2×#107—531no—
328C2×#107—531no—
329C2×#107—531no—
330C2×#106—531no—
331C2×#106—531no—
332C2×#106—531no—
333C2×#103—531no—
334C2×#103—531no—
335C2×#103—531no—
336C2×#102—531no—
337C2×#102—531no—
338C2×#102—531no—
339C2×C22⋊C8—531no—
340C2×#97—531no—
341C2×C4×C8—531no—
342#399—1062no—
343#397—22no—
344#391C2×D5311yescharacteristic
345#391C2×D5311yescharacteristic
346#390—531no—
347#390—531no—
348#387—22no—
349#384—531no—
350#384—531no—
351#384C2×D5311yescharacteristic
352#382C2×D5311yescharacteristic
353C4×C22⋊C4C2×D5311yescharacteristic
354#336—531no—
355#336—531no—
356C53#2794411yescharacteristic, complemented, Sylow
357C22×Q8C22×D5311yescharacteristic
358C22×D4—531no—
359C22×D4—1061no—
360C23×C4—1061no—
361C23×C4C22×D5311yescharacteristic
362C2×SD16—1062no—
363C2×SD16—1062no—
364C2×SD16—1062no—
365C2×M16—534no—
366C2×M16—1062no—
367C2×M16—1062no—
368C22×C8—531no—
369C22×C8—531no—
370C22×C8—531no—
371C22×C8—531no—
372#123—2128no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375#121—1062no—
376#121—1062no—
377#117—22no—
378#117—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—1062no—
385C2×C4⋊C4—1062no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4—1062no—
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1062no—
396C2×C22⋊C4—1062no—
397C2×C22⋊C4C22⋊D5312yes—
398C2×C42—21no—
399C2×C42C22×D5311yescharacteristic
400C2×C42C22×D5311yescharacteristic
401#107—1062no—
402#107—1062no—
403#107—1062no—
404#107—1062no—
405#107—1062no—
406#107—1062no—
407#106—1062no—
408#106—1062no—
409#106—1062no—
410#103—1062no—
411#103—1062no—
412#103—1062no—
413#102—1062no—
414#102—1062no—
415#102—1062no—
416C22⋊C8—534no—
417#97—1062no—
418C4×C8—1062no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95C22⋊D5312yes—
423C24—1061no—
424C24C2×C22⋊D5311yescharacteristic
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×D4—1061no—
432C2×D4—1061no—
433C2×D4—1061no—
434C2×D4—1061no—
435C2×D4—1061no—
436C2×D4—1061no—
437C2×D4—2124no—
438C2×D4—2122no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—2124no—
451C22×C4—2122no—
452C22×C4D4×D5312yes—
453C22×C4D4×D5312yes—
454C22×C4C2×C22⋊D5312yes—
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C2×C22⋊D5312yes—
457C22×C4C23×D5311yescharacteristic
458SD16—2124no—
459M16—1068no—
460C2×C8—532no—
461C2×C8—532no—
462C2×C8—532no—
463C2×C8—532no—
464C2×C8—532no—
465C2×C8—532no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—2124no—
487C4⋊C4—2122no—
488C4⋊C4—2124no—
489C22⋊C4—24no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—2124no—
493C22⋊C4—2122no—
494C22⋊C4—2124no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—42no—
500C23—21no—
501C23—21no—
502C23—22no—
503C23—22no—
504C23—2121no—
505C23—2121no—
506C23—2122no—
507C23—2121no—
508C23—2122no—
509C23#1313321312yes—
510C23C22×C22⋊D5311yescharacteristic
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515D4—4244no—
516D4—2121no—
517D4—2121no—
518D4—2121no—
519D4—2121no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—4244no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4#1313321312yes—
548C2×C4#1313324214yes—
549C2×C4C2×D4×D5311yescharacteristic
550C2×C4C2×D4×D5311yescharacteristic
551C2×C4#41×D5311yescharacteristic
552C2×C4#41×D5311yescharacteristic
553C2×C4#1313326411yescharacteristic
554C2×C4#1313326411yescharacteristic
555C8—534no—
556C8—1062no—
557C8—1062no—
558C8—1062no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—42no—
563C22—42no—
564C22—4241no—
565C22—4242no—
566C22—4241no—
567C22—4242no—
568C22—4241no—
569C22C2×#1313321311yescharacteristic
570C22#1681127211yescharacteristic
571C22#1681127211yescharacteristic
572C22C2×#1313324212yes—
573C22#1681143812yes—
574C4—22no—
575C4—24no—
576C4—42no—
577C4—81no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—4241no—
582C4—4241no—
583C4—4241no—
584C4#123×D5311yescharacteristic
585C4#1681119511yescharacteristic
586C4#1681127111yescharacteristic
587C4#1681127111yescharacteristic
588C2—42no—
589C2—8481no—
590C2#2027918411yescharacteristic
591C2#2028516811yescharacteristic
592C2#2028520611yescharacteristic
593C2#2028715512yes—
594C2#2028729312yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2499C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2499D53
hypocenterC53#27944
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2499

Complements

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

NH ≅ G/Ntype
1. C2×#16810956C2semidirect
4. C2×#16810666C2semidirect
6. #20276258C2semidirect
7. #20276253C2semidirect
8. #20276244C2semidirect
9. #20276239C2semidirect
10. #20275599C2semidirect
11. #20275572C2semidirect
12. #20274884C2semidirect
13. #20274874C2semidirect
14. #20274499C2semidirect
15. C53×#2499C2semidirect
260. #88322#123semidirect
296. #2499D53semidirect
356. C53#27944semidirect

Maximal subgroups

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

subgroupindex
C2×#168109562
C2×#168109472
C2×#168108712
C2×#168106662
#202791862
#202762582
#202762532
#202762442
#202762392
#202755992
#202755722
#202748842
#202748742
#202744992
C53×#24992
#2794453