Finite Group Databasebeta

← #27469532

Subgroups — #27469532

Order 13568.

888 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810941C211yesmaximal, characteristic
2C2×#16810961C211yesmaximal, characteristic, complemented
3C2×#16810871C211yesmaximal, characteristic
4C2×#16810663C211yesmaximal, characteristic, complemented
5#20279177C211yesmaximal, characteristic
6#20276263C211yesmaximal, characteristic, complemented
7#20276258C211yesmaximal, characteristic, complemented
8#20276239C211yesmaximal, characteristic, complemented
9#20276239C211yesmaximal, characteristic, complemented
10#20275599C211yesmaximal, characteristic, complemented
11#20275572C211yesmaximal, characteristic, complemented
12#20274884C211yesmaximal, characteristic, complemented
13#20274877C211yesmaximal, characteristic, complemented
14#20274499C211yesmaximal, characteristic, complemented
15C53×#2503C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×#13133191C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20#16810961—24no—
21#16810941—24no—
22#16810871—24no—
23#16810663—24no—
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28C2×#13133148C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31C2×#13133148C2211yescharacteristic
32C2×#13133148C2211yescharacteristic
33C2×#13133141C2211yescharacteristic
34C2×#13133141C2211yescharacteristic
35C2×#13133141C2211yescharacteristic
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×C22⋊Q8×C53C2211yescharacteristic
51C22×C4⋊C4×C53C2211yescharacteristic
52C53×#400—22no—
53C53×#391C2211yescharacteristic
54C53×#391C2211yescharacteristic
55C53×#382C2211yescharacteristic
56C53×#382C2211yescharacteristic
57C53×#381—22no—
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C2×#11719924—22no—
62C2×#11719924—22no—
63C2×#11719924—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#13133148—22no—
89#13133148—22no—
90#13133148—22no—
91#13133148—22no—
92#13133148—22no—
93#13133141—22no—
94#13133141—22no—
95#13133141—22no—
96#13133137—22no—
97#13133137—22no—
98#13133137—22no—
99C22⋊#88322D414yes—
100#13133128—22no—
101C4×#88322—22no—
102#13133124—22no—
103C22×Q8×C53C2311yescharacteristic
104C23×C4×C53—21no—
105C23×C4×C53C2311yescharacteristic
106C22⋊Q8×C53—22no—
107C22⋊Q8×C53—22no—
108C2×C4⋊C4×C53—21no—
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×C53D412yes—
117C2×C42×C53C2311yescharacteristic
118C2×C42×C53C2311yescharacteristic
119#95×C53—22no—
120#95×C53—22no—
121#95×C53—22no—
122#95×C53—22no—
123#95×C53D412yes—
124C24×C53C2×D411yescharacteristic
125C23×D53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C22×C4×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—44no—
141C22×C4×C53—42no—
142C22×C4×C53C2×D412yes—
143C22×C4×C53C2×D412yes—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D412yes—
147C22×C4×C53C2411yescharacteristic
148C2×C22⋊D53—42no—
149C2×C22⋊D53—82no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—42no—
166C4⋊C4×C53—42no—
167C22⋊C4×C53—42no—
168C22⋊C4×C53—42no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—21no—
173C22⋊Dic53—42no—
174#11719924—44no—
175#11719921—44no—
176#11719921—42no—
177#11719921—82no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—82no—
185C4×Dic53—42no—
186#11719917—28no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322C2×D412yes—
194C2×#88322C2×D412yes—
195C2×#88322C2×D412yes—
196C2×#88322C2×D412yes—
197C2×#88322#4112yes—
198C2×#88322#4112yes—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—22no—
202C23×C53—22no—
203C23×C53#12712yes—
204C23×C53C22×D411yescharacteristic
205C22×D53—41no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—42no—
209C22×D53—42no—
210Q8×C53—41no—
211Q8×C53—41no—
212Q8×C53—41no—
213Q8×C53—41no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
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—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—84no—
236C2×C4×C53#12712yes—
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C532−1+411yescharacteristic
242C2×C4×C532−1+411yescharacteristic
243C22⋊D53—84no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250D212—41no—
251D212—41no—
252D212—41no—
253D212—41no—
254#88322—22no—
255#88322—22no—
256#88322—22no—
257#88322#12314yes—
258#27946—531nomaximal, Sylow
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53C2×#12711yescharacteristic
265C22×C53#54711yescharacteristic
266C22×C53#54711yescharacteristic
267C22×C53#57512yes—
268C22×C53#57712yes—
269C2×D53—82no—
270C2×D53—82no—
271C2×D53—81no—
272C2×D53—81no—
273C2×D53—81no—
274C4×C53—22no—
275C4×C53—42no—
276C4×C53—42no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53—81no—
280C4×C53—81no—
281C4×C53C2×#12311yescharacteristic
282C4×C53#53611yescharacteristic
283C4×C53#54711yescharacteristic
284C4×C53#54711yescharacteristic
285Dic53—81no—
286Dic53—81no—
287Dic53—81no—
288C2×#481—531no—
289C2×#485—531no—
290C2×#468—531no—
291C2×#428—531no—
292#2503D5311yescharacteristic, complemented
293#2500—531no—
294#2097—531no—
295#2096—531no—
296#2092—531no—
297#2092—531no—
298#1973—531no—
299#1968—531no—
300#1847—531no—
301#1846—531no—
302#1753—531no—
303C2×C53—42no—
304C2×C53#250111yescharacteristic
305C2×C53#315311yescharacteristic
306C2×C53#315411yescharacteristic
307C2×C53#334812yes—
308C2×C53#335112yes—
309D53—161no—
310C22×SD16—531no—
311C22×M16—531no—
312C2×C22⋊Q8C2×D5311yescharacteristic
313C2×#121—531no—
314C2×#117—531no—
315C22×C4⋊C4C2×D5311yescharacteristic
316#485—1064no—
317#481—1064no—
318#468—1064no—
319#428—1064no—
320C2×#107—531no—
321C2×#107—531no—
322C2×#107—531no—
323C2×#107—531no—
324C2×#106—531no—
325C2×#106—531no—
326C2×#106—531no—
327C2×#106—531no—
328C2×#106—531no—
329C2×#103—531no—
330C2×#103—531no—
331C2×#103—531no—
332C2×#102—531no—
333C2×#102—531no—
334C2×#102—531no—
335C2×C22⋊C8—531no—
336C2×#97—531no—
337C2×C4×C8—531no—
338#400—22no—
339#394—1062no—
340#391C2×D5311yescharacteristic
341#391C2×D5311yescharacteristic
342#390—531no—
343#390—531no—
344#388—1062no—
345#384—531no—
346#382—531no—
347#382C2×D5311yescharacteristic
348#382C2×D5311yescharacteristic
349#381—22no—
350C4×C22⋊C4C2×D5311yescharacteristic
351#336—531no—
352#336—531no—
353C53#2794611yescharacteristic, complemented, Sylow
354C22×Q8C22×D5311yescharacteristic
355C22×D4—531no—
356C22×D4—1061no—
357C23×C4—21no—
358C23×C4C22×D5311yescharacteristic
359C2×SD16—1062no—
360C2×SD16—1062no—
361C2×SD16—1062no—
362C2×M16—534no—
363C2×M16—1062no—
364C2×M16—1062no—
365C22×C8—531no—
366C22×C8—531no—
367C22×C8—531no—
368C22×C8—531no—
369C22⋊Q8—22no—
370C22⋊Q8—22no—
371#121—1062no—
372#121—1062no—
373#117—1062no—
374#117—1062no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—1062no—
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1062no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1062no—
391C2×C22⋊C4C22⋊D5312yes—
392C2×C42—1061no—
393C2×C42C22×D5311yescharacteristic
394C2×C42C22×D5311yescharacteristic
395#107—1062no—
396#107—1062no—
397#107—1062no—
398#107—1062no—
399#106—1062no—
400#106—1062no—
401#106—1062no—
402#106—1062no—
403#106—1062no—
404#103—1062no—
405#103—1062no—
406#103—1062no—
407#102—1062no—
408#102—1062no—
409#102—1062no—
410C22⋊C8—534no—
411#97—1062no—
412C4×C8—1062no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—1062no—
418#95C22⋊D5312yes—
419C24—1061no—
420C24C2×C22⋊D5311yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×D4—1061no—
428C2×D4—1061no—
429C2×D4—1061no—
430C2×D4—1061no—
431C2×D4—1061no—
432C2×D4—1061no—
433C2×D4—4242no—
434C2×D4—2122no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4D4×D5312yes—
449C22×C4D4×D5312yes—
450C22×C4C2×C22⋊D5312yes—
451C22×C4C2×C22⋊D5312yes—
452C22×C4C2×C22⋊D5311yescharacteristic
453C22×C4C23×D5311yescharacteristic
454SD16—2124no—
455M16—1068no—
456C2×C8—532no—
457C2×C8—532no—
458C2×C8—532no—
459C2×C8—532no—
460C2×C8—532no—
461C2×C8—532no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—4242no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—4242no—
486C22⋊C4—2124no—
487C22⋊C4—2122no—
488C22⋊C4—2122no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—2122no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—2121no—
499C23—2121no—
500C23—2121no—
501C23—2122no—
502C23—2122no—
503C23#1313321312yes—
504C23C22×C22⋊D5311yescharacteristic
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509D4—4244no—
510D4—2121no—
511D4—2121no—
512D4—2121no—
513D4—2121no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—84no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4#1313321312yes—
543C2×C4C2×D4×D5311yescharacteristic
544C2×C4C2×D4×D5311yescharacteristic
545C2×C4#41×D5311yescharacteristic
546C2×C4#41×D5311yescharacteristic
547C2×C4#1313326411yescharacteristic
548C2×C4#1313326411yescharacteristic
549C8—534no—
550C8—1062no—
551C8—1062no—
552C8—1062no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—42no—
557C22—42no—
558C22—4241no—
559C22—4242no—
560C22—4241no—
561C22—4242no—
562C22—4241no—
563C22C2×#1313321311yescharacteristic
564C22#1681127211yescharacteristic
565C22#1681127211yescharacteristic
566C22#1681142112yes—
567C22#1681143112yes—
568C4—22no—
569C4—42no—
570C4—42no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—4241no—
576C4—4241no—
577C4—4241no—
578C4#123×D5311yescharacteristic
579C4#1681119511yescharacteristic
580C4#1681127111yescharacteristic
581C4#1681127111yescharacteristic
582C2—42no—
583C2—8481no—
584C2#2027918411yescharacteristic
585C2#2028517911yescharacteristic
586C2#2028519311yescharacteristic
587C2#2028722412yes—
588C2#2028727512yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2503C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2503D53
hypocenterC53#27946
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
2. C2×#16810961C2semidirect
4. C2×#16810663C2semidirect
6. #20276263C2semidirect
7. #20276258C2semidirect
8. #20276239C2semidirect
9. #20276239C2semidirect
10. #20275599C2semidirect
11. #20275572C2semidirect
12. #20274884C2semidirect
13. #20274877C2semidirect
14. #20274499C2semidirect
15. C53×#2503C2semidirect
292. #2503D53semidirect
353. C53#27946semidirect

Maximal subgroups

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

subgroupindex
C2×#168109412
C2×#168109612
C2×#168108712
C2×#168106632
#202791772
#202762632
#202762582
#202762392
#202755992
#202755722
#202748842
#202748772
#202744992
C53×#25032
#2794653