Finite Group Databasebeta

← #27469525

Subgroups — #27469525

Order 13568.

904 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
1C2×#16810964C211yesmaximal, characteristic, complemented
2C2×#16810934C211yesmaximal, characteristic
3C2×#16810867C211yesmaximal, characteristic
4C2×#16810666C211yesmaximal, characteristic, complemented
5#20279193C211yesmaximal, characteristic, complemented
6#20276266C211yesmaximal, characteristic, complemented
7#20276254C211yesmaximal, characteristic, complemented
8#20276248C211yesmaximal, characteristic, complemented
9#20276243C211yesmaximal, characteristic, complemented
10#20275595C211yesmaximal, characteristic, complemented
11#20275568C211yesmaximal, characteristic, complemented
12#20274884C211yesmaximal, characteristic, complemented
13#20274874C211yesmaximal, characteristic, complemented
14#20274499C211yesmaximal, characteristic, complemented
15C53×#2500C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20#16810964—24no—
21#16810934—24no—
22#16810867—24no—
23#16810666—24no—
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28C2×#13133150C2211yescharacteristic
29C2×#13133150C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31C2×#13133148C2211yescharacteristic
32C2×#13133148C2211yescharacteristic
33C2×#13133140C2211yescharacteristic
34C2×#13133140C2211yescharacteristic
35C2×#13133140C2211yescharacteristic
36C2×#13133138C2211yescharacteristic
37C2×#13133138C2211yescharacteristic
38C2×#13133138C2211yescharacteristic
39C2×C22⋊#88322C2211yescharacteristic
40C2×#13133128C2211yescharacteristic
41C2×C4×#88322C2211yescharacteristic
42#16810555—22no—
43#16810529C2211yescharacteristic
44#16810529C2211yescharacteristic
45#16810498—22no—
46#16810493C2211yescharacteristic
47#16810493C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#121×C53C2211yescharacteristic
51C2×#117×C53C2211yescharacteristic
52C53×#394—22no—
53C53×#390C2211yescharacteristic
54C53×#390C2211yescharacteristic
55C53×#388—22no—
56C53×#384C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C23×Dic53—21no—
61C2×#11719923—22no—
62C2×#11719923—22no—
63C2×#11719923—22no—
64C2×#11719917—22no—
65C2×#11719917—22no—
66C2×#11719917D414yes—
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71#13133196—22no—
72#13133196—22no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53—22no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83#13133150—22no—
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#13133140—22no—
93#13133140—22no—
94#13133140—22no—
95#13133138—22no—
96#13133138—22no—
97#13133138—22no—
98C22⋊#88322D414yes—
99#13133128—22no—
100C4×#88322—22no—
101#13133124—22no—
102#13133124—22no—
103C22×D4×C53—21no—
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53C2311yescharacteristic
106#121×C53—22no—
107#121×C53—22no—
108#117×C53—22no—
109#117×C53—22no—
110C2×C4⋊C4×C53—22no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—22no—
117C2×C22⋊C4×C53—22no—
118C2×C22⋊C4×C53D412yes—
119C2×C42×C53—21no—
120C2×C42×C53C2311yescharacteristic
121C2×C42×C53C2311yescharacteristic
122#95×C53—22no—
123#95×C53D412yes—
124C24×C53—21no—
125C24×C53C2×D411yescharacteristic
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—42no—
133C2×D4×C53—82no—
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
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—42no—
152C22×Dic53—82no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—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—82no—
166C22⋊C4×C53—24no—
167C22⋊C4×C53—42no—
168C22⋊C4×C53—42no—
169C22⋊C4×C53—44no—
170C22⋊C4×C53—82no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—42no—
176C22⋊Dic53—42no—
177#11719923—44no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—42no—
185C4⋊Dic53—82no—
186C4⋊Dic53—82no—
187#11719919—42no—
188#11719917—28no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322C2×D412yes—
196C2×#88322C2×D412yes—
197C2×#88322C2×D412yes—
198C2×#88322C2×D412yes—
199C2×#88322#4112yes—
200C2×#88322#4112yes—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—22no—
204C23×C53—22no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—42no—
209C23×C53—42no—
210C23×C53#12712yes—
211C23×C53C22×D411yescharacteristic
212D4×C53—41no—
213D4×C53—41no—
214D4×C53—41no—
215D4×C53—41no—
216D4×C53—84no—
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—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53#12712yes—
235C2×C4×C53#13714yes—
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C532−1+411yescharacteristic
241C2×C4×C532−1+411yescharacteristic
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—84no—
252Dic106—41no—
253Dic106—41no—
254Dic106—41no—
255Dic106—41no—
256#88322—22no—
257#88322—22no—
258#88322—22no—
259#88322#12314yes—
260#27945—531nomaximal, Sylow
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—82no—
267C22×C53—82no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53C2×#12711yescharacteristic
272C22×C53#54711yescharacteristic
273C22×C53#54711yescharacteristic
274C22×C53C2×#13712yes—
275C22×C53#57612yes—
276C4×C53—24no—
277C4×C53—22no—
278C4×C53—42no—
279C4×C53—81no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53C2×#12311yescharacteristic
283C4×C53#53611yescharacteristic
284C4×C53#54711yescharacteristic
285C4×C53#54711yescharacteristic
286Dic53—81no—
287Dic53—81no—
288Dic53—81no—
289Dic53—81no—
290C2×#485—531no—
291C2×#480—531no—
292C2×#468—531no—
293C2×#428—531no—
294#2502—531no—
295#2500D5311yescharacteristic
296#2097—531no—
297#2095—531no—
298#2094—531no—
299#2093—531no—
300#1973—531no—
301#1968—531no—
302#1847—531no—
303#1845—531no—
304#1753—531no—
305C2×C53—42no—
306C2×C53—161no—
307C2×C53#250111yescharacteristic
308C2×C53#315211yescharacteristic
309C2×C53#315511yescharacteristic
310C2×C53#334612yes—
311C2×C53#335712yes—
312C22×SD16—531no—
313C22×M16—531no—
314C2×C22⋊Q8—531no—
315C2×#121C2×D5311yescharacteristic
316C2×#117C2×D5311yescharacteristic
317C22×C4⋊C4—531no—
318#485—1064no—
319#480—1064no—
320#468—1064no—
321#428—1064no—
322C2×#107—531no—
323C2×#107—531no—
324C2×#107—531no—
325C2×#107—531no—
326C2×#107—531no—
327C2×#107—531no—
328C2×#106—531no—
329C2×#106—531no—
330C2×#106—531no—
331C2×#103—531no—
332C2×#103—531no—
333C2×#103—531no—
334C2×#102—531no—
335C2×#102—531no—
336C2×#102—531no—
337C2×C22⋊C8—531no—
338C2×#97—531no—
339C2×C4×C8—531no—
340#398—1062no—
341#394—22no—
342#391—531no—
343#391—531no—
344#390C2×D5311yescharacteristic
345#390C2×D5311yescharacteristic
346#388—22no—
347#385—1062no—
348#384—531no—
349#384—531no—
350#384C2×D5311yescharacteristic
351#382C2×D5311yescharacteristic
352C4×C22⋊C4C2×D5311yescharacteristic
353#336—531no—
354#336—531no—
355C53#2794511yescharacteristic, complemented, Sylow
356C22×Q8—531no—
357C22×D4—21no—
358C22×D4C22×D5311yescharacteristic
359C23×C4—1061no—
360C23×C4C22×D5311yescharacteristic
361C2×SD16—1062no—
362C2×SD16—1062no—
363C2×SD16—1062no—
364C2×M16—534no—
365C2×M16—1062no—
366C2×M16—1062no—
367C22×C8—531no—
368C22×C8—531no—
369C22×C8—531no—
370C22×C8—531no—
371C22⋊Q8—1062no—
372C22⋊Q8—1062no—
373#121—22no—
374#121—22no—
375#117—22no—
376#117—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—1062no—
382C2×C4⋊C4—1062no—
383C2×C4⋊C4—1062no—
384C2×C4⋊C4—1062no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4C22⋊D5312yes—
396C2×C42—21no—
397C2×C42C22×D5311yescharacteristic
398C2×C42C22×D5311yescharacteristic
399#107—1062no—
400#107—1062no—
401#107—1062no—
402#107—1062no—
403#107—1062no—
404#107—1062no—
405#106—1062no—
406#106—1062no—
407#106—1062no—
408#103—1062no—
409#103—1062no—
410#103—1062no—
411#102—1062no—
412#102—1062no—
413#102—1062no—
414C22⋊C8—534no—
415#97—1062no—
416C4×C8—1062no—
417#95—22no—
418#95—1062no—
419#95—1062no—
420#95C22⋊D5312yes—
421C24—21no—
422C24C2×C22⋊D5311yescharacteristic
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C2×Q8—1061no—
427C2×Q8—1061no—
428C2×Q8—1061no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—42no—
436C2×D4—82no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—4242no—
449C22×C4—2122no—
450C22×C4D4×D5312yes—
451C22×C4D4×D5312yes—
452C22×C4C2×C22⋊D5312yes—
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C2×C22⋊D5312yes—
455C22×C4C23×D5311yescharacteristic
456SD16—2124no—
457M16—1068no—
458C2×C8—532no—
459C2×C8—532no—
460C2×C8—532no—
461C2×C8—532no—
462C2×C8—532no—
463C2×C8—532no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—82no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—4242no—
484C4⋊C4—4242no—
485C4⋊C4—2122no—
486C4⋊C4—2122no—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—44no—
491C22⋊C4—82no—
492C22⋊C4—2122no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—42no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23#1313321312yes—
508C23C22×C22⋊D5311yescharacteristic
509Q8—2121no—
510Q8—2121no—
511Q8—2121no—
512Q8—2121no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—84no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—4244no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4#1313321312yes—
546C2×C4#1313324714yes—
547C2×C4C2×D4×D5311yescharacteristic
548C2×C4C2×D4×D5311yescharacteristic
549C2×C4#41×D5311yescharacteristic
550C2×C4#41×D5311yescharacteristic
551C2×C4#1313326411yescharacteristic
552C2×C4#1313326411yescharacteristic
553C8—534no—
554C8—1062no—
555C8—1062no—
556C8—1062no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—42no—
561C22—42no—
562C22—81no—
563C22—82no—
564C22—81no—
565C22—82no—
566C22—81no—
567C22C2×#1313321311yescharacteristic
568C22#1681127211yescharacteristic
569C22#1681127211yescharacteristic
570C22C2×#1313324712yes—
571C22#1681142612yes—
572C4—22no—
573C4—24no—
574C4—42no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—4241no—
579C4—4241no—
580C4—4241no—
581C4—4241no—
582C4#123×D5311yescharacteristic
583C4#1681119511yescharacteristic
584C4#1681127111yescharacteristic
585C4#1681127111yescharacteristic
586C2—42no—
587C2—161no—
588C2#2027918411yescharacteristic
589C2#2028517511yescharacteristic
590C2#2028519811yescharacteristic
591C2#2028720012yes—
592C2#2028733512yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2500C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2500D53
hypocenterC53#27945
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2500

Complements

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

NH ≅ G/Ntype
1. C2×#16810964C2semidirect
4. C2×#16810666C2semidirect
5. #20279193C2semidirect
6. #20276266C2semidirect
7. #20276254C2semidirect
8. #20276248C2semidirect
9. #20276243C2semidirect
10. #20275595C2semidirect
11. #20275568C2semidirect
12. #20274884C2semidirect
13. #20274874C2semidirect
14. #20274499C2semidirect
355. C53#27945semidirect

Maximal subgroups

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

subgroupindex
C2×#168109642
C2×#168109342
C2×#168108672
C2×#168106662
#202791932
#202762662
#202762542
#202762482
#202762432
#202755952
#202755682
#202748842
#202748742
#202744992
C53×#25002
#2794553