Finite Group Databasebeta

← #27466341

Subgroups — #27466341

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279169C211yesmaximal, characteristic
2#20278400C211yesmaximal, characteristic
3#20276286C211yesmaximal, characteristic, complemented
4#20276275C211yesmaximal, characteristic, complemented
5#20276136C211yesmaximal, characteristic, complemented
6#20275854C211yesmaximal, characteristic, complemented
7#20275674C211yesmaximal, characteristic, complemented
8#20275393C211yesmaximal, characteristic, complemented
9#20275388C211yesmaximal, characteristic, complemented
10#20275139C211yesmaximal, characteristic
11#20274844C211yesmaximal, characteristic, complemented
12#20274811C211yesmaximal, characteristic
13#20274489C211yesmaximal, characteristic
14#20274431C211yesmaximal, characteristic
15C53×#1873C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133141C2211yescharacteristic
20C2×#13133141C2211yescharacteristic
21C2×C8×Dic53C2211yescharacteristic
22C2×C8×Dic53C2211yescharacteristic
23#16810551—21no—
24#16810551—21no—
25#16810546—21no—
26#16810535—21no—
27#16810530C2211yescharacteristic
28#16810530C2211yescharacteristic
29#16810529—21no—
30#16810522C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810507—21no—
33#16810504—21no—
34#16810504—21no—
35#16810492C2211yescharacteristic
36#16810488C2211yescharacteristic
37#16810488C2211yescharacteristic
38#16810483C2211yescharacteristic
39C4×C22⋊Dic53C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C22⋊Q8×C53C2211yescharacteristic
53C2×#117×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C22×#88322—22no—
61C22×#88322—22no—
62C22×#88322—41no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133216—22no—
66#13133216—41no—
67#13133176—22no—
68#13133176—22no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53D411yescharacteristic
81C2×C22⋊Dic53D411yescharacteristic
82C2×C4×Dic53—21no—
83C2×C4×Dic53C2311yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85#13133141—22no—
86#13133141—41no—
87C8×Dic53—22no—
88C8×Dic53—41no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124D411yescharacteristic
101#13133124D411yescharacteristic
102C22×Q8×C53C2311yescharacteristic
103C23×C4×C53C2311yescharacteristic
104C22×C8×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22⋊Q8×C53—22no—
107C22⋊Q8×C53—22no—
108#117×C53—22no—
109#117×C53—41no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C42×C53—21no—
117#103×C53—22no—
118#103×C53—41no—
119C22⋊C8×C53—22no—
120C24×C53C2×D411yescharacteristic
121C2×Q8×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53#4112yes—
137C22×C4×C53#4112yes—
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C2×Dic106—42no—
148C2×Dic106—42no—
149C2×Dic106—81no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—42no—
162C4⋊C4×C53#4112yes—
163C22⋊C4×C53—42no—
164C22⋊C4×C53—42no—
165C42×C53—42no—
166C22⋊Dic53—22no—
167C22⋊Dic53—22no—
168C22⋊Dic53—42no—
169C22⋊Dic53—81no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—81no—
179#11719919—42no—
180#11719919—42no—
181#11719919—42no—
182#11719919—42no—
183C4×Dic53—21no—
184C4×Dic53—21no—
185C4×Dic53—21no—
186C4×Dic53—21no—
187C4×Dic53—42no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—44no—
194C2×#88322#4112yes—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53#12412yes—
202C23×C53C22×D411yescharacteristic
203Q8×C53—41no—
204Q8×C53—41no—
205Q8×C53—41no—
206Q8×C53—41no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53#12412yes—
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53H(5,𝔽2)11yescharacteristic
231C2×C4×C532−1+411yescharacteristic
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53#13512yes—
255C2×Dic53#13512yes—
256Dic106—84no—
257C8×C53—41no—
258C8×C53—41no—
259#88322—22no—
260#88322—41no—
261#27698—531nomaximal, Sylow
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—22no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53C2×#12411yescharacteristic
270C22×C53#54611yescharacteristic
271C22×C53#54711yescharacteristic
272C22×C53C2×#13511yescharacteristic
273C22×C53C2×#13511yescharacteristic
274C22×C53#57711yescharacteristic
275C22×C53#57811yescharacteristic
276C4×C53—22no—
277C4×C53—42no—
278C4×C53—81no—
279C4×C53—81no—
280C4×C53—81no—
281C4×C53#52911yescharacteristic
282C4×C53#53911yescharacteristic
283C4×C53#55111yescharacteristic
284C4×C53#55211yescharacteristic
285Dic53—22no—
286Dic53—22no—
287Dic53—41no—
288Dic53—41no—
289Dic53—81no—
290Dic53—81no—
291Dic53—81no—
292Dic53—81no—
293#2499—531no—
294#2439—531no—
295#2101—531no—
296#2099—531no—
297#2074—531no—
298#2024—531no—
299#1990—531no—
300#1933—531no—
301#1933—531no—
302#1892—531no—
303#1873D5311yescharacteristic
304#1835—531no—
305#1825—531no—
306C8×C22⋊C4—531no—
307#1735—531no—
308C2×C53—42no—
309C2×C53#248811yescharacteristic
310C2×C53#313111yescharacteristic
311C2×C53#313711yescharacteristic
312C2×C53#329611yescharacteristic
313C2×C53#330011yescharacteristic
314C2×C53#334011yescharacteristic
315C2×C53#334211yescharacteristic
316C23×C8—531no—
317C2×C4⋊Q8—531no—
318C2×C22⋊Q8C2×D5311yescharacteristic
319C2×#117—531no—
320C2×#117C2×D5311yescharacteristic
321C2×#103—531no—
322C2×#103—531no—
323C2×#103C2×D5311yescharacteristic
324C2×#103C2×D5311yescharacteristic
325C2×C22⋊C8C2×D5311yescharacteristic
326C2×C4×C8—531no—
327C2×C4×C8—531no—
328#397—1061no—
329#397—1061no—
330#396—1061no—
331#393—1061no—
332#391—531no—
333#391—531no—
334#391—1061no—
335#389—531no—
336#389—531no—
337#387—1061no—
338#387—1061no—
339#387—1061no—
340#384—531no—
341#382—531no—
342#382—531no—
343#380—531no—
344C4×C22⋊C4—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340C2×D5311yescharacteristic
356#340C2×D5311yescharacteristic
357#336—531no—
358#336—531no—
359C53#2769811yescharacteristic, complemented, Sylow
360C22×Q8—1061no—
361C22×Q8C22×D5311yescharacteristic
362C23×C4C22×D5311yescharacteristic
363C22×C8—531no—
364C22×C8—531no—
365C22×C8—1062no—
366C22×C8—1062no—
367C22×C8—2121no—
368C22×C8C22×D5311yescharacteristic
369C22×C8C22×D5311yescharacteristic
370C4⋊Q8—1062no—
371C4⋊Q8—2121no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374#117—22no—
375#117—41no—
376#117—1062no—
377#117—1062no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—531no—
394C2×C22⋊C4—531no—
395C2×C22⋊C4—1061no—
396C2×C22⋊C4—1061no—
397C2×C42—21no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—1061no—
401#103—22no—
402#103—41no—
403#103—1062no—
404#103—2121no—
405C22⋊C8—22no—
406C4×C8—1062no—
407C4×C8—2121no—
408#95—531no—
409#95—531no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421C24D4×D5311yescharacteristic
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—4241no—
429C2×Q8—2122no—
430C2×Q8—2122no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4#1171994512yes—
448C22×C4D4×D5311yescharacteristic
449C22×C4#1171994712yes—
450C22×C4C23×D5311yescharacteristic
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—532no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—2124no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—4241no—
479C4⋊C4—2122no—
480C4⋊C4—2122no—
481C4⋊C4—2122no—
482C4⋊C4—2122no—
483C4⋊C4#1171994712yes—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—1062no—
487C22⋊C4—1062no—
488C22⋊C4—4241no—
489C22⋊C4—2122no—
490C42—42no—
491C42—1061no—
492C42—1061no—
493C42—1061no—
494C42—1061no—
495C42—2122no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23#1313320312yes—
503C23C2×D4×D5311yescharacteristic
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—4244no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—532no—
528C2×C4—532no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4#1313320312yes—
552C2×C4#41×D5311yescharacteristic
553C2×C4C2×#1171994711yescharacteristic
554C2×C4C2×#1171994711yescharacteristic
555C2×C4C2×#1171994511yescharacteristic
556C2×C4#1313326211yescharacteristic
557C2×C4#1313326411yescharacteristic
558C8—41no—
559C8—41no—
560C8—1062no—
561C8—2121no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22C2×#1313320311yescharacteristic
570C22#1681126011yescharacteristic
571C22#1681127111yescharacteristic
572C22#135×D5311yescharacteristic
573C22#135×D5311yescharacteristic
574C22#1681143211yescharacteristic
575C22#1681143911yescharacteristic
576C4—22no—
577C4—42no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—1062no—
582C4—1062no—
583C4—4241no—
584C4—4241no—
585C4—4241no—
586C4—4241no—
587C4—2121no—
588C4—2121no—
589C4#1681115411yescharacteristic
590C4#1681121811yescharacteristic
591C4#1681130211yescharacteristic
592C4#1681131311yescharacteristic
593C2—42no—
594C2#2027903911yescharacteristic
595C2#2028503511yescharacteristic
596C2#2028507611yescharacteristic
597C2#2028654311yescharacteristic
598C2#2028659511yescharacteristic
599C2#2028712111yescharacteristic
600C2#2028714011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1873C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1873D53
hypocenterC53#27698
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
3. #20276286C2semidirect
4. #20276275C2semidirect
5. #20276136C2semidirect
6. #20275854C2semidirect
7. #20275674C2semidirect
8. #20275393C2semidirect
9. #20275388C2semidirect
11. #20274844C2semidirect
359. C53#27698semidirect

Maximal subgroups

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

subgroupindex
#202791692
#202784002
#202762862
#202762752
#202761362
#202758542
#202756742
#202753932
#202753882
#202751392
#202748442
#202748112
#202744892
#202744312
C53×#18732
#2769853