Finite Group Databasebeta

← #27466369

Subgroups — #27466369

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279164C211yesmaximal, characteristic
2#20278429C211yesmaximal, characteristic
3#20276288C211yesmaximal, characteristic, complemented
4#20276275C211yesmaximal, characteristic, complemented
5#20276127C211yesmaximal, characteristic, complemented
6#20275865C211yesmaximal, characteristic, complemented
7#20275678C211yesmaximal, characteristic, complemented
8#20275392C211yesmaximal, characteristic, complemented
9#20275389C211yesmaximal, characteristic, complemented
10#20275136C211yesmaximal, characteristic
11#20275025C211yesmaximal, characteristic
12#20274841C211yesmaximal, characteristic, complemented
13#20274489C211yesmaximal, characteristic
14#20274431C211yesmaximal, characteristic
15C53×#1825C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133208C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20C2×#13133142C2211yescharacteristic
21C2×#13133142C2211yescharacteristic
22C2×#13133140C2211yescharacteristic
23C2×#13133140C2211yescharacteristic
24C2×C8×Dic53C2211yescharacteristic
25C2×C8×Dic53C2211yescharacteristic
26#16810559—21no—
27#16810551—21no—
28#16810551—21no—
29#16810546—21no—
30#16810528C2211yescharacteristic
31#16810528C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810520—21no—
34#16810519C2211yescharacteristic
35#16810510—21no—
36#16810507—21no—
37#16810507—21no—
38#16810492C2211yescharacteristic
39#16810492C2211yescharacteristic
40#16810488C2211yescharacteristic
41C4×C22⋊Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C53×#380C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×#88322—22no—
61C22×#88322—22no—
62C22×#88322—41no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133208—22no—
66#13133208—41no—
67#13133196—22no—
68#13133196—22no—
69#13133176—22no—
70#13133176—41no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53D411yescharacteristic
86C2×C22⋊Dic53D411yescharacteristic
87C2×C4×Dic53—21no—
88C2×C4×Dic53C2311yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90#13133142—22no—
91#13133142—41no—
92#13133140—22no—
93#13133140—41no—
94C8×Dic53—22no—
95C8×Dic53—41no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124D411yescharacteristic
105#13133124D411yescharacteristic
106C23×C4×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109#117×C53—22no—
110#117×C53—22no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C42×C53—21no—
118C22⋊C8×C53—22no—
119#95×C53—21no—
120C24×C53C2×D411yescharacteristic
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53#4112yes—
131C22×C4×C53#4112yes—
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C2×Dic106—21no—
142C2×Dic106—21no—
143C2×Dic106—21no—
144C2×Dic106—21no—
145C2×Dic106—21no—
146C2×Dic106—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C22⋊C4×C53—42no—
162C22⋊C4×C53—81no—
163C42×C53—42no—
164C22⋊Dic53—22no—
165C22⋊Dic53—22no—
166C22⋊Dic53—42no—
167C22⋊Dic53—42no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—42no—
174C4⋊Dic53—81no—
175C4⋊Dic53#4112yes—
176#11719919—42no—
177#11719919—42no—
178#11719919—42no—
179#11719919—42no—
180#11719919—42no—
181#11719919—42no—
182#11719919—81no—
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
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
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#12412yes—
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53H(5,𝔽2)11yescharacteristic
227C2×C4×C532−1+411yescharacteristic
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—22no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
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#13512yes—
251C2×Dic53#13512yes—
252Dic106—41no—
253Dic106—41no—
254Dic106—41no—
255Dic106—41no—
256C8×C53—41no—
257C8×C53—41no—
258#88322—22no—
259#88322—41no—
260#27700—531nomaximal, Sylow
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53C2×#12411yescharacteristic
269C22×C53#54611yescharacteristic
270C22×C53#54711yescharacteristic
271C22×C53C2×#13511yescharacteristic
272C22×C53C2×#13511yescharacteristic
273C22×C53#57711yescharacteristic
274C22×C53#57811yescharacteristic
275C4×C53—22no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53—81no—
280C4×C53#52911yescharacteristic
281C4×C53#53911yescharacteristic
282C4×C53#55111yescharacteristic
283C4×C53#55211yescharacteristic
284Dic53—22no—
285Dic53—22no—
286Dic53—41no—
287Dic53—42no—
288Dic53—41no—
289Dic53—81no—
290Dic53—81no—
291Dic53—81no—
292#2499—531no—
293#2441—531no—
294#2101—531no—
295#2099—531no—
296#2072—531no—
297#2026—531no—
298#1991—531no—
299#1933—531no—
300#1933—531no—
301#1892—531no—
302#1873—531no—
303#1834—531no—
304#1825D5311yescharacteristic
305C8×C22⋊C4—531no—
306#1735—531no—
307C2×C53—42no—
308C2×C53#248811yescharacteristic
309C2×C53#313111yescharacteristic
310C2×C53#313711yescharacteristic
311C2×C53#330211yescharacteristic
312C2×C53#330411yescharacteristic
313C2×C53#333611yescharacteristic
314C2×C53#333711yescharacteristic
315C23×C8—531no—
316C2×#125—531no—
317C2×C22⋊Q8—531no—
318C2×#117—531no—
319C2×#117C2×D5311yescharacteristic
320C2×#103—531no—
321C2×#103—531no—
322C2×#103—531no—
323C2×#103—531no—
324C2×C22⋊C8C2×D5311yescharacteristic
325C2×C4×C8—531no—
326C2×C4×C8—531no—
327#399—1061no—
328#397—1061no—
329#397—1061no—
330#396—1061no—
331#391—531no—
332#391—531no—
333#389—531no—
334#389—531no—
335#389—1061no—
336#387—1061no—
337#387—1061no—
338#387—1061no—
339#384—531no—
340#384—531no—
341#382—531no—
342#380C2×D5311yescharacteristic
343C4×C22⋊C4—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340C2×D5311yescharacteristic
353#340C2×D5311yescharacteristic
354#340C2×D5311yescharacteristic
355#340C2×D5311yescharacteristic
356#336—531no—
357#336—531no—
358C53#2770011yescharacteristic, complemented, Sylow
359C22×Q8—531no—
360C23×C4C22×D5311yescharacteristic
361C22×C8—531no—
362C22×C8—531no—
363C22×C8—1062no—
364C22×C8—1062no—
365C22×C8—2121no—
366C22×C8C22×D5311yescharacteristic
367C22×C8C22×D5311yescharacteristic
368#125—1062no—
369#125—2121no—
370C22⋊Q8—1062no—
371C22⋊Q8—1062no—
372#117—22no—
373#117—22no—
374#117—1062no—
375#117—2121no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
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×C4⋊C4C22×D5311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—531no—
395C2×C22⋊C4—531no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C42—21no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—1061no—
402#103—1062no—
403#103—1062no—
404#103—2121no—
405#103—2121no—
406C22⋊C8—22no—
407C4×C8—1062no—
408C4×C8—2121no—
409#95—21no—
410#95—531no—
411#95—531no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420C24D4×D5311yescharacteristic
421C2×Q8—1061no—
422C2×Q8—1061no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C2×Q8—1061no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4#1171994512yes—
444C22×C4D4×D5311yescharacteristic
445C22×C4#1171994712yes—
446C22×C4C23×D5311yescharacteristic
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—532no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—2124no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—532no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—4241no—
475C4⋊C4—4241no—
476C4⋊C4—2122no—
477C4⋊C4—2122no—
478C4⋊C4—2122no—
479C4⋊C4—2122no—
480C4⋊C4—2122no—
481C4⋊C4—2122no—
482C4⋊C4—2122no—
483C22⋊C4—42no—
484C22⋊C4—81no—
485C22⋊C4—1062no—
486C22⋊C4—1062no—
487C22⋊C4—2122no—
488C22⋊C4—2122no—
489C42—42no—
490C42—1061no—
491C42—1061no—
492C42—1061no—
493C42—1061no—
494C42—2122no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23#1313320312yes—
502C23C2×D4×D5311yescharacteristic
503Q8—2121no—
504Q8—2121no—
505Q8—2121no—
506Q8—2121no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
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—532no—
526C2×C4—532no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1062no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—2121no—
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#1313320312yes—
550C2×C4#41×D5311yescharacteristic
551C2×C4#41×D5311yescharacteristic
552C2×C4C2×#1171994711yescharacteristic
553C2×C4C2×#1171994511yescharacteristic
554C2×C4#1313326311yescharacteristic
555C2×C4#1313326411yescharacteristic
556C8—41no—
557C8—41no—
558C8—1062no—
559C8—2121no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22C2×#1313320311yescharacteristic
568C22#1681126511yescharacteristic
569C22#1681127111yescharacteristic
570C22#135×D5311yescharacteristic
571C22#135×D5311yescharacteristic
572C22#1681143311yescharacteristic
573C22#1681143911yescharacteristic
574C4—22no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—1062no—
580C4—1062no—
581C4—4241no—
582C4—4241no—
583C4—4241no—
584C4—2121no—
585C4—2122no—
586C4—2121no—
587C4#1681115011yescharacteristic
588C4#1681121711yescharacteristic
589C4#1681129711yescharacteristic
590C4#1681130611yescharacteristic
591C2—42no—
592C2#2027903711yescharacteristic
593C2#2028503511yescharacteristic
594C2#2028508311yescharacteristic
595C2#2028663011yescharacteristic
596C2#2028665211yescharacteristic
597C2#2028707611yescharacteristic
598C2#2028708711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1825C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1825D53
hypocenterC53#27700
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
3. #20276288C2semidirect
4. #20276275C2semidirect
5. #20276127C2semidirect
6. #20275865C2semidirect
7. #20275678C2semidirect
8. #20275392C2semidirect
9. #20275389C2semidirect
12. #20274841C2semidirect
358. C53#27700semidirect

Maximal subgroups

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

subgroupindex
#202791642
#202784292
#202762882
#202762752
#202761272
#202758652
#202756782
#202753922
#202753892
#202751362
#202750252
#202748412
#202744892
#202744312
C53×#18252
#2770053