Finite Group Databasebeta

← #27369604

Subgroups — #27369604

Order 13568.

594 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285876C211yesmaximal, characteristic, complemented
2#20284877C211yesmaximal, characteristic
3#20284786C211yesmaximal, characteristic
4#20274173C211yesmaximal, characteristic, complemented
5#20274166C211yesmaximal, characteristic, complemented
6#20274145C211yesmaximal, characteristic, complemented
7#20274138C211yesmaximal, characteristic, complemented
8#20274109C211yesmaximal, characteristic, complemented
9#20274104C211yesmaximal, characteristic, complemented
10#20274097C211yesmaximal, characteristic, complemented
11#20274090C211yesmaximal, characteristic, complemented
12#20273495C211yesmaximal, characteristic, complemented
13#20273413C211yesmaximal, characteristic, complemented
14#20273373C211yesmaximal, characteristic, complemented
15C53×#2988C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811180C2211yescharacteristic
18#16811028C2211yescharacteristic
19#16811021C2211yescharacteristic
20#16810974C2211yescharacteristic
21#16810974C2211yescharacteristic
22#16810965—21no—
23#16810965—21no—
24#16810964—21no—
25#16810964—21no—
26#16810931—21no—
27#16810931—21no—
28#16810918C2211yescharacteristic
29#16810913C2211yescharacteristic
30#16810908C2211yescharacteristic
31#16810903C2211yescharacteristic
32#16810833C2211yescharacteristic
33#16810833C2211yescharacteristic
34#16810828C2211yescharacteristic
35#16810828C2211yescharacteristic
36#16810785—21no—
37#16810785—21no—
38#16810728—21no—
39#16810723—21no—
40C4×#11719925—21no—
41C4×#11719923—21no—
42#16810680C2211yescharacteristic
43#16810676C2211yescharacteristic
44#16810376C2211yescharacteristic
45#16810376C2211yescharacteristic
46#16810374C2211yescharacteristic
47#16810374C2211yescharacteristic
48#16810371C2211yescharacteristic
49#16810371C2211yescharacteristic
50#16810370C2211yescharacteristic
51#16810370C2211yescharacteristic
52#16810368C2211yescharacteristic
53#16810368C2211yescharacteristic
54#16810368C2211yescharacteristic
55#16810368C2211yescharacteristic
56C4×#41×C53C2211yescharacteristic
57C8×Q8×C53C2211yescharacteristic
58C8×Q8×C53C2211yescharacteristic
59C8×D4×C53—21no—
60C8×D4×C53—21no—
61C8×D4×C53C2211yescharacteristic
62C8×D4×C53C2211yescharacteristic
63C53×#431—21no—
64C53×#431C2211yescharacteristic
65C2×C4⋊C8×C53—21no—
66C2×C4⋊C8×C53C2211yescharacteristic
67C2×#11719925—21no—
68C2×#11719925—21no—
69C2×#11719923—21no—
70C2×#11719923—21no—
71#13133217C2311yescharacteristic
72#13133217C2311yescharacteristic
73#13133217C2311yescharacteristic
74#13133217C2311yescharacteristic
75#13133209—21no—
76#13133209—21no—
77#13133196—21no—
78#13133196—21no—
79#13133196—21no—
80#13133196—21no—
81C4×Dic106—21no—
82C4×Dic106—21no—
83C4×Dic106—21no—
84C4×Dic106—21no—
85#13133176—21no—
86#13133176—21no—
87#13133151—21no—
88#13133151—21no—
89#13133150—21no—
90#13133150—21no—
91#13133149—21no—
92#13133149—21no—
93#13133148—21no—
94#13133148—21no—
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97C4⋊#88322C2311yescharacteristic
98C4⋊#88322C2311yescharacteristic
99#13133142—21no—
100#13133142—21no—
101#13133142—21no—
102#13133142—21no—
103#13133141—21no—
104#13133141—21no—
105#13133140—21no—
106#13133140—21no—
107#13133140—21no—
108#13133140—21no—
109#13133138—21no—
110#13133138—21no—
111C22⋊#88322—21no—
112C22⋊#88322—21no—
113#13133128—21no—
114C4×#88322—21no—
115C2×#41×C53D411yescharacteristic
116C22×C8×C53—21no—
117C22×C8×C53—21no—
118C22×C8×C53—21no—
119C4×Q8×C53D411yescharacteristic
120C4×Q8×C53C2311yescharacteristic
121C4×D4×C53—21no—
122C4×D4×C53—21no—
123C4×D4×C53D411yescharacteristic
124C4×D4×C53C2311yescharacteristic
125#117×C53—21no—
126#117×C53D411yescharacteristic
127C2×C42×C53—21no—
128C2×C42×C53C2311yescharacteristic
129C4⋊C8×C53—21no—
130C4⋊C8×C53—21no—
131C4⋊C8×C53—21no—
132C4⋊C8×C53D411yescharacteristic
133C4⋊C8×C53D411yescharacteristic
134C4⋊C8×C53C2311yescharacteristic
135C4⋊C8×C53C2311yescharacteristic
136C22⋊C8×C53—21no—
137C22⋊C8×C53—21no—
138C22⋊C8×C53D411yescharacteristic
139C22⋊C8×C53D411yescharacteristic
140C4×C8×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144#41×C53—22no—
145#41×C53—22no—
146C2×Q8×C53C2×D411yescharacteristic
147C2×D4×C53—21no—
148C2×D4×C53C2×D411yescharacteristic
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53#4111yescharacteristic
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—81no—
162C2×C8×C53—81no—
163C2×C8×C53—81no—
164C2×C8×C53C2×D411yescharacteristic
165C2×C8×C53C2×D411yescharacteristic
166C2×C8×C53C2×D411yescharacteristic
167C2×C8×C53C2×D411yescharacteristic
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53C2×D411yescharacteristic
171C4⋊C4×C53C2×D411yescharacteristic
172C22⋊C4×C53—21no—
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53C2×D411yescharacteristic
179C42×C53C2×D411yescharacteristic
180C42×C53#4111yescharacteristic
181C42×C53C2411yescharacteristic
182C22⋊Dic53—41no—
183C22⋊Dic53—41no—
184#11719925—42no—
185#11719925—42no—
186#11719923—42no—
187#11719923—42no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196#11719919—41no—
197#11719919—41no—
198#11719919—41no—
199#11719919—41no—
200C4×Dic53—41no—
201C4×Dic53—41no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C23×C53—21no—
207C23×C53#12111yescharacteristic
208Q8×C53—21no—
209Q8×C53#13512yes—
210D4×C53—24no—
211D4×C53—21no—
212D4×C53#13512yes—
213C2×C4×C53—24no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—81no—
225C2×C4×C53#12111yescharacteristic
226C2×C4×C53#12111yescharacteristic
227C2×C4×C53#12111yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235Dic106—41no—
236Dic106—41no—
237Dic106—41no—
238Dic106—41no—
239Dic106—41no—
240Dic106—41no—
241Dic106—81no—
242C8×C53—21no—
243C8×C53—21no—
244C8×C53—21no—
245C8×C53—21no—
246C8×C53—41no—
247C8×C53—41no—
248#88322—41no—
249#88322—41no—
250#88322—41no—
251#88322—41no—
252#19418—531nomaximal, Sylow
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—22no—
256C22×C53—21no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53C2×#12111yescharacteristic
260C4×C53—21no—
261C4×C53—21no—
262C4×C53—22no—
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—41no—
266C4×C53—41no—
267C4×C53C2×#13511yescharacteristic
268C4×C53C2×#13511yescharacteristic
269C4×C53#57711yescharacteristic
270C4×C53#57811yescharacteristic
271Dic53—81no—
272Dic53—81no—
273Dic53—81no—
274Dic53—81no—
275#3231—531no—
276#3116—531no—
277#3107—531no—
278#2988D5311yescharacteristic
279#1684—531no—
280#1683—531no—
281#1680—531no—
282#1679—531no—
283#1674—531no—
284#1673—531no—
285#1672—531no—
286#1671—531no—
287#1572—531no—
288#1552—531no—
289#1542—531no—
290C2×C53—41no—
291C2×C53—42no—
292C2×C53#304611yescharacteristic
293C2×C53#305011yescharacteristic
294C2×C53#306911yescharacteristic
295#533—531no—
296#533—531no—
297C4×#41C2×D5311yescharacteristic
298#500—531no—
299#498—531no—
300#487—531no—
301#487—531no—
302#485—1061no—
303#485—1061no—
304#485—1061no—
305#485—1061no—
306#479—1061no—
307#479—1061no—
308#477—531no—
309#476—531no—
310#475—531no—
311#474—531no—
312#462—531no—
313#462—531no—
314#461—531no—
315#461—531no—
316#452—1061no—
317#452—1061no—
318C8×Q8C2×D5311yescharacteristic
319C8×Q8C2×D5311yescharacteristic
320#441—1061no—
321#440—1061no—
322C4×Q16—1061no—
323C4×SD16—1061no—
324C8×D4—21no—
325C8×D4—21no—
326C8×D4C2×D5311yescharacteristic
327C8×D4C2×D5311yescharacteristic
328#432—531no—
329#431—21no—
330#431—531no—
331#431C2×D5311yescharacteristic
332C2×C4⋊C8—21no—
333C2×C4⋊C8C2×D5311yescharacteristic
334#335—531no—
335#335—531no—
336#334—531no—
337#334—531no—
338#333—531no—
339#333—531no—
340#332—531no—
341#332—531no—
342#332—531no—
343#332—531no—
344#332—531no—
345#332—531no—
346C53#1941811yescharacteristic, complemented, Sylow
347C2×#41D21211yescharacteristic
348C2×Q16—1061no—
349C2×Q16—1061no—
350C2×SD16—1061no—
351C2×SD16—1061no—
352C22×C8—21no—
353C22×C8—21no—
354C22×C8—21no—
355C4⋊Q8—531no—
356C4⋊Q8—531no—
357C4⋊Q8—531no—
358C4⋊Q8—531no—
359#125—1061no—
360#125—1061no—
361C22⋊Q8—1061no—
362C22⋊Q8—1061no—
363C22⋊Q8—1061no—
364C22⋊Q8—1061no—
365C4×Q8—1061no—
366C4×Q8—1061no—
367C4×Q8—1061no—
368C4×Q8—1061no—
369C4×Q8D21211yescharacteristic
370C4×Q8C22×D5311yescharacteristic
371C4×D4—21no—
372C4×D4—21no—
373C4×D4D21211yescharacteristic
374C4×D4C22×D5311yescharacteristic
375#117—21no—
376#117—1061no—
377#117—1061no—
378#117D21211yescharacteristic
379C2×C42—21no—
380C2×C42C22×D5311yescharacteristic
381#107—1061no—
382#107—1061no—
383#107—1061no—
384#107—1061no—
385#106—1061no—
386#106—1061no—
387#106—1061no—
388#106—1061no—
389C4⋊C8—21no—
390C4⋊C8—21no—
391C4⋊C8—21no—
392C4⋊C8—531no—
393C4⋊C8—531no—
394C4⋊C8—531no—
395C4⋊C8—531no—
396C4⋊C8C22⋊D5311yescharacteristic
397C4⋊C8C22⋊D5311yescharacteristic
398C4⋊C8C22×D5311yescharacteristic
399C4⋊C8C22×D5311yescharacteristic
400#103—1061no—
401#103—1061no—
402#103—1061no—
403#103—1061no—
404#103—1061no—
405#103—1061no—
406#103—1061no—
407#103—1061no—
408#103—1061no—
409#103—1061no—
410#102—1061no—
411#102—1061no—
412C22⋊C8—21no—
413C22⋊C8—21no—
414C22⋊C8—1061no—
415C22⋊C8—1061no—
416C22⋊C8C22⋊D5311yescharacteristic
417C22⋊C8C22⋊D5311yescharacteristic
418#97—1061no—
419C4×C8—21no—
420C4×C8—21no—
421C4×C8—1061no—
422C4×C8C22×D5311yescharacteristic
423C4×C8C22×D5311yescharacteristic
424#41—22no—
425#41—22no—
426C2×Q8—1061no—
427C2×Q8—1061no—
428C2×Q8—1061no—
429C2×Q8—1061no—
430C2×Q8C2×D21211yescharacteristic
431C2×D4—21no—
432C2×D4C2×D21211yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4C2×D21211yescharacteristic
437C22×C4#1171994511yescharacteristic
438C22×C4C2×C22⋊D5311yescharacteristic
439Q16—2122no—
440Q16—2122no—
441SD16—2122no—
442SD16—2122no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—81no—
446C2×C8—81no—
447C2×C8—81no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8C2×C22⋊D5311yescharacteristic
453C2×C8C2×C22⋊D5311yescharacteristic
454C2×C8C2×C22⋊D5311yescharacteristic
455C2×C8C2×C22⋊D5311yescharacteristic
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—2121no—
467C4⋊C4—2121no—
468C4⋊C4—2121no—
469C4⋊C4—2121no—
470C4⋊C4C2×D21211yescharacteristic
471C4⋊C4C2×D21211yescharacteristic
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—2121no—
476C22⋊C4—2121no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—2121no—
481C42—2121no—
482C42C2×D21211yescharacteristic
483C42#1171994511yescharacteristic
484C42C2×C22⋊D5311yescharacteristic
485C42C23×D5311yescharacteristic
486C23—21no—
487C23#1313319111yescharacteristic
488Q8—21no—
489Q8—4241no—
490Q8—2121no—
491Q8—2121no—
492Q8—2121no—
493Q8—2121no—
494Q8—2121no—
495Q8—2121no—
496Q8#1313323912yes—
497D4—24no—
498D4—21no—
499D4#1313323912yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—81no—
512C2×C4—2121no—
513C2×C4—2121no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4#1313319111yescharacteristic
517C2×C4#1313319111yescharacteristic
518C2×C4#1313319111yescharacteristic
519C2×C4C22×C22⋊D5311yescharacteristic
520C2×C4C22×D21211yescharacteristic
521C2×C4C2×#1171994511yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—2121no—
529C8—2121no—
530C8—2121no—
531C8—2121no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—22no—
536C22—41no—
537C22—41no—
538C22C2×#1313319111yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—22no—
544C4—41no—
545C4—41no—
546C4—4241no—
547C4—4241no—
548C4—4241no—
549C4—4241no—
550C4C2×#1313323911yescharacteristic
551C4C2×#1313323911yescharacteristic
552C4#1681143411yescharacteristic
553C4#1681143811yescharacteristic
554C2—41no—
555C2—42no—
556C2#2028426511yescharacteristic
557C2#2028429611yescharacteristic
558C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2988C2
socleC22×C53C2×#121
prime radicalC2×#41×C53D4
hypercenter#2988D53
hypocenterC53#19418
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
1. #20285876C2semidirect
4. #20274173C2semidirect
5. #20274166C2semidirect
6. #20274145C2semidirect
7. #20274138C2semidirect
8. #20274109C2semidirect
9. #20274104C2semidirect
10. #20274097C2semidirect
11. #20274090C2semidirect
12. #20273495C2semidirect
13. #20273413C2semidirect
14. #20273373C2semidirect
346. C53#19418semidirect

Maximal subgroups

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

subgroupindex
#202858762
#202848772
#202847862
#202741732
#202741662
#202741452
#202741382
#202741092
#202741042
#202740972
#202740902
#202734952
#202734132
#202733732
C53×#29882
#1941853