Finite Group Databasebeta

← #27369168

Subgroups — #27369168

Order 13568.

584 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285739C211yesmaximal, characteristic
2#20285262C211yesmaximal, characteristic, complemented
3#20283720C211yesmaximal, characteristic
4#20274231C211yesmaximal, characteristic, complemented
5#20274226C211yesmaximal, characteristic, complemented
6#20274223C211yesmaximal, characteristic, complemented
7#20274216C211yesmaximal, characteristic, complemented
8#20274107C211yesmaximal, characteristic, complemented
9#20274102C211yesmaximal, characteristic, complemented
10#20274099C211yesmaximal, characteristic, complemented
11#20274092C211yesmaximal, characteristic, complemented
12#20273480C211yesmaximal, characteristic, complemented
13#20273425C211yesmaximal, characteristic, complemented
14#20273388C211yesmaximal, characteristic, complemented
15C53×#3252C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17#16811033—21no—
18#16811033—21no—
19#16811026C2211yescharacteristic
20#16811026C2211yescharacteristic
21#16811019C2211yescharacteristic
22#16811019C2211yescharacteristic
23#16811018—21no—
24#16810993—21no—
25#16810833C2211yescharacteristic
26#16810833C2211yescharacteristic
27#16810828C2211yescharacteristic
28#16810828C2211yescharacteristic
29#16810778—21no—
30#16810761—21no—
31#16810754C2211yescharacteristic
32Q8×#88322C2211yescharacteristic
33#16810701—21no—
34#16810701C2211yescharacteristic
35#16810695—21no—
36D4×#88322C2211yescharacteristic
37#16810684—21no—
38#16810680C2211yescharacteristic
39#16810676C2211yescharacteristic
40#16810666—21no—
41#16810666—21no—
42#16810579—21no—
43C4×#11719917C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810370C2211yescharacteristic
51#16810370C2211yescharacteristic
52#16810370C2211yescharacteristic
53#16810370C2211yescharacteristic
54#16810369C2211yescharacteristic
55#16810369C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C53×#529C2211yescharacteristic
58C53×#479—21no—
59C53×#477C2211yescharacteristic
60C53×#476C2211yescharacteristic
61C53×#470—21no—
62C53×#468—21no—
63C53×#464—21no—
64C53×#462C2211yescharacteristic
65C53×#461C2211yescharacteristic
66C53×#423C2211yescharacteristic
67C22×Dic106—21no—
68C2×#11719925—21no—
69C2×#11719925—21no—
70C2×#11719923—21no—
71C2×#11719923—21no—
72C2×#11719917—21no—
73C2×#11719917D411yescharacteristic
74C2×#11719917D411yescharacteristic
75C22×#88322—21no—
76#13133217—21no—
77#13133217—21no—
78#13133217—21no—
79#13133217C2311yescharacteristic
80#13133217C2311yescharacteristic
81C2×C4⋊Dic53—21no—
82C2×C4⋊Dic53—21no—
83#13133150—21no—
84#13133150—21no—
85#13133150—21no—
86#13133150—21no—
87#13133148—21no—
88#13133148—21no—
89#13133148—21no—
90#13133148—21no—
91C4⋊#88322—21no—
92C4⋊#88322—21no—
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97#13133140—21no—
98#13133140—21no—
99#13133140—21no—
100#13133140—21no—
101C22⋊#88322—21no—
102C22⋊#88322—21no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105#13133128—21no—
106#13133128—21no—
107#13133128D411yescharacteristic
108#13133128D411yescharacteristic
109C4×#88322—21no—
110C4×#88322—21no—
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C2×Q16×C53—21no—
114C2×SD16×C53—21no—
115C2×M16×C53—21no—
116C4⋊Q8×C53C2311yescharacteristic
117C4⋊Q8×C53C2311yescharacteristic
118#126×C53—21no—
119#125×C53—21no—
120#125×C53D411yescharacteristic
121#124×C53D411yescharacteristic
122#123×C53—21no—
123C22⋊Q8×C53—21no—
124C22⋊Q8×C53—21no—
125C22⋊Q8×C53D411yescharacteristic
126#121×C53D411yescharacteristic
127C4×Q8×C53—21no—
128C4×Q8×C53—21no—
129C4×Q8×C53C2311yescharacteristic
130C4×D4×C53—21no—
131C4×D4×C53C2311yescharacteristic
132#117×C53—21no—
133#117×C53—21no—
134C2×C42×C53C2311yescharacteristic
135#107×C53—21no—
136#106×C53—21no—
137C4⋊C8×C53—21no—
138C4⋊C8×C53C2311yescharacteristic
139C4⋊C8×C53C2311yescharacteristic
140#103×C53—21no—
141#103×C53—21no—
142#103×C53—21no—
143#102×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53C2×D411yescharacteristic
147C2×D4×C53—21no—
148C2×D4×C53C2×D411yescharacteristic
149C22×C4×C53—21no—
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53#4111yescharacteristic
153C22×Dic53—41no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—41no—
157C2×Dic106—81no—
158C2×Dic106—81no—
159Q16×C53—81no—
160SD16×C53—81no—
161M16×C53—81no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—41no—
173C4⋊C4×C53—41no—
174C4⋊C4×C53C2×D411yescharacteristic
175C4⋊C4×C53C2×D411yescharacteristic
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—21no—
181C22⋊C4×C53—41no—
182C42×C53—21no—
183C42×C53—41no—
184C42×C53C2×D411yescharacteristic
185C42×C53C2×D411yescharacteristic
186C42×C53#4111yescharacteristic
187C42×C53C2411yescharacteristic
188#11719925—81no—
189#11719925—81no—
190#11719923—81no—
191#11719923—81no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—41no—
197C4⋊Dic53—41no—
198#11719917—22no—
199#11719917—22no—
200#11719917—42no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—42no—
206C2×#88322C2×D411yescharacteristic
207C2×#88322C2×D411yescharacteristic
208C2×#88322C2×D411yescharacteristic
209C2×#88322C2×D411yescharacteristic
210C23×C53—21no—
211C23×C53#12111yescharacteristic
212Q8×C53—21no—
213Q8×C53—21no—
214Q8×C53—41no—
215Q8×C53—41no—
216Q8×C53—81no—
217D4×C53—21no—
218D4×C53—21no—
219D4×C53—81no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53#12111yescharacteristic
233C2×C4×C53#12111yescharacteristic
234C2×C4×C53#12111yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—82no—
241Dic106—41no—
242Dic106—41no—
243Dic106—41no—
244Dic106—41no—
245Dic106—82no—
246C8×C53—41no—
247C8×C53—41no—
248#88322—21no—
249#88322—21no—
250#88322—41no—
251#88322—41no—
252#88322—41no—
253#88322—41no—
254#88322#13712yes—
255#88322#13712yes—
256#19378—531nomaximal, Sylow
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53C2×#12111yescharacteristic
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—21no—
267C4×C53—21no—
268C4×C53—41no—
269C4×C53—41no—
270C4×C53—41no—
271C4×C53—81no—
272C4×C53—81no—
273C4×C53C2×#13711yescharacteristic
274C4×C53C2×#13711yescharacteristic
275C4×C53#57611yescharacteristic
276C4×C53#57711yescharacteristic
277Dic53—81no—
278Dic53—81no—
279#3252D5311yescharacteristic
280#3218—531no—
281#3165—531no—
282#2969—531no—
283#1694—531no—
284#1693—531no—
285#1692—531no—
286#1691—531no—
287#1674—531no—
288#1673—531no—
289#1672—531no—
290#1671—531no—
291#1569—531no—
292#1554—531no—
293#1546—531no—
294C2×C53—41no—
295C2×C53—81no—
296C2×C53#303911yescharacteristic
297C2×C53#306211yescharacteristic
298C2×C53#306911yescharacteristic
299#533C2×D5311yescharacteristic
300C2×C4⋊Q8—531no—
301#529C2×D5311yescharacteristic
302#501—1061no—
303#501—1061no—
304#500—531no—
305#500—531no—
306#498—531no—
307#498—531no—
308#497—1061no—
309#491—1061no—
310#479—21no—
311#477C2×D5311yescharacteristic
312#476C2×D5311yescharacteristic
313#470—21no—
314#468—21no—
315#464—21no—
316#462—531no—
317#462—531no—
318#462C2×D5311yescharacteristic
319#461—531no—
320#461—531no—
321#461C2×D5311yescharacteristic
322#451—1061no—
323#448—1061no—
324#446—531no—
325C8×Q8—531no—
326#436—531no—
327#436—1061no—
328#435—1061no—
329C8×D4—531no—
330#433—1061no—
331#432—531no—
332#431—531no—
333#428—1061no—
334#428—1061no—
335#423C2×D5311yescharacteristic
336#405—1061no—
337C4×M16—531no—
338#335—531no—
339#335—531no—
340#334—531no—
341#334—531no—
342#333—531no—
343#333—531no—
344#332—531no—
345#332—531no—
346#332—531no—
347#332—531no—
348#332—531no—
349#332—531no—
350C53#1937811yescharacteristic, complemented, Sylow
351C22×Q8—1061no—
352C2×Q16—21no—
353C2×Q16—1061no—
354C2×Q16—1061no—
355C2×SD16—21no—
356C2×SD16—1061no—
357C2×SD16—1061no—
358C2×M16—21no—
359C2×M16—531no—
360C2×M16—531no—
361C2×M16—1061no—
362C22×C8—1061no—
363C4⋊Q8—531no—
364C4⋊Q8—531no—
365C4⋊Q8—1061no—
366C4⋊Q8—1061no—
367C4⋊Q8—1061no—
368C4⋊Q8C22×D5311yescharacteristic
369C4⋊Q8C22×D5311yescharacteristic
370#126—21no—
371#125—21no—
372#125C22⋊D5311yescharacteristic
373#124C22⋊D5311yescharacteristic
374#123—21no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8C22⋊D5311yescharacteristic
378#121C22⋊D5311yescharacteristic
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8C22×D5311yescharacteristic
382C4×D4—21no—
383C4×D4C22×D5311yescharacteristic
384#117—21no—
385#117—21no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C42C22×D5311yescharacteristic
389#107—21no—
390#107—1061no—
391#107—1061no—
392#107—1061no—
393#107—1061no—
394#106—21no—
395#106—1061no—
396#106—1061no—
397#106—1061no—
398#106—1061no—
399C4⋊C8—21no—
400C4⋊C8—531no—
401C4⋊C8—531no—
402C4⋊C8—531no—
403C4⋊C8—531no—
404C4⋊C8—1061no—
405C4⋊C8—1061no—
406C4⋊C8C22×D5311yescharacteristic
407C4⋊C8C22×D5311yescharacteristic
408#103—21no—
409#103—21no—
410#103—21no—
411#103—1061no—
412#103—1061no—
413#103—1061no—
414#103—1061no—
415#102—21no—
416C22⋊C8—1061no—
417C22⋊C8—1061no—
418C22⋊C8—1061no—
419C22⋊C8—1061no—
420#97—531no—
421#97—531no—
422#97—1061no—
423#97—1061no—
424C4×C8—531no—
425C4×C8—531no—
426C4×C8—1061no—
427C4×C8—1061no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—1061no—
431C2×Q8—1061no—
432C2×Q8—4241no—
433C2×Q8—4241no—
434C2×Q8—2121no—
435C2×Q8C2×C22⋊D5311yescharacteristic
436C2×D4—21no—
437C2×D4C2×C22⋊D5311yescharacteristic
438C22×C4—21no—
439C22×C4—2121no—
440C22×C4D4×D5311yescharacteristic
441C22×C4#1171994711yescharacteristic
442C22×C4C2×C22⋊D5311yescharacteristic
443Q16—81no—
444Q16—4241no—
445Q16—4241no—
446SD16—81no—
447SD16—4241no—
448SD16—4241no—
449M16—81no—
450M16—1062no—
451M16—1062no—
452M16—2122no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—531no—
456C2×C8—531no—
457C2×C8—531no—
458C2×C8—531no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—2122no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—2121no—
479C4⋊C4—2121no—
480C4⋊C4C2×C22⋊D5311yescharacteristic
481C4⋊C4C2×C22⋊D5311yescharacteristic
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—41no—
488C42—21no—
489C42—41no—
490C42D4×D5311yescharacteristic
491C42#1171994711yescharacteristic
492C42C2×C22⋊D5311yescharacteristic
493C42C23×D5311yescharacteristic
494C23—21no—
495C23#1313318911yescharacteristic
496Q8—21no—
497Q8—21no—
498Q8—41no—
499Q8—41no—
500Q8—81no—
501Q8—4242no—
502Q8—2121no—
503Q8—2121no—
504Q8—2121no—
505Q8—2121no—
506D4—21no—
507D4—21no—
508D4—81no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—4242no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4#1313318911yescharacteristic
525C2×C4#1313318911yescharacteristic
526C2×C4#1313318911yescharacteristic
527C2×C4C2×D4×D5311yescharacteristic
528C2×C4C22×C22⋊D5311yescharacteristic
529C2×C4C2×#1171994711yescharacteristic
530C8—41no—
531C8—41no—
532C8—532no—
533C8—532no—
534C8—1061no—
535C8—1061no—
536C8—2121no—
537C8—2121no—
538C8—2121no—
539C8—2121no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22C2×#1313318911yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—81no—
555C4—81no—
556C4—4241no—
557C4—4241no—
558C4#137×D5311yescharacteristic
559C4#137×D5311yescharacteristic
560C4#1681142611yescharacteristic
561C4#1681143111yescharacteristic
562C2—41no—
563C2—81no—
564C2#2028420011yescharacteristic
565C2#2028439711yescharacteristic
566C2#2028445411yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3252C2
socleC22×C53C2×#121
prime radical#121×C53D4
hypercenter#3252D53
hypocenterC53#19378
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3252

Complements

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

NH ≅ G/Ntype
2. #20285262C2semidirect
4. #20274231C2semidirect
5. #20274226C2semidirect
6. #20274223C2semidirect
7. #20274216C2semidirect
8. #20274107C2semidirect
9. #20274102C2semidirect
10. #20274099C2semidirect
11. #20274092C2semidirect
12. #20273480C2semidirect
13. #20273425C2semidirect
14. #20273388C2semidirect
350. C53#19378semidirect

Maximal subgroups

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

subgroupindex
#202857392
#202852622
#202837202
#202742312
#202742262
#202742232
#202742162
#202741072
#202741022
#202740992
#202740922
#202734802
#202734252
#202733882
C53×#32522
#1937853