Finite Group Databasebeta

← #27354928

Subgroups — #27354928

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285322C211yesmaximal, characteristic
2#20283864C211yesmaximal, characteristic
3#20283812C211yesmaximal, characteristic, complemented
4#20273596C211yesmaximal, characteristic, complemented
5#20273589C211yesmaximal, characteristic, complemented
6#20273583C211yesmaximal, characteristic, complemented
7#20273569C211yesmaximal, characteristic, complemented
8#20273568C211yesmaximal, characteristic, complemented
9#20273562C211yesmaximal, characteristic, complemented
10#20273561C211yesmaximal, characteristic, complemented
11#20273548C211yesmaximal, characteristic, complemented
12#20273483C211yesmaximal, characteristic, complemented
13#20273171C211yesmaximal, characteristic, complemented
14#20273004C211yesmaximal, characteristic, complemented
15C53×#1494C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811033C411yescharacteristic
18#16811018C411yescharacteristic
19#16810993C411yescharacteristic
20#16810982C411yescharacteristic
21#16810879C411yescharacteristic
22#16810879C411yescharacteristic
23#16810867C411yescharacteristic
24#16810867C411yescharacteristic
25#16810754C2211yescharacteristic
26#16810751—21no—
27#16810751—21no—
28#16810750C2211yescharacteristic
29#16810750C2211yescharacteristic
30Q8×#88322C2211yescharacteristic
31#16810747C2211yescharacteristic
32#16810747C2211yescharacteristic
33C4×#11719925C2211yescharacteristic
34C4×#11719925C2211yescharacteristic
35C4×#11719923C2211yescharacteristic
36C4×#11719923C2211yescharacteristic
37#16810701—21no—
38#16810701C2211yescharacteristic
39#16810695—21no—
40D4×#88322C2211yescharacteristic
41#16810684—21no—
42#16810681—21no—
43#16810681—21no—
44#16810676C2211yescharacteristic
45#16810650—21no—
46#16810647C2211yescharacteristic
47C4×#11719917C2211yescharacteristic
48#16810375C2211yescharacteristic
49#16810375C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810373C2211yescharacteristic
52#16810355C2211yescharacteristic
53#16810355C2211yescharacteristic
54#16810355C2211yescharacteristic
55#16810355C2211yescharacteristic
56#16810345C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810344C2211yescharacteristic
59#16810344C2211yescharacteristic
60C53×#529C2211yescharacteristic
61C53×#432C2211yescharacteristic
62C2×C4×C8×C53C2211yescharacteristic
63C53×Q8⋊C8C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×D4⋊C8C2211yescharacteristic
66C53×D4⋊C8C2211yescharacteristic
67C2×#11719925C2×C411yescharacteristic
68C2×#11719925C2×C411yescharacteristic
69C2×#11719923C2×C411yescharacteristic
70C2×#11719923C2×C411yescharacteristic
71C2×#11719917—21no—
72C2×#11719917—21no—
73C2×#11719917C2×C411yescharacteristic
74C2×#11719917C2×C411yescharacteristic
75C22×#88322—21no—
76#13133217C2×C411yescharacteristic
77#13133217C2×C411yescharacteristic
78#13133209—21no—
79#13133196—21no—
80#13133196C2×C411yescharacteristic
81#13133196C2×C411yescharacteristic
82C4×Dic106—21no—
83C4×Dic106C2311yescharacteristic
84C4×Dic106C2311yescharacteristic
85#13133176—21no—
86#13133150C2×C411yescharacteristic
87#13133150C2×C411yescharacteristic
88#13133148C2×C411yescharacteristic
89#13133148C2×C411yescharacteristic
90C4⋊#88322—21no—
91C4⋊#88322—21no—
92C4⋊#88322—21no—
93C4⋊#88322D411yescharacteristic
94C4⋊#88322D411yescharacteristic
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97#13133146—21no—
98#13133146—21no—
99#13133146—21no—
100#13133146—21no—
101#13133141C2×C411yescharacteristic
102#13133141C2×C411yescharacteristic
103#13133140C2×C411yescharacteristic
104#13133140C2×C411yescharacteristic
105#13133140C2×C411yescharacteristic
106#13133140C2×C411yescharacteristic
107#13133138C2×C411yescharacteristic
108#13133138C2×C411yescharacteristic
109C22⋊#88322—21no—
110C22⋊#88322—21no—
111C22⋊#88322D411yescharacteristic
112C22⋊#88322D411yescharacteristic
113#13133128—21no—
114#13133128C2×C411yescharacteristic
115#13133128C2×C411yescharacteristic
116#13133127—21no—
117#13133127—21no—
118C4×#88322—21no—
119C4×#88322C2311yescharacteristic
120C4×#88322C2311yescharacteristic
121C4×#88322C2311yescharacteristic
122C4×#88322C2311yescharacteristic
123C8×Dic53—21no—
124C8×Dic53—21no—
125C22×C8×C53—21no—
126#126×C53—21no—
127#125×C53C2×C411yescharacteristic
128#124×C53C2×C411yescharacteristic
129#123×C53—21no—
130C22⋊Q8×C53C2×C411yescharacteristic
131#121×C53C2×C411yescharacteristic
132C4×Q8×C53C2311yescharacteristic
133C4×D4×C53—21no—
134C4×D4×C53C2311yescharacteristic
135#117×C53—21no—
136C2×C42×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138C4⋊C8×C53C2311yescharacteristic
139C22⋊C8×C53—21no—
140#97×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×D4×C53—21no—
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53—21no—
148C22×C4×C53C22×C411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53#4111yescharacteristic
151C2×Dic106—21no—
152C2×Dic106C22×C411yescharacteristic
153C2×Dic106C22×C411yescharacteristic
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—42no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53C22×C411yescharacteristic
164C4⋊C4×C53C22×C411yescharacteristic
165C22⋊C4×C53—21no—
166C22⋊C4×C53—21no—
167C22⋊C4×C53—21no—
168C22⋊C4×C53—21no—
169C22⋊C4×C53—21no—
170C42×C53—21no—
171C42×C53C22×C411yescharacteristic
172C42×C53C2×D411yescharacteristic
173C42×C53#4111yescharacteristic
174C42×C53C2411yescharacteristic
175C22⋊Dic53—21no—
176C22⋊Dic53—21no—
177#11719925—41no—
178#11719925—41no—
179#11719923—41no—
180#11719923—41no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53C22×C411yescharacteristic
185C4⋊Dic53C22×C411yescharacteristic
186#11719919—21no—
187#11719919—21no—
188#11719919—21no—
189#11719919—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192#11719917—41no—
193#11719917—41no—
194#11719917—81no—
195#11719917—81no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—81no—
199C2×#88322C22×C411yescharacteristic
200C2×#88322C22×C411yescharacteristic
201C2×#88322C22×C411yescharacteristic
202C2×#88322C22×C411yescharacteristic
203C2×#88322C2×D411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207C23×C53—21no—
208C23×C53C4×D411yescharacteristic
209Q8×C53—21no—
210Q8×C53—21no—
211D4×C53—21no—
212D4×C53—21no—
213D4×C53—81no—
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×C53C4×D411yescharacteristic
225C2×C4×C53C4×D411yescharacteristic
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53C23×C411yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234Dic106—21no—
235Dic106—21no—
236Dic106—21no—
237Dic106—21no—
238Dic106—81no—
239C8×C53—22no—
240C8×C53—41no—
241C8×C53—41no—
242C8×C53—41no—
243#88322—21no—
244#88322—21no—
245#88322—21no—
246#88322—21no—
247#88322—21no—
248#88322—21no—
249#88322—21no—
250#88322—21no—
251#88322—41no—
252#88322—41no—
253#17988—531nomaximal, Sylow
254C22×C53—21no—
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53C2×C4×D411yescharacteristic
261C4×C53—21no—
262C4×C53—21no—
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—41no—
266C4×C53—41no—
267C4×C53—41no—
268C4×C53#56811yescharacteristic
269C4×C53#56811yescharacteristic
270C4×C53#57711yescharacteristic
271C4×C53#57811yescharacteristic
272Dic53—41no—
273Dic53—41no—
274Dic53—41no—
275Dic53—41no—
276#3172—531no—
277#2993—531no—
278#2985—531no—
279#1588—531no—
280#1587—531no—
281#1587—531no—
282#1585—531no—
283#1584—531no—
284#1584—531no—
285#1583—531no—
286#1582—531no—
287#1569—531no—
288#1494D5311yescharacteristic
289#1493—531no—
290#1451—531no—
291C2×C53—41no—
292C2×C53—81no—
293C2×C53#293211yescharacteristic
294C2×C53#294711yescharacteristic
295C2×C53#295611yescharacteristic
296#533—531no—
297#529C2×D5311yescharacteristic
298#501—531no—
299#497—531no—
300#491—531no—
301#489—531no—
302#470—531no—
303#470—531no—
304#468—531no—
305#468—531no—
306#446—531no—
307#446—531no—
308#446—531no—
309#446—1061no—
310#446—1061no—
311C8×Q8—531no—
312C8×Q8—531no—
313C8×Q8—531no—
314C4×Q16—531no—
315C4×Q16—531no—
316C4×SD16—531no—
317C4×SD16—531no—
318#436—531no—
319#436—1061no—
320#435—1061no—
321C8×D4—531no—
322#433—1061no—
323#433—1061no—
324#433—1061no—
325#432C2×D5311yescharacteristic
326#431—531no—
327#424—1061no—
328#423—531no—
329C4×M16—531no—
330C2×C4×C8C2×D5311yescharacteristic
331#335—531no—
332#335—531no—
333#334—531no—
334#334—531no—
335Q8⋊C8—531no—
336Q8⋊C8—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8C2×D5311yescharacteristic
340Q8⋊C8C2×D5311yescharacteristic
341D4⋊C8C2×D5311yescharacteristic
342D4⋊C8C2×D5311yescharacteristic
343#322—531no—
344#322—531no—
345#322—531no—
346#322—531no—
347C53#1798811yescharacteristic, complemented, Sylow
348C2×Q16—531no—
349C2×Q16—531no—
350C2×SD16—531no—
351C2×SD16—531no—
352C2×M16—531no—
353C2×M16—531no—
354C2×M16—1061no—
355C2×M16—1061no—
356C22×C8—21no—
357C22×C8—1061no—
358C4⋊Q8—531no—
359C4⋊Q8—531no—
360#126—21no—
361#125—1061no—
362#125C4×D5311yescharacteristic
363#124C4×D5311yescharacteristic
364#123—21no—
365C22⋊Q8—531no—
366C22⋊Q8—531no—
367C22⋊Q8—1061no—
368C22⋊Q8C4×D5311yescharacteristic
369#121C4×D5311yescharacteristic
370C4×Q8—531no—
371C4×Q8—531no—
372C4×Q8—1061no—
373C4×Q8C22×D5311yescharacteristic
374C4×D4—21no—
375C4×D4C22×D5311yescharacteristic
376#117—21no—
377#117—1061no—
378C2×C42C22×D5311yescharacteristic
379#107—531no—
380#107—531no—
381#106—531no—
382#106—531no—
383C4⋊C8—531no—
384C4⋊C8—531no—
385C4⋊C8—531no—
386C4⋊C8—531no—
387C4⋊C8—1061no—
388C4⋊C8—1061no—
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8C22×D5311yescharacteristic
395C4⋊C8C22×D5311yescharacteristic
396#103—531no—
397#103—531no—
398#103—531no—
399#103—531no—
400#103—531no—
401#103—531no—
402#102—531no—
403#102—531no—
404C22⋊C8—21no—
405C22⋊C8—531no—
406C22⋊C8—531no—
407C22⋊C8—1061no—
408C22⋊C8—1061no—
409#97—21no—
410#97—531no—
411#97—531no—
412#97—1061no—
413#97—1061no—
414#97—1061no—
415C4×C8—21no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—1061no—
421C4×C8—1061no—
422C4×C8—1061no—
423C4×C8C22×D5311yescharacteristic
424C4×C8C22×D5311yescharacteristic
425C2×Q8—531no—
426C2×Q8—531no—
427C2×Q8—1061no—
428C2×Q8C2×C4×D5311yescharacteristic
429C2×D4—21no—
430C2×D4C2×C4×D5311yescharacteristic
431C22×C4—21no—
432C22×C4C2×C4×D5311yescharacteristic
433C22×C4D4×D5311yescharacteristic
434C22×C4#1171994711yescharacteristic
435Q16—2121no—
436Q16—2121no—
437SD16—2121no—
438SD16—2121no—
439M16—4241no—
440M16—4241no—
441M16—2121no—
442M16—2121no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—42no—
448C2×C8—531no—
449C2×C8—531no—
450C2×C8—531no—
451C2×C8—531no—
452C2×C8—531no—
453C2×C8—531no—
454C2×C8—531no—
455C2×C8—531no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—4241no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—531no—
464C4⋊C4—531no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4C2×C4×D5311yescharacteristic
473C4⋊C4C2×C4×D5311yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—1061no—
480C22⋊C4—1061no—
481C42—21no—
482C42—1061no—
483C42—1061no—
484C42C2×C4×D5311yescharacteristic
485C42D4×D5311yescharacteristic
486C42#1171994711yescharacteristic
487C42C23×D5311yescharacteristic
488C23—21no—
489C23#1313317711yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—1061no—
493Q8—1061no—
494Q8—1061no—
495Q8—1061no—
496Q8—4241no—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—1061no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4#1313317711yescharacteristic
515C2×C4#1313317711yescharacteristic
516C2×C4#1313317711yescharacteristic
517C2×C4C22×C4×D5311yescharacteristic
518C2×C4C2×D4×D5311yescharacteristic
519C2×C4C2×#1171994711yescharacteristic
520C8—22no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—1061no—
525C8—1061no—
526C8—1061no—
527C8—1061no—
528C8—1061no—
529C8—1061no—
530C8—1061no—
531C8—1061no—
532C8—2121no—
533C8—2121no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22C2×#1313317711yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—2121no—
549C4—2121no—
550C4—2121no—
551C4—2121no—
552C4#1681138811yescharacteristic
553C4#1681138811yescharacteristic
554C4#1681143011yescharacteristic
555C4#1681143711yescharacteristic
556C2—41no—
557C2—81no—
558C2#2028343011yescharacteristic
559C2#2028356111yescharacteristic
560C2#2028362111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1494C2
socleC22×C53C2×C4×D4
prime radical#121×C53C2×C4
cosocleC42×C53C24
hypercenter#1494D53
hypocenterC53#17988
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1494

Complements

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

NH ≅ G/Ntype
3. #20283812C2semidirect
4. #20273596C2semidirect
5. #20273589C2semidirect
6. #20273583C2semidirect
7. #20273569C2semidirect
8. #20273568C2semidirect
9. #20273562C2semidirect
10. #20273561C2semidirect
11. #20273548C2semidirect
12. #20273483C2semidirect
13. #20273171C2semidirect
14. #20273004C2semidirect
347. C53#17988semidirect

Maximal subgroups

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

subgroupindex
#202853222
#202838642
#202838122
#202735962
#202735892
#202735832
#202735692
#202735682
#202735622
#202735612
#202735482
#202734832
#202731712
#202730042
C53×#14942
#1798853