Finite Group Databasebeta

← #28317166

Subgroups — #28317166

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#20291660C211yesmaximal, characteristic
2#20291619C211yesmaximal, characteristic, complemented
3#20290342C211yesmaximal, characteristic, complemented
4#20290341C211yesmaximal, characteristic, complemented
5#20290340C211yesmaximal, characteristic, complemented
6#20290331C211yesmaximal, characteristic, complemented
7#20290328C211yesmaximal, characteristic, complemented
8#20290326C211yesmaximal, characteristic, complemented
9#20290323C211yesmaximal, characteristic, complemented
10#20290321C211yesmaximal, characteristic, complemented
11#20290279C211yesmaximal, characteristic, complemented
12#20290068C211yesmaximal, characteristic
13#20290063C211yesmaximal, characteristic, complemented
14#20289922C211yesmaximal, characteristic, complemented
15#20285316C211yesmaximal, characteristic
16#16811675—21no—
17#16811674C2211yescharacteristic
18#16811674C2211yescharacteristic
19#16811673—21no—
20#16811673C2211yescharacteristic
21#16811672C2211yescharacteristic
22Q8×#88324C2211yescharacteristic
23Q8×#88324C2211yescharacteristic
24#16811657C2211yescharacteristic
25#16811656—21no—
26#16811655—21no—
27#16811651C2211yescharacteristic
28#16811648—21no—
29#16811647—21no—
30#16811647—21no—
31#16811644C2211yescharacteristic
32#16811639C2211yescharacteristic
33#16811630—21no—
34#16811626C2211yescharacteristic
35C2×C4×#88324C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46#16811471C2211yescharacteristic
47#16811471C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#88324⋊C8C2211yescharacteristic
51#88324⋊C8C2211yescharacteristic
52#16811175C2211yescharacteristic
53#16811159C2211yescharacteristic
54#16811030C411yescharacteristic
55#16811016C411yescharacteristic
56#16810990C411yescharacteristic
57#16810984C411yescharacteristic
58#16810879C411yescharacteristic
59#16810871C411yescharacteristic
60#16810869C411yescharacteristic
61Q16×Dic53C2211yescharacteristic
62#16810715C2211yescharacteristic
63#16810710C2211yescharacteristic
64#16810708C2211yescharacteristic
65M16×Dic53C2211yescharacteristic
66C53×#470C411yescharacteristic
67C2×#11719936—21no—
68C2×#11719940—21no—
69C2×D53⋊C8—21no—
70C22×#88324—21no—
71#13133284—21no—
72#13133284—21no—
73#13133284—21no—
74#13133284—21no—
75C4⋊#88324—21no—
76C4⋊#88324—21no—
77C4⋊#88324—21no—
78C4⋊#88324D411yescharacteristic
79C4⋊#88324D411yescharacteristic
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84#13133274—21no—
85#13133274—21no—
86C22⋊#88324—21no—
87C22⋊#88324D411yescharacteristic
88C22⋊#88324D411yescharacteristic
89#13133272—21no—
90#13133272—21no—
91#13133272—21no—
92#13133272—21no—
93C4×#88324—21no—
94C4×#88324—21no—
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×#11719925C2×C411yescharacteristic
102C2×#11719913C2×C411yescharacteristic
103C2×#11719924C2×C411yescharacteristic
104C2×#11719917C2×C411yescharacteristic
105#13133216C2×C411yescharacteristic
106#13133215C2×C411yescharacteristic
107#13133212—21no—
108#13133208C2×C411yescharacteristic
109#13133207—21no—
110#13133206C2×C411yescharacteristic
111#13133201—21no—
112#13133196C2×C411yescharacteristic
113#13133194—21no—
114#13133191C2×C411yescharacteristic
115Q8×Dic53C2311yescharacteristic
116Q8×Dic53C2311yescharacteristic
117#13133182—21no—
118#13133182C2311yescharacteristic
119#13133179C2311yescharacteristic
120#13133177—21no—
121#13133175—21no—
122#13133173—21no—
123C2×C4×Dic53C2311yescharacteristic
124#13133151C2×C411yescharacteristic
125#13133150C2×C411yescharacteristic
126#13133148C2×C411yescharacteristic
127#13133142C2×C411yescharacteristic
128#13133141C2×C411yescharacteristic
129#13133141C2×C411yescharacteristic
130#13133140C2×C411yescharacteristic
131#13133139C2×C411yescharacteristic
132#13133137C2×C411yescharacteristic
133#13133127C2×C411yescharacteristic
134#13133127C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C2×Q16×C53C2×C411yescharacteristic
138C2×M16×C53C2×C411yescharacteristic
139C22⋊Q8×C53C2×C411yescharacteristic
140C22⋊Q8×C53C2×C411yescharacteristic
141#106×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143#103×C53C2×C411yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146C22×C4×C53C22×C411yescharacteristic
147C2×C22⋊D53—21no—
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53#4111yescharacteristic
150C2×D212C22×C411yescharacteristic
151C2×C4×D53—21no—
152C2×Dic106—21no—
153C2×Dic106C22×C411yescharacteristic
154#11719940—81no—
155C2×#88324—21no—
156C2×#88324—21no—
157C2×#88324—21no—
158C2×#88324—21no—
159C2×#88324—21no—
160C2×#88324—21no—
161C2×#88324—42no—
162C2×#88324C2×D411yescharacteristic
163C2×#88324C2×D411yescharacteristic
164C2×#88324C2×D411yescharacteristic
165C2×#88324C2×D411yescharacteristic
166#11719936—81no—
167D53⋊C8—81no—
168Q16×C53—41no—
169M16×C53—41no—
170C2×C8×C53C22×C411yescharacteristic
171C2×C8×C53C22×C411yescharacteristic
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53C22×C411yescharacteristic
175C4⋊C4×C53C22×C411yescharacteristic
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C22⋊Dic53—21no—
179#11719925—41no—
180#11719924—41no—
181#11719921—21no—
182#11719921—21no—
183#11719921—21no—
184#11719921—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53C22×C411yescharacteristic
189C4⋊Dic53C22×C411yescharacteristic
190#11719919—21no—
191#11719919—21no—
192#11719919—21no—
193#11719919—21no—
194#11719919—21no—
195#11719919—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53C22×C411yescharacteristic
200C4×Dic53C2×D411yescharacteristic
201C4×Dic53#4111yescharacteristic
202C4×Dic53C2411yescharacteristic
203#11719917—41no—
204C2×#88322C22×C411yescharacteristic
205C2×#88322C22×C411yescharacteristic
206#11719913—41no—
207C23×C53C4×D411yescharacteristic
208C22×D53—21no—
209Q8×C53—21no—
210Q8×C53—21no—
211Q8×C53—21no—
212Q8×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53C4×D411yescharacteristic
220C2×C4×C53C23×C411yescharacteristic
221C22⋊D53—81no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—41no—
228C2×Dic53C4×D411yescharacteristic
229C2×Dic53C4×D411yescharacteristic
230C2×Dic53C22×D411yescharacteristic
231C2×Dic53C2×#4111yescharacteristic
232D212—21no—
233D212—21no—
234C4×D53—41no—
235C4×D53—41no—
236Dic106—21no—
237Dic106—21no—
238Dic106—81no—
239#88324—22no—
240#88324—21no—
241#88324—21no—
242#88324—21no—
243#88324—21no—
244#88324—41no—
245#88324—41no—
246#88324—41no—
247#88324—41no—
248#88324—41no—
249C8×C53—21no—
250C8×C53—21no—
251#88322—21no—
252#88322—21no—
253#17988—531nomaximal, Sylow
254C22×C53—21no—
255C22×C53—21no—
256C22×C53—21no—
257C22×C53C2×C4×D411yescharacteristic
258C2×D53—41no—
259C2×D53—41no—
260C2×D53—41no—
261C4×C53—21no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53#56811yescharacteristic
267C4×C53#56811yescharacteristic
268Dic53—21no—
269Dic53—21no—
270Dic53—21no—
271Dic53—41no—
272Dic53—41no—
273Dic53—41no—
274Dic53#57711yescharacteristic
275Dic53#57811yescharacteristic
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#1494—531no—
289#1493—531no—
290#1451—531no—
291C2×C53—41no—
292C2×C53#293211yescharacteristic
293C2×C53#294711yescharacteristic
294C2×C53#295611yescharacteristic
295D53—81no—
296#533—531no—
297#529—531no—
298#501—531no—
299#497—531no—
300#491—531no—
301#489—531no—
302#470—531no—
303#470#615711yescharacteristic
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#432—531no—
326#431—531no—
327#424—1061no—
328#423—531no—
329C4×M16—531no—
330C2×C4×C8—531no—
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⋊C8—531no—
340Q8⋊C8—531no—
341D4⋊C8—531no—
342D4⋊C8—531no—
343#322—531no—
344#322—531no—
345#322—531no—
346#322—531no—
347C53#1798811yescharacteristic, complemented, Sylow
348C2×Q16—531no—
349C2×Q16C2×#615711yescharacteristic
350C2×SD16—531no—
351C2×SD16—531no—
352C2×M16—531no—
353C2×M16—1061no—
354C2×M16—1061no—
355C2×M16C2×#615711yescharacteristic
356C22×C8—1061no—
357C22×C8—1061no—
358C4⋊Q8—531no—
359C4⋊Q8—531no—
360#126—1061no—
361#125—531no—
362#125—1061no—
363#124—531no—
364#123—1061no—
365C22⋊Q8—531no—
366C22⋊Q8—1061no—
367C22⋊Q8C2×#615711yescharacteristic
368C22⋊Q8C2×#615711yescharacteristic
369#121—531no—
370C4×Q8—531no—
371C4×Q8—531no—
372C4×Q8—531no—
373C4×Q8—1061no—
374C4×D4—531no—
375C4×D4—1061no—
376#117—1061no—
377#117—1061no—
378C2×C42—531no—
379#107—531no—
380#107—531no—
381#106—531no—
382#106C2×#615711yescharacteristic
383C4⋊C8—531no—
384C4⋊C8—531no—
385C4⋊C8—531no—
386C4⋊C8—531no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396#103—531no—
397#103—531no—
398#103—531no—
399#103—531no—
400#103C2×#615711yescharacteristic
401#103C2×#615711yescharacteristic
402#102—531no—
403#102—531no—
404C22⋊C8—531no—
405C22⋊C8—531no—
406C22⋊C8—1061no—
407C22⋊C8—1061no—
408C22⋊C8—1061no—
409#97—531no—
410#97—531no—
411#97—1061no—
412#97—1061no—
413#97—1061no—
414#97—1061no—
415C4×C8—531no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—1061no—
422C4×C8—1061no—
423C4×C8—1061no—
424C4×C8—1061no—
425C2×Q8—531no—
426C2×Q8—1061no—
427C2×Q8C22×#615711yescharacteristic
428C2×Q8C22×#615711yescharacteristic
429C2×D4—531no—
430C2×D4—1061no—
431C22×C4—531no—
432C22×C4—531no—
433C22×C4—1061no—
434C22×C4C22×#615711yescharacteristic
435Q16—41no—
436Q16—2121no—
437SD16—2121no—
438SD16—2121no—
439M16—41no—
440M16—4241no—
441M16—4241no—
442M16—2121no—
443C2×C8—531no—
444C2×C8—531no—
445C2×C8—531no—
446C2×C8—531no—
447C2×C8—531no—
448C2×C8—531no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—4241no—
456C2×C8—2122no—
457C2×C8C22×#615711yescharacteristic
458C2×C8C22×#615711yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—531no—
462C4⋊C4—531no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4C22×#615711yescharacteristic
473C4⋊C4C22×#615711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—1061no—
477C22⋊C4—1061no—
478C22⋊C4—1061no—
479C22⋊C4—1061no—
480C22⋊C4—1061no—
481C42—531no—
482C42—531no—
483C42—531no—
484C42—531no—
485C42—1061no—
486C42—1061no—
487C42—1061no—
488C23—1061no—
489C23D4×#615711yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—1061no—
495Q8—1061no—
496Q8—4241no—
497D4—1061no—
498D4—1061no—
499D4—4241no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—531no—
507C2×C4—531no—
508C2×C4—531no—
509C2×C4—531no—
510C2×C4—1061no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—2121no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4D4×#615711yescharacteristic
519C2×C4C23×#615711yescharacteristic
520C8—21no—
521C8—21no—
522C8—1061no—
523C8—1061no—
524C8—1061no—
525C8—1061no—
526C8—1062no—
527C8—1061no—
528C8—1061no—
529C8—2121no—
530C8—2121no—
531C8—2121no—
532C8—2121no—
533C8—2121no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—2121no—
538C22—2121no—
539C22—2121no—
540C22C2×D4×#615711yescharacteristic
541C4—21no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—531no—
547C4—531no—
548C4—1061no—
549C4—1061no—
550C4—1061no—
551C4—2121no—
552C4—2121no—
553C4—2121no—
554C4#1681170111yescharacteristic
555C4#1681170211yescharacteristic
556C2—41no—
557C2—4241no—
558C2#2029144311yescharacteristic
559C2#2029147311yescharacteristic
560C2#2029149011yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#470C4
socleC22×C53C2×C4×D4
prime radical#13133191C2×C4
cosocleC4×Dic53C24
hypercenter#470#6157
hypocenterC53#17988
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #20291619C2semidirect
3. #20290342C2semidirect
4. #20290341C2semidirect
5. #20290340C2semidirect
6. #20290331C2semidirect
7. #20290328C2semidirect
8. #20290326C2semidirect
9. #20290323C2semidirect
10. #20290321C2semidirect
11. #20290279C2semidirect
13. #20290063C2semidirect
14. #20289922C2semidirect
347. C53#17988semidirect

Maximal subgroups

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

subgroupindex
#202916602
#202916192
#202903422
#202903412
#202903402
#202903312
#202903282
#202903262
#202903232
#202903212
#202902792
#202900682
#202900632
#202899222
#202853162
#1798853