Finite Group Databasebeta

← #28316401

Subgroups — #28316401

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291631C211yesmaximal, characteristic
2#20291609C211yesmaximal, characteristic, complemented
3#20290310C211yesmaximal, characteristic, complemented
4#20290309C211yesmaximal, characteristic, complemented
5#20290306C211yesmaximal, characteristic, complemented
6SD16⋊#88324C211yesmaximal, characteristic, complemented
7#20290303C211yesmaximal, characteristic, complemented
8#20290302C211yesmaximal, characteristic, complemented
9#20290298C211yesmaximal, characteristic, complemented
10SD16×#88324C211yesmaximal, characteristic, complemented
11#20290267C211yesmaximal, characteristic, complemented
12#20290007C211yesmaximal, characteristic
13D53⋊(Q8⋊C8)C211yesmaximal, characteristic, complemented
14#20289913C211yesmaximal, characteristic, complemented
15#20283537C211yesmaximal, characteristic
16#16811672—21no—
17#16811672—21no—
18#16811672C2211yescharacteristic
19#16811672C2211yescharacteristic
20#16811672C2211yescharacteristic
21#16811672C2211yescharacteristic
22Q8×#88324C2211yescharacteristic
23Q8×#88324C2211yescharacteristic
24#16811650—21no—
25#16811650—21no—
26D4×#88324C2211yescharacteristic
27D4×#88324C2211yescharacteristic
28#16811642—21no—
29#16811642C2211yescharacteristic
30#16811641—21no—
31#16811641C2211yescharacteristic
32#16811640—21no—
33C2×#13133284—21no—
34D53⋊(C4⋊C8)C2211yescharacteristic
35C4×D53⋊C8C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42#16811473C2211yescharacteristic
43#16811473C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#16811464C2211yescharacteristic
49#16811464C2211yescharacteristic
50C8×#88324C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52C2×#13133246C411yescharacteristic
53C4×#11719947C2211yescharacteristic
54C4×Q8×D53C2211yescharacteristic
55#16810731C411yescharacteristic
56#16810728C411yescharacteristic
57#16810723C411yescharacteristic
58#16810717C2211yescharacteristic
59#16810715C2211yescharacteristic
60#16810714C2211yescharacteristic
61SD16×Dic53C2211yescharacteristic
62#16810664C411yescharacteristic
63#16810630C411yescharacteristic
64#16810621C411yescharacteristic
65#16810575C2211yescharacteristic
66C53×#440C411yescharacteristic
67C2×D53⋊C8—21no—
68C2×D53⋊C8—21no—
69C22×#88324—21no—
70C22×#88324—21no—
71#13133284—21no—
72#13133284—21no—
73#13133284—21no—
74#13133284—21no—
75#13133284—21no—
76#13133284D411yescharacteristic
77#13133284D411yescharacteristic
78C4⋊#88324—21no—
79C4⋊#88324—21no—
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84#13133274—21no—
85#13133274D411yescharacteristic
86#13133274D411yescharacteristic
87C22⋊#88324—21no—
88C22⋊#88324—21no—
89C4×#88324—21no—
90C4×#88324—21no—
91C4×#88324—21no—
92C4×#88324—21no—
93C4×#88324—21no—
94C4×#88324—21no—
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×#11719947C2×C411yescharacteristic
102C2×Q8×D53C2×C411yescharacteristic
103#13133246C818yescomplemented
104C2×Dic212C2×C411yescharacteristic
105C2×#11719925C2×C411yescharacteristic
106C2×#11719923C2×C411yescharacteristic
107C2×#11719912C2×C411yescharacteristic
108C2×#11719912C2×C411yescharacteristic
109Q8×Dic53C2311yescharacteristic
110C4×Dic106—21no—
111C4×Dic106C2×C411yescharacteristic
112C4×Dic106C2×C411yescharacteristic
113#13133182—21no—
114#13133182C2311yescharacteristic
115#13133182C2311yescharacteristic
116D4×Dic53C2311yescharacteristic
117C4×C22⋊D53—21no—
118#13133177—21no—
119#13133175C2×C411yescharacteristic
120#13133173—21no—
121C4⋊C4×D53—21no—
122C4⋊C4×D53C2×C411yescharacteristic
123C2×C4×Dic53—21no—
124C42×D53—21no—
125C42×D53C2311yescharacteristic
126#13133150C2×C411yescharacteristic
127#13133149C2×C411yescharacteristic
128#13133148C2×C411yescharacteristic
129#13133142C2×C411yescharacteristic
130#13133142C2×C411yescharacteristic
131#13133141C2×C411yescharacteristic
132#13133140C2×C411yescharacteristic
133#13133140C2×C411yescharacteristic
134#13133138C2×C411yescharacteristic
135#13133128C2×C411yescharacteristic
136C8×Dic53C2311yescharacteristic
137C8×Dic53C2311yescharacteristic
138C2×SD16×C53C2×C411yescharacteristic
139C4×Q8×C53C2×C411yescharacteristic
140C4×D4×C53C2×C411yescharacteristic
141#107×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143#102×C53C2×C411yescharacteristic
144#97×C53C2×C411yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53—21no—
148C2×C22⋊D53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151Q8×D53—22no—
152Q8×D53C2×C814yescomplemented
153#11719947—22no—
154#11719947C2×C814yes—
155C2×C4×D53—21no—
156C2×C4×D53—21no—
157C2×C4×D53C22×C411yescharacteristic
158C2×C4×D53C2×D411yescharacteristic
159C2×C4×D53#4111yescharacteristic
160C2×Dic106—21no—
161C2×Dic106C22×C411yescharacteristic
162C2×Dic106C22×C411yescharacteristic
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—21no—
166C2×#88324—21no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—81no—
170C2×#88324—81no—
171C2×#88324C2×D411yescharacteristic
172C2×#88324C2×D411yescharacteristic
173C2×#88324C2×D411yescharacteristic
174C2×#88324C2×D411yescharacteristic
175D53⋊C8—42no—
176D53⋊C8—81no—
177SD16×C53C2×C814yescomplemented
178C2×C8×C53C22×C411yescharacteristic
179C2×C8×C53C22×C411yescharacteristic
180C4⋊C4×C53—21no—
181C4⋊C4×C53C22×C411yescharacteristic
182C4⋊C4×C53C22×C411yescharacteristic
183C22⋊C4×C53—21no—
184C42×C53—21no—
185C42×C53C22×C411yescharacteristic
186C22⋊Dic53—21no—
187#11719925C2×C814yescomplemented
188#11719923C2×C814yescomplemented
189#11719921—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53C22×C411yescharacteristic
192C4⋊Dic53C22×C411yescharacteristic
193#11719919—21no—
194#11719919—21no—
195#11719919—21no—
196#11719919—21no—
197#11719919—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53C2×D411yescharacteristic
205C4×Dic53#4111yescharacteristic
206C4×Dic53C2411yescharacteristic
207C2×#88322C22×C411yescharacteristic
208C2×#88322C22×C411yescharacteristic
209Dic212C2×C814yescomplemented
210#11719912—22no—
211#11719912C2×C814yescomplemented
212C23×C53—21no—
213C22×D53C4×D411yescharacteristic
214Q8×C53—21no—
215Q8×C53C22×C812yes—
216D4×C53—21no—
217D4×C53C22×C812yes—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—81no—
222C2×C4×C53C4×D411yescharacteristic
223C2×C4×C53C4×D411yescharacteristic
224C2×C4×C53C23×C411yescharacteristic
225C22⋊D53—24no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—24no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53C4×D411yescharacteristic
234C2×Dic53C22×D411yescharacteristic
235C2×Dic53C2×#4111yescharacteristic
236C4×D53—24no—
237C4×D53—21no—
238C4×D53—21no—
239C4×D53—41no—
240C4×D53—81no—
241C4×D53C22×C812yes—
242C4×D53#13112yes—
243Dic106—21no—
244Dic106—21no—
245Dic106—24no—
246Dic106C22×C812yes—
247Dic106C22×C812yes—
248#88324—22no—
249#88324—21no—
250#88324—21no—
251#88324—21no—
252#88324—21no—
253#88324—41no—
254#88324—41no—
255#88324—41no—
256#88324—41no—
257#88324—41no—
258C8×C53—21no—
259C8×C53C22×C812yes—
260#88322—21no—
261#88322C22×C812yes—
262#17853—531nomaximal, Sylow
263C22×C53—22no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53C2×C4×D411yescharacteristic
267C2×D53—21no—
268C2×D53—21no—
269C2×D53C8×D412yes—
270C4×C53—22no—
271C4×C53—21no—
272C4×C53—21no—
273C4×C53—41no—
274C4×C53—41no—
275C4×C53C23×C811yescharacteristic
276C4×C53C2×#13111yescharacteristic
277Dic53—22no—
278Dic53—22no—
279Dic53—21no—
280Dic53—41no—
281Dic53—41no—
282Dic53C8×D412yes—
283Dic53#57711yescharacteristic
284Dic53#57811yescharacteristic
285#2988—531no—
286#2982—531no—
287C4×#137—531no—
288#1579—531no—
289#1579—531no—
290SD16⋊C8—531no—
291SD16⋊C8—531no—
292C8×Q16—531no—
293C8×Q16—531no—
294C8×SD16—531no—
295C8×SD16—531no—
296#1565—531no—
297#1476—531no—
298C2×Q8⋊C8—531no—
299C8×M16—531no—
300C2×C53—42no—
301C2×C53C2×C8×D411yescharacteristic
302C2×C53#294711yescharacteristic
303C2×C53#295611yescharacteristic
304D53—22no—
305C2×#137—531no—
306C4×#41—531no—
307C2×C4×Q8—531no—
308C8×Q8—531no—
309C8×Q8—531no—
310C8×Q8—531no—
311C8×Q8—531no—
312C8×Q8—531no—
313C8×Q8—531no—
314C8×Q8—1061no—
315C8×Q8—1061no—
316#441—531no—
317#441—531no—
318#440—531no—
319#440#615711yescharacteristic
320C4×Q16—531no—
321C4×Q16—531no—
322C4×SD16—531no—
323C4×SD16—531no—
324C8×D4—531no—
325C8×D4—531no—
326C8×D4—1061no—
327C8×D4—1061no—
328#431—531no—
329#431—531no—
330#431—1061no—
331#431—1061no—
332#431—1061no—
333#428—531no—
334C2×C4⋊C8—531no—
335C2×C4⋊C8—1061no—
336#419—531no—
337#417—531no—
338C4×M16—531no—
339C2×C4×C8—531no—
340#335—531no—
341#335—531no—
342#334—531no—
343#334—531no—
344Q8⋊C8—531no—
345Q8⋊C8—531no—
346Q8⋊C8—531no—
347Q8⋊C8—531no—
348Q8⋊C8—531no—
349Q8⋊C8—531no—
350D4⋊C8—531no—
351D4⋊C8—531no—
352#322—531no—
353#322—531no—
354C82—531no—
355C82—531no—
356C53#1785311yescharacteristic, complemented, Sylow
357C2×#41—531no—
358C22×Q8—531no—
359#137—538no—
360C2×Q16—531no—
361C2×Q16—531no—
362C2×SD16—531no—
363C2×SD16C2×#615711yescharacteristic
364C2×M16—531no—
365C2×M16—531no—
366C22×C8—1061no—
367C22×C8—1061no—
368C22×C8—1061no—
369C22×C8—1061no—
370C4×Q8—531no—
371C4×Q8—531no—
372C4×Q8—531no—
373C4×Q8—531no—
374C4×Q8—531no—
375C4×Q8—1061no—
376C4×Q8—1061no—
377C4×Q8C2×#615711yescharacteristic
378C4×D4—531no—
379C4×D4—1061no—
380C4×D4—1061no—
381C4×D4C2×#615711yescharacteristic
382#117—531no—
383#117—1061no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—1061no—
386C2×C42—531no—
387C2×C42—1061no—
388C2×C42—1061no—
389#107—531no—
390#107C2×#615711yescharacteristic
391#106—531no—
392#106—531no—
393C4⋊C8—531no—
394C4⋊C8—531no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—1061no—
400C4⋊C8—1061no—
401C4⋊C8—1061no—
402C4⋊C8—1061no—
403C4⋊C8—1061no—
404C4⋊C8—1061no—
405C4⋊C8—1061no—
406#103—531no—
407#103—531no—
408#103—531no—
409#103—531no—
410#103—531no—
411#103C2×#615711yescharacteristic
412#102—531no—
413#102C2×#615711yescharacteristic
414C22⋊C8—531no—
415C22⋊C8—531no—
416C22⋊C8—1061no—
417C22⋊C8—1061no—
418C22⋊C8—1061no—
419#97—531no—
420#97C2×#615711yescharacteristic
421C4×C8—531no—
422C4×C8—531no—
423C4×C8—531no—
424C4×C8—531no—
425C4×C8—531no—
426C4×C8—531no—
427C4×C8—1061no—
428C4×C8—1061no—
429C4×C8—1061no—
430C4×C8—1061no—
431C4×C8—1061no—
432C4×C8—1061no—
433C4×C8—1061no—
434C4×C8—1061no—
435#41—534no—
436#41—1062no—
437C2×Q8—534no—
438C2×Q8—531no—
439C2×Q8—531no—
440C2×Q8—1061no—
441C2×Q8—1062no—
442C2×Q8C22×#615711yescharacteristic
443C2×D4—1061no—
444C2×D4C22×#615711yescharacteristic
445C22×C4—21no—
446C22×C4—531no—
447C22×C4—531no—
448C22×C4—531no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453Q16—534no—
454Q16—534no—
455SD16—534no—
456SD16D53⋊C814yescomplemented
457M16—534no—
458M16—1062no—
459C2×C8—531no—
460C2×C8—531no—
461C2×C8—531no—
462C2×C8—531no—
463C2×C8—531no—
464C2×C8—531no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—4241no—
472C2×C8—4241no—
473C2×C8—4241no—
474C2×C8—2122no—
475C2×C8C22×#615711yescharacteristic
476C2×C8C22×#615711yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—531no—
479C4⋊C4—531no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4C22×#615711yescharacteristic
487C4⋊C4C22×#615711yescharacteristic
488C22⋊C4—21no—
489C22⋊C4—1061no—
490C22⋊C4—1061no—
491C42—21no—
492C42—531no—
493C42—531no—
494C42—531no—
495C42—1061no—
496C42—1061no—
497C42—1061no—
498C42—1061no—
499C42—1061no—
500C42—1061no—
501C42C22×#615711yescharacteristic
502C23—21no—
503C23—531no—
504Q8—21no—
505Q8—532no—
506Q8—532no—
507Q8—1064no—
508Q8—1061no—
509Q8—1061no—
510Q8C2×D53⋊C812yes—
511D4—21no—
512D4—1064no—
513D4C2×D53⋊C812yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—81no—
518C2×C4—531no—
519C2×C4—531no—
520C2×C4—532no—
521C2×C4—532no—
522C2×C4—531no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—1064no—
526C2×C4—1064no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—4241no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4D4×#615711yescharacteristic
536C2×C4D4×#615711yescharacteristic
537C2×C4C23×#615711yescharacteristic
538C8—21no—
539C8—532no—
540C8—1061no—
541C8—1061no—
542C8—1061no—
543C8—1061no—
544C8—1062no—
545C8—1061no—
546C8—2121no—
547C8—2121no—
548C8—2121no—
549C8—2121no—
550C8—2121no—
551C8C2×D53⋊C812yes—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—532no—
556C22—1061no—
557C22—1061no—
558C22C2×D4×#615711yescharacteristic
559C4—22no—
560C4—21no—
561C4—21no—
562C4—41no—
563C4—41no—
564C4—532no—
565C4—531no—
566C4—531no—
567C4—1062no—
568C4—1062no—
569C4—1061no—
570C4—2121no—
571C4—2121no—
572C4C22×D53⋊C811yescharacteristic
573C4D53⋊#13111yescharacteristic
574C2—42no—
575C2—1062no—
576C2D4×D53⋊C811yescharacteristic
577C2#2029147311yescharacteristic
578C2#2029148911yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC4×C53C23×C8
FrattiniC2×C4C23×#6157
FittingC53×#440C4
socleC22×C53C2×C4×D4
prime radicalC2×#11719947C2×C4
cosocleC4×Dic53C24
hypercenter#440#6157
hypocenterC53#17853
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #20291609C2semidirect
3. #20290310C2semidirect
4. #20290309C2semidirect
5. #20290306C2semidirect
6. SD16⋊#88324C2semidirect
7. #20290303C2semidirect
8. #20290302C2semidirect
9. #20290298C2semidirect
10. SD16×#88324C2semidirect
11. #20290267C2semidirect
13. D53⋊(Q8⋊C8)C2semidirect
14. #20289913C2semidirect
103. #13133246C8semidirect
152. Q8×D53C2×C8semidirect
177. SD16×C53C2×C8semidirect
187. #11719925C2×C8semidirect
188. #11719923C2×C8semidirect
209. Dic212C2×C8semidirect
211. #11719912C2×C8semidirect
356. C53#17853semidirect
456. SD16D53⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202916312
#202916092
#202903102
#202903092
#202903062
SD16⋊#883242
#202903032
#202903022
#202902982
SD16×#883242
#202902672
#202900072
D53⋊(Q8⋊C8)2
#202899132
#202835372
#1785353