Finite Group Databasebeta

← #28316708

Subgroups — #28316708

Order 13568.

626 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#20291531C211yesmaximal, characteristic, complemented
2#41×#88324C211yesmaximal, characteristic, complemented
3#20290342C211yesmaximal, characteristic, complemented
4#20290340C211yesmaximal, characteristic, complemented
5#20290334C211yesmaximal, characteristic, complemented
6#20290332C211yesmaximal, characteristic, complemented
7#20290312C211yesmaximal, characteristic, complemented
8#20290308C211yesmaximal, characteristic, complemented
9#20290305C211yesmaximal, characteristic, complemented
10SD16⋊#88324C211yesmaximal, characteristic, complemented
11#20290286C211yesmaximal, characteristic, complemented
12#20290046C211yesmaximal, characteristic, complemented
13#20290028C211yesmaximal, characteristic, complemented
14#20289930C211yesmaximal, characteristic, complemented
15#20284734C211yesmaximal, characteristic
16#16811674C2211yescharacteristic
17Q8×#88324C2211yescharacteristic
18Q8×#88324C2211yescharacteristic
19Q8×#88324C2211yescharacteristic
20#16811657C2211yescharacteristic
21#16811652—21no—
22#16811651C2211yescharacteristic
23#16811650—21no—
24D4×#88324—21no—
25D4×#88324—21no—
26D4×#88324C2211yescharacteristic
27D4×#88324C2211yescharacteristic
28#16811648—21no—
29#16811644C2211yescharacteristic
30#16811640C2211yescharacteristic
31#16811639—21no—
32#16811639C2211yescharacteristic
33#16811588—21no—
34C2×C4×#88324—21no—
35C2×C4×#88324C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40Q8⋊#88324C2211yescharacteristic
41Q8⋊#88324C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44#16811471C2211yescharacteristic
45#16811471C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
51#16811464C2211yescharacteristic
52#16811159C2211yescharacteristic
53#41×Dic53C2211yescharacteristic
54#16810961C411yescharacteristic
55#16810948C411yescharacteristic
56#16810928C411yescharacteristic
57#16810921C411yescharacteristic
58#16810847C411yescharacteristic
59#16810840C411yescharacteristic
60#16810768C411yescharacteristic
61Q16×Dic53C2211yescharacteristic
62SD16×Dic53C2211yescharacteristic
63#16810708C2211yescharacteristic
64#16810703C2211yescharacteristic
65#16810673C2211yescharacteristic
66C53×#452C411yescharacteristic
67C2×#11719936—21no—
68C2×D53⋊C8—21no—
69C22×#88324—21no—
70C22×#88324—21no—
71C22×#88324D411yescharacteristic
72C22×#88324D411yescharacteristic
73#13133284—21no—
74#13133284—21no—
75C4⋊#88324—21no—
76C4⋊#88324—21no—
77C4⋊#88324—21no—
78C4⋊#88324C2311yescharacteristic
79C4⋊#88324C2311yescharacteristic
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82#13133274—21no—
83#13133274—21no—
84C22⋊#88324—21no—
85C22⋊#88324—21no—
86C22⋊#88324—21no—
87C22⋊#88324—21no—
88C22⋊#88324—21no—
89#13133272—21no—
90#13133272—21no—
91C4×#88324—21no—
92C4×#88324—21no—
93C4×#88324—21no—
94C4×#88324—21no—
95C4×#88324D411yescharacteristic
96C4×#88324D411yescharacteristic
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×#11719924C2×C411yescharacteristic
102C2×#11719922C2×C411yescharacteristic
103#13133212—21no—
104#13133208C2×C411yescharacteristic
105#13133206C2×C411yescharacteristic
106#13133201—21no—
107#13133191C2×C411yescharacteristic
108Q8×Dic53C2×C411yescharacteristic
109Q8×Dic53C2311yescharacteristic
110Q8×Dic53C2311yescharacteristic
111D4×Dic53—21no—
112D4×Dic53—21no—
113D4×Dic53C2×C411yescharacteristic
114D4×Dic53C2311yescharacteristic
115#13133179C2311yescharacteristic
116#13133177—21no—
117#13133176—21no—
118#13133176C2×C411yescharacteristic
119#13133175—21no—
120C2×C4×Dic53—21no—
121C2×C4×Dic53C2311yescharacteristic
122#13133151C2×C411yescharacteristic
123#13133150C2×C411yescharacteristic
124#13133149C2×C411yescharacteristic
125#13133148C2×C411yescharacteristic
126#13133146C2×C411yescharacteristic
127#13133146C2×C411yescharacteristic
128#13133141C2×C411yescharacteristic
129#13133141C2×C411yescharacteristic
130#13133141C2×C411yescharacteristic
131#13133139C2×C411yescharacteristic
132#13133139C2×C411yescharacteristic
133#13133138C2×C411yescharacteristic
134C22⋊#88322C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C2×#41×C53C2×C411yescharacteristic
138C2×Q16×C53C2×C411yescharacteristic
139C2×SD16×C53C2×C411yescharacteristic
140C22⋊Q8×C53C2×C411yescharacteristic
141#103×C53C2×C411yescharacteristic
142#102×C53C2×C411yescharacteristic
143C22⋊C8×C53C2×C411yescharacteristic
144#41×C53—22no—
145#41×C53—22no—
146C2×Q8×C53C22×C411yescharacteristic
147C2×Q8×C53C22×C411yescharacteristic
148C2×D4×C53—21no—
149C2×D4×C53C22×C411yescharacteristic
150C22×C4×C53—21no—
151C22×C4×C53C22×C411yescharacteristic
152C2×C22⋊D53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53#4111yescharacteristic
157C2×D212C22×C411yescharacteristic
158C2×C4×D53—21no—
159C2×#88324—21no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—41no—
166C2×#88324—41no—
167C2×#88324—42no—
168C2×#88324—42no—
169C2×#88324C2×D411yescharacteristic
170C2×#88324C2×D411yescharacteristic
171C2×#88324C2×D411yescharacteristic
172C2×#88324C2×D411yescharacteristic
173#11719936—42no—
174D53⋊C8—42no—
175Q16×C53—22no—
176SD16×C53—22no—
177C2×C8×C53C22×C411yescharacteristic
178C2×C8×C53C22×C411yescharacteristic
179C4⋊C4×C53—21no—
180C4⋊C4×C53C22×C411yescharacteristic
181C22⋊C4×C53—21no—
182C22⋊Dic53—21no—
183C22⋊Dic53—21no—
184C22⋊Dic53—21no—
185#11719924—22no—
186#11719922—22no—
187#11719921—21no—
188#11719921—21no—
189#11719921—21no—
190#11719921—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53C22×C411yescharacteristic
196C4⋊Dic53C22×C411yescharacteristic
197C4⋊Dic53C22×C411yescharacteristic
198#11719919—21no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53C22×C411yescharacteristic
204C4×Dic53C2×D411yescharacteristic
205C4×Dic53#4111yescharacteristic
206C4×Dic53C2411yescharacteristic
207C2×#88322C22×C411yescharacteristic
208C2×#88322C22×C411yescharacteristic
209C23×C53—21no—
210C23×C53C4×D411yescharacteristic
211C22×D53—21no—
212Q8×C53—21no—
213Q8×C53—21no—
214Q8×C53—21no—
215Q8×C53#13112yes—
216D4×C53—24no—
217D4×C53—21no—
218D4×C53#13112yes—
219C2×C4×C53—24no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53C4×D411yescharacteristic
226C2×C4×C53C23×C411yescharacteristic
227C22⋊D53—42no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—41no—
234C2×Dic53—42no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53C4×D411yescharacteristic
238C2×Dic53C4×D411yescharacteristic
239C2×Dic53C22×D411yescharacteristic
240C2×Dic53C2×#4111yescharacteristic
241D212—21no—
242D212—21no—
243C4×D53—41no—
244C4×D53—41no—
245#88324—21no—
246#88324—21no—
247#88324—21no—
248#88324—21no—
249#88324—41no—
250#88324—41no—
251#88324—41no—
252#88324—41no—
253#88324—41no—
254#88324—41no—
255C8×C53—21no—
256C8×C53—21no—
257#88322—21no—
258#88322—21no—
259#17909—531nomaximal, Sylow
260C22×C53—21no—
261C22×C53—22no—
262C22×C53—21no—
263C22×C53—21no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53C2×C4×D411yescharacteristic
267C2×D53—41no—
268C2×D53—41no—
269C2×D53—41no—
270C4×C53—21no—
271C4×C53—22no—
272C4×C53—41no—
273C4×C53—41no—
274C4×C53C2×#13111yescharacteristic
275C4×C53C2×#13111yescharacteristic
276Dic53—21no—
277Dic53—21no—
278Dic53—21no—
279Dic53—41no—
280Dic53—41no—
281Dic53—41no—
282Dic53#57511yescharacteristic
283Dic53#57511yescharacteristic
284#3102—531no—
285#2966—531no—
286C8×#41—531no—
287#1588—531no—
288#1587—531no—
289#1586—531no—
290#1585—531no—
291#1580—531no—
292#1579—531no—
293SD16⋊C8—531no—
294SD16⋊C8—531no—
295#1572—531no—
296#1488—531no—
297#1482—531no—
298#1452—531no—
299C2×C53—41no—
300C2×C53—42no—
301C2×C53#292911yescharacteristic
302C2×C53#294311yescharacteristic
303C2×C53#295511yescharacteristic
304D53—81no—
305#529—531no—
306C4×#41—531no—
307#485—531no—
308#482—531no—
309#479—531no—
310#478—531no—
311#464—531no—
312#463—531no—
313#452#615711yescharacteristic
314#449—531no—
315#446—531no—
316C8×Q8—531no—
317C8×Q8—531no—
318C8×Q8—531no—
319C4×Q16—531no—
320C4×SD16—531no—
321C4×SD16—531no—
322C4×D8—531no—
323#436—531no—
324#435—1061no—
325C8×D4—531no—
326C8×D4—531no—
327C8×D4—531no—
328C8×D4—1061no—
329C8×D4—1061no—
330C8×D4—1061no—
331#433—1061no—
332#432—531no—
333#431—531no—
334#431—531no—
335#431—531no—
336#431—1061no—
337#405—1061no—
338C2×C4×C8—531no—
339C2×C4×C8—1061no—
340#335—531no—
341#335—531no—
342#334—531no—
343#334—531no—
344Q8⋊C8—531no—
345Q8⋊C8—531no—
346Q8⋊C8—531no—
347Q8⋊C8—531no—
348D4⋊C8—531no—
349D4⋊C8—531no—
350D4⋊C8—531no—
351D4⋊C8—531no—
352#322—531no—
353#322—531no—
354#322—531no—
355#322—531no—
356C53#1790911yescharacteristic, complemented, Sylow
357C2×#41C2×#615711yescharacteristic
358C2×Q16C2×#615711yescharacteristic
359C2×SD16—531no—
360C2×SD16C2×#615711yescharacteristic
361C2×D8—531no—
362C2×M16—1061no—
363C22×C8—531no—
364C22×C8—531no—
365C22×C8—1061no—
366C22×C8—1061no—
367C22×C8—1061no—
368#126—1061no—
369#125—531no—
370#124—531no—
371#123—1061no—
372C22⋊Q8C2×#615711yescharacteristic
373#121—531no—
374C4×Q8—531no—
375C4×Q8—531no—
376C4×Q8—531no—
377C4×D4—531no—
378C4×D4—531no—
379C4×D4—531no—
380C4×D4—1061no—
381C4×D4—1061no—
382C4×D4—1061no—
383#117—531no—
384#117—1061no—
385#117—1061no—
386C2×C42—531no—
387C2×C42—1061no—
388#107—531no—
389#107—531no—
390#106—531no—
391#106—531no—
392C4⋊C8—531no—
393C4⋊C8—531no—
394C4⋊C8—531no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—1061no—
399C4⋊C8—1061no—
400C4⋊C8—1061no—
401C4⋊C8—1061no—
402C4⋊C8—1061no—
403#103—531no—
404#103—531no—
405#103—531no—
406#103C2×#615711yescharacteristic
407#102—531no—
408#102—531no—
409#102—531no—
410#102C2×#615711yescharacteristic
411C22⋊C8—531no—
412C22⋊C8—1061no—
413C22⋊C8—1061no—
414C22⋊C8—1061no—
415C22⋊C8—1061no—
416C22⋊C8—1061no—
417C22⋊C8—1061no—
418C22⋊C8—1061no—
419C22⋊C8C2×#615711yescharacteristic
420#97—1061no—
421#97—1061no—
422C4×C8—531no—
423C4×C8—531no—
424C4×C8—531no—
425C4×C8—531no—
426C4×C8—531no—
427C4×C8—531no—
428C4×C8—531no—
429C4×C8—531no—
430C4×C8—1061no—
431C4×C8—1061no—
432C4×C8—1061no—
433C4×C8—1061no—
434#41—22no—
435#41—22no—
436C2×Q8C22×#615711yescharacteristic
437C2×Q8C22×#615711yescharacteristic
438C2×D4—21no—
439C2×D4—531no—
440C2×D4—1061no—
441C2×D4C22×#615711yescharacteristic
442C22×C4—21no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4C22×#615711yescharacteristic
449Q16—22no—
450SD16—22no—
451SD16—1062no—
452D8—1062no—
453M16—2122no—
454C2×C8—531no—
455C2×C8—531no—
456C2×C8—531no—
457C2×C8—531no—
458C2×C8—531no—
459C2×C8—531no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—2121no—
467C2×C8—2121no—
468C2×C8—2122no—
469C2×C8—2122no—
470C2×C8—2122no—
471C2×C8C22×#615711yescharacteristic
472C2×C8C22×#615711yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—531no—
475C4⋊C4—531no—
476C4⋊C4—531no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4C22×#615711yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—1061no—
485C22⋊C4—1061no—
486C22⋊C4—1061no—
487C22⋊C4—1061no—
488C22⋊C4—1061no—
489C22⋊C4—1061no—
490C22⋊C4—1061no—
491C42—531no—
492C42—531no—
493C42—531no—
494C42—531no—
495C42—1061no—
496C42—1061no—
497C42—1061no—
498C42—1061no—
499C23—21no—
500C23—1061no—
501C23D4×#615711yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8#1313330412yes—
506D4—24no—
507D4—21no—
508D4—1061no—
509D4—1061no—
510D4—2122no—
511D4#1313330412yes—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—531no—
519C2×C4—531no—
520C2×C4—531no—
521C2×C4—531no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2122no—
533C2×C4D4×#615711yescharacteristic
534C2×C4C23×#615711yescharacteristic
535C8—21no—
536C8—21no—
537C8—1061no—
538C8—1061no—
539C8—1061no—
540C8—1061no—
541C8—1061no—
542C8—1061no—
543C8—2121no—
544C8—2121no—
545C8—2121no—
546C8—2121no—
547C8—2121no—
548C8—2121no—
549C22—21no—
550C22—22no—
551C22—21no—
552C22—21no—
553C22—41no—
554C22—41no—
555C22—2121no—
556C22—2121no—
557C22—2121no—
558C22C2×D4×#615711yescharacteristic
559C4—21no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—531no—
564C4—531no—
565C4—1061no—
566C4—1061no—
567C4—1061no—
568C4—2121no—
569C4—2121no—
570C4—2121no—
571C4C2×#1313330411yescharacteristic
572C4C2×#1313330411yescharacteristic
573C2—41no—
574C2—42no—
575C2—4241no—
576C2#2029143011yescharacteristic
577C2#2029146911yescharacteristic
578C2#2029148811yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#452C4
socleC22×C53C2×C4×D4
prime radical#16810768C4
cosocleC4×Dic53C24
hypercenter#452#6157
hypocenterC53#17909
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #20291531C2semidirect
2. #41×#88324C2semidirect
3. #20290342C2semidirect
4. #20290340C2semidirect
5. #20290334C2semidirect
6. #20290332C2semidirect
7. #20290312C2semidirect
8. #20290308C2semidirect
9. #20290305C2semidirect
10. SD16⋊#88324C2semidirect
11. #20290286C2semidirect
12. #20290046C2semidirect
13. #20290028C2semidirect
14. #20289930C2semidirect
356. C53#17909semidirect

Maximal subgroups

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

subgroupindex
#202915312
#41×#883242
#202903422
#202903402
#202903342
#202903322
#202903122
#202903082
#202903052
SD16⋊#883242
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