Finite Group Databasebeta

← #28316685

Subgroups — #28316685

Order 13568.

634 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#20291529C211yesmaximal, characteristic, complemented
2#20291520C211yesmaximal, characteristic, complemented
3#20290329C211yesmaximal, characteristic, complemented
4#20290325C211yesmaximal, characteristic, complemented
5#20290320C211yesmaximal, characteristic, complemented
6#20290316C211yesmaximal, characteristic, complemented
7#20290302C211yesmaximal, characteristic, complemented
8#20290298C211yesmaximal, characteristic, complemented
9SD16×#88324C211yesmaximal, characteristic, complemented
10D8×#88324C211yesmaximal, characteristic, complemented
11#20290287C211yesmaximal, characteristic, complemented
12#20290047C211yesmaximal, characteristic, complemented
13#20290027C211yesmaximal, characteristic, complemented
14#20289921C211yesmaximal, characteristic, complemented
15#20284730C211yesmaximal, characteristic
16#16811674C2211yescharacteristic
17#16811672C2211yescharacteristic
18#16811672C2211yescharacteristic
19Q8×#88324C2211yescharacteristic
20#16811658C2211yescharacteristic
21#16811655—21no—
22#16811650—21no—
23#16811650—21no—
24#16811650—21no—
25D4×#88324C2211yescharacteristic
26D4×#88324C2211yescharacteristic
27D4×#88324C2211yescharacteristic
28#16811647—21no—
29#16811645C2211yescharacteristic
30#16811642C2211yescharacteristic
31#16811642C2211yescharacteristic
32#16811640—21no—
33#16811587—21no—
34C4×D53⋊C8C2211yescharacteristic
35C2×C4×#88324—21no—
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44D4⋊#88324C2211yescharacteristic
45D4⋊#88324C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50C8×#88324C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811152C2211yescharacteristic
53C4×#11719947C2211yescharacteristic
54#16810963C411yescharacteristic
55#16810946C411yescharacteristic
56#16810931C411yescharacteristic
57#16810923C411yescharacteristic
58#16810842C411yescharacteristic
59#16810782C411yescharacteristic
60#16810765C411yescharacteristic
61#16810715C2211yescharacteristic
62#16810714C2211yescharacteristic
63SD16×Dic53C2211yescharacteristic
64D8×Dic53C2211yescharacteristic
65#16810675C2211yescharacteristic
66C53×#463C411yescharacteristic
67C2×#11719940—21no—
68C2×D53⋊C8D411yescharacteristic
69C2×D53⋊C8D411yescharacteristic
70C22×#88324—21no—
71C22×#88324—21no—
72C22×#88324—21no—
73#13133284—21no—
74#13133284—21no—
75#13133284—21no—
76C4⋊#88324—21no—
77C4⋊#88324—21no—
78C4⋊#88324C2311yescharacteristic
79C4⋊#88324C2311yescharacteristic
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82#13133274—21no—
83#13133274—21no—
84#13133274—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×#11719947C2×C411yescharacteristic
102C2×#11719925C2×C411yescharacteristic
103C2×#11719923C2×C411yescharacteristic
104#13133210—21no—
105#13133209C2×C411yescharacteristic
106#13133199—21no—
107#13133193C2×C411yescharacteristic
108#13133188C2×C411yescharacteristic
109Q8×Dic53C2311yescharacteristic
110C4×Dic106C2×C411yescharacteristic
111#13133182C2311yescharacteristic
112D4×Dic53C2311yescharacteristic
113D4×Dic53C2311yescharacteristic
114C4×C22⋊D53—21no—
115#13133177—21no—
116#13133177—21no—
117#13133175C2×C411yescharacteristic
118#13133173—21no—
119#13133173—21no—
120C2×C4×Dic53—21no—
121C42×D53C2311yescharacteristic
122#13133151C2×C411yescharacteristic
123#13133150C2×C411yescharacteristic
124#13133149C2×C411yescharacteristic
125#13133148C2×C411yescharacteristic
126C4⋊#88322C2×C411yescharacteristic
127#13133142C2×C411yescharacteristic
128#13133142C2×C411yescharacteristic
129#13133141C2×C411yescharacteristic
130#13133138C2×C411yescharacteristic
131#13133138C2×C411yescharacteristic
132#13133138C2×C411yescharacteristic
133#13133129C2×C411yescharacteristic
134#13133129C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C2×SD16×C53C2×C411yescharacteristic
138C2×D8×C53C2×C411yescharacteristic
139#124×C53C2×C411yescharacteristic
140C4×D4×C53C2×C411yescharacteristic
141C4⋊C8×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143#102×C53C2×C411yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×D4×C53C22×C411yescharacteristic
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53—21no—
148C2×C22⋊D53—21no—
149C2×C22⋊D53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153#11719947—22no—
154#11719947—22no—
155C2×C4×D53C22×C411yescharacteristic
156C2×C4×D53C2×D411yescharacteristic
157C2×C4×D53#4111yescharacteristic
158C2×Dic106C22×C411yescharacteristic
159#11719940—42no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—21no—
166C2×#88324—42no—
167C2×#88324—42no—
168C2×#88324—42no—
169C2×#88324C2×D411yescharacteristic
170C2×#88324C2×D411yescharacteristic
171C2×#88324C2×D411yescharacteristic
172C2×#88324C2×D411yescharacteristic
173D53⋊C8—22no—
174D53⋊C8—22no—
175SD16×C53—22no—
176D8×C53—22no—
177C2×C8×C53C22×C411yescharacteristic
178C2×C8×C53C22×C411yescharacteristic
179C4⋊C4×C53C22×C411yescharacteristic
180C22⋊C4×C53—21no—
181C22⋊C4×C53—21no—
182C22⋊C4×C53—21no—
183C42×C53C22×C411yescharacteristic
184C22⋊Dic53—21no—
185C22⋊Dic53—21no—
186C22⋊Dic53—21no—
187#11719925—22no—
188#11719923—22no—
189#11719921—21no—
190#11719921—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53C22×C411yescharacteristic
193C4⋊Dic53C22×C411yescharacteristic
194C4⋊Dic53C22×C411yescharacteristic
195#11719919—21no—
196#11719919—21no—
197#11719919—21no—
198#11719919—21no—
199#11719919—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
209C23×C53—21no—
210C23×C53—21no—
211C22×D53C4×D411yescharacteristic
212Q8×C53—21no—
213Q8×C53—21no—
214D4×C53—21no—
215D4×C53—21no—
216D4×C53—21no—
217D4×C53#13112yes—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—42no—
221C2×C4×C53C4×D411yescharacteristic
222C2×C4×C53C4×D411yescharacteristic
223C2×C4×C53C23×C411yescharacteristic
224C22⋊D53—24no—
225C22⋊D53—42no—
226C2×Dic53—24no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53C4×D411yescharacteristic
236C2×Dic53C22×D411yescharacteristic
237C2×Dic53C2×#4111yescharacteristic
238C4×D53—21no—
239C4×D53—21no—
240C4×D53—21no—
241C4×D53—21no—
242C4×D53—41no—
243Dic106—21no—
244Dic106#13112yes—
245#88324—21no—
246#88324—21no—
247#88324—41no—
248#88324—41no—
249#88324—41no—
250#88324—41no—
251#88324—41no—
252#88324—41no—
253#88324#13512yes—
254#88324#13512yes—
255C8×C53—21no—
256C8×C53—21no—
257#88322—21no—
258#88322—21no—
259#17904—531nomaximal, Sylow
260C22×C53—22no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53C2×C4×D411yescharacteristic
267C2×D53—21no—
268C2×D53—21no—
269C2×D53—21no—
270C4×C53—21no—
271C4×C53—21no—
272C4×C53—41no—
273C4×C53—41no—
274C4×C53C2×#13111yescharacteristic
275C4×C53C2×#13111yescharacteristic
276Dic53—22no—
277Dic53—21no—
278Dic53—21no—
279Dic53—41no—
280Dic53—41no—
281Dic53—41no—
282Dic53C2×#13511yescharacteristic
283Dic53C2×#13511yescharacteristic
284#3102—531no—
285#2966—531no—
286C8×#41—531no—
287#1584—531no—
288#1583—531no—
289#1582—531no—
290#1581—531no—
291C8×Q16—531no—
292C8×SD16—531no—
293C8×SD16—531no—
294C8×D8—531no—
295#1572—531no—
296#1488—531no—
297#1482—531no—
298#1450—531no—
299C2×C53—42no—
300C2×C53—81no—
301C2×C53#292911yescharacteristic
302C2×C53C4×#13511yescharacteristic
303C2×C53#295511yescharacteristic
304D53—41no—
305#529—531no—
306C4×#41—531no—
307#485—531no—
308#482—531no—
309#479—531no—
310#478—531no—
311#464—531no—
312#463#615711yescharacteristic
313#452—531no—
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—
354C82—531no—
355C82—531no—
356C53#1790411yescharacteristic, complemented, Sylow
357C2×#41—531no—
358C2×Q16—531no—
359C2×SD16—531no—
360C2×SD16C2×#615711yescharacteristic
361C2×D8C2×#615711yescharacteristic
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#124C2×#615711yescharacteristic
371#123—1061no—
372C22⋊Q8—531no—
373#121—531no—
374C4×Q8—531no—
375C4×Q8—531no—
376C4×Q8—531no—
377C4×D4—531no—
378C4×D4—531no—
379C4×D4—1061no—
380C4×D4—1061no—
381C4×D4—1061no—
382C4×D4C2×#615711yescharacteristic
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—1061no—
398C4⋊C8—1061no—
399C4⋊C8—1061no—
400C4⋊C8—1061no—
401C4⋊C8—1061no—
402C4⋊C8C2×#615711yescharacteristic
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—531no—
413C22⋊C8—1061no—
414C22⋊C8—1061no—
415C22⋊C8—1061no—
416C22⋊C8—1061no—
417C22⋊C8—1061no—
418C22⋊C8—1061no—
419C22⋊C8—1061no—
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—1062no—
435#41—1062no—
436C2×Q8—531no—
437C2×Q8C22×#615711yescharacteristic
438C2×D4—1061no—
439C2×D4—1061no—
440C2×D4C22×#615711yescharacteristic
441C2×D4C22×#615711yescharacteristic
442C22×C4—21no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—531no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449Q16—1062no—
450SD16—22no—
451SD16—1062no—
452D8—22no—
453M16—2122no—
454C2×C8—531no—
455C2×C8—531no—
456C2×C8—531no—
457C2×C8—531no—
458C2×C8—531no—
459C2×C8—531no—
460C2×C8—1062no—
461C2×C8—1062no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—2122no—
469C2×C8—2122no—
470C2×C8—2122no—
471C2×C8C22×#615711yescharacteristic
472C2×C8C22×#615711yescharacteristic
473C4⋊C4—531no—
474C4⋊C4—531no—
475C4⋊C4—531no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4C22×#615711yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—1061no—
487C22⋊C4—1061no—
488C22⋊C4—1061no—
489C22⋊C4—1061no—
490C22⋊C4—1061no—
491C42—531no—
492C42—531no—
493C42—531no—
494C42—1061no—
495C42—1061no—
496C42—1061no—
497C42—1061no—
498C42C22×#615711yescharacteristic
499C23—21no—
500C23—21no—
501C23—531no—
502Q8—21no—
503Q8—21no—
504Q8—532no—
505Q8—1061no—
506D4—21no—
507D4—21no—
508D4—21no—
509D4—1064no—
510D4—2122no—
511D4#1313330412yes—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—42no—
515C2×C4—531no—
516C2×C4—531no—
517C2×C4—531no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1064no—
521C2×C4—1061no—
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×C4D4×#615711yescharacteristic
533C2×C4D4×#615711yescharacteristic
534C2×C4C23×#615711yescharacteristic
535C8—21no—
536C8—21no—
537C8—532no—
538C8—532no—
539C8—1061no—
540C8—1061no—
541C8—1061no—
542C8—1061no—
543C8—2121no—
544C8—2121no—
545C8—2121no—
546C8—2121no—
547C8—2121no—
548C8—2121no—
549C22—22no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—1061no—
556C22—1061no—
557C22—1061no—
558C22C2×D4×#615711yescharacteristic
559C4—21no—
560C4—21no—
561C4—41no—
562C4—41no—
563C4—531no—
564C4—531no—
565C4—1061no—
566C4—1062no—
567C4—1061no—
568C4—2121no—
569C4—2121no—
570C4—2121no—
571C4C2×#1313330411yescharacteristic
572C4D53⋊#13111yescharacteristic
573C2—42no—
574C2—81no—
575C2—2121no—
576C2#2029142911yescharacteristic
577C2#135×#615711yescharacteristic
578C2#2029148711yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#463C4
socleC22×C53C2×C4×D4
prime radical#16810765C4
cosocleC4×Dic53C24
hypercenter#463#6157
hypocenterC53#17904
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #20291529C2semidirect
2. #20291520C2semidirect
3. #20290329C2semidirect
4. #20290325C2semidirect
5. #20290320C2semidirect
6. #20290316C2semidirect
7. #20290302C2semidirect
8. #20290298C2semidirect
9. SD16×#88324C2semidirect
10. D8×#88324C2semidirect
11. #20290287C2semidirect
12. #20290047C2semidirect
13. #20290027C2semidirect
14. #20289921C2semidirect
356. C53#17904semidirect

Maximal subgroups

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

subgroupindex
#202915292
#202915202
#202903292
#202903252
#202903202
#202903162
#202903022
#202902982
SD16×#883242
D8×#883242
#202902872
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