Finite Group Databasebeta

← #28316980

Subgroups — #28316980

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291657C211yesmaximal, characteristic, complemented
2#20291634C211yesmaximal, characteristic, complemented
3#20290365C211yesmaximal, characteristic, complemented
4#20290360C211yesmaximal, characteristic, complemented
5#20290352C211yesmaximal, characteristic, complemented
6#20290351C211yesmaximal, characteristic, complemented
7#20290341C211yesmaximal, characteristic, complemented
8#20290336C211yesmaximal, characteristic, complemented
9#20290325C211yesmaximal, characteristic, complemented
10#20290320C211yesmaximal, characteristic, complemented
11#20290282C211yesmaximal, characteristic, complemented
12#20290050C211yesmaximal, characteristic, complemented
13#20290030C211yesmaximal, characteristic, complemented
14#20289932C211yesmaximal, characteristic, complemented
15#20284726C211yesmaximal, characteristic
16#16811674C2211yescharacteristic
17#16811674C2211yescharacteristic
18#16811673C2211yescharacteristic
19#16811672C2211yescharacteristic
20#16811658—21no—
21#16811658C2211yescharacteristic
22#16811658C2211yescharacteristic
23#16811658C2211yescharacteristic
24#16811654—21no—
25#16811654—21no—
26#16811654—21no—
27D4×#88324C2211yescharacteristic
28#16811647—21no—
29#16811644—21no—
30#16811639C2211yescharacteristic
31#16811629—21no—
32#16811626—21no—
33#16811626C2211yescharacteristic
34C2×C4⋊#88324C2211yescharacteristic
35C4×#11719940C2211yescharacteristic
36#16811485C2211yescharacteristic
37#16811485C2211yescharacteristic
38#16811484C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44D4⋊#88324C2211yescharacteristic
45D4⋊#88324C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811154C2211yescharacteristic
53#41×Dic53C2211yescharacteristic
54#16810965C411yescharacteristic
55#16810944C411yescharacteristic
56#16810932C411yescharacteristic
57#16810925C411yescharacteristic
58#16810843C411yescharacteristic
59#16810836C411yescharacteristic
60#16810785C411yescharacteristic
61#16810715C2211yescharacteristic
62#16810714C2211yescharacteristic
63SD16×Dic53C2211yescharacteristic
64D8×Dic53C2211yescharacteristic
65#16810673C2211yescharacteristic
66C53×#449C411yescharacteristic
67C2×#11719940—21no—
68C2×#11719940—21no—
69C2×#11719940—21no—
70C2×#11719940—21no—
71C2×#11719940—21no—
72C22×#88324—21no—
73C22×#88324—21no—
74#13133284—21no—
75#13133284—21no—
76#13133284—21no—
77#13133284—21no—
78C4⋊#88324—21no—
79C4⋊#88324—21no—
80C4⋊#88324D411yescharacteristic
81C4⋊#88324D411yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84C4⋊#88324C2311yescharacteristic
85C4⋊#88324C2311yescharacteristic
86C22⋊#88324—21no—
87C22⋊#88324—21no—
88C22⋊#88324—21no—
89C22⋊#88324—21no—
90C22⋊#88324D411yescharacteristic
91C22⋊#88324D411yescharacteristic
92#13133272—21no—
93#13133272—21no—
94#13133272—21no—
95#13133272—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×#11719925C2×C411yescharacteristic
102C2×#11719923C2×C411yescharacteristic
103#13133210—21no—
104#13133208C2×C411yescharacteristic
105#13133204C2×C411yescharacteristic
106#13133198—21no—
107#13133196C2×C411yescharacteristic
108Q8×Dic53C2×C411yescharacteristic
109Q8×Dic53C2311yescharacteristic
110#13133182C2311yescharacteristic
111D4×Dic53—21no—
112D4×Dic53—21no—
113D4×Dic53—21no—
114D4×Dic53C2×C411yescharacteristic
115D4×Dic53C2311yescharacteristic
116D4×Dic53C2311yescharacteristic
117#13133176—21no—
118#13133176C2×C411yescharacteristic
119#13133173—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#13133142C2×C411yescharacteristic
129#13133142C2×C411yescharacteristic
130#13133141C2×C411yescharacteristic
131#13133138C2×C411yescharacteristic
132#13133138C2×C411yescharacteristic
133#13133138C2×C411yescharacteristic
134C22⋊#88322C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C2×#41×C53C2×C411yescharacteristic
138C2×SD16×C53C2×C411yescharacteristic
139C2×D8×C53C2×C411yescharacteristic
140#121×C53C2×C411yescharacteristic
141#103×C53C2×C411yescharacteristic
142#102×C53C2×C411yescharacteristic
143C22⋊C8×C53C2×C411yescharacteristic
144#41×C53—41no—
145#41×C53—41no—
146C2×Q8×C53C22×C411yescharacteristic
147C2×D4×C53—21no—
148C2×D4×C53—21no—
149C2×D4×C53C22×C411yescharacteristic
150C2×D4×C53C22×C411yescharacteristic
151C22×C4×C53—21no—
152C22×C4×C53C22×C411yescharacteristic
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53#4111yescharacteristic
158C2×Dic106C22×C411yescharacteristic
159#11719940—42no—
160#11719940—81no—
161#11719940—81no—
162#11719940—81no—
163#11719940—81no—
164C2×#88324—21no—
165C2×#88324—21no—
166C2×#88324—21no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—21no—
170C2×#88324—81no—
171C2×#88324—81no—
172C2×#88324C2×D411yescharacteristic
173C2×#88324C2×D411yescharacteristic
174C2×#88324C2×D411yescharacteristic
175C2×#88324C2×D411yescharacteristic
176SD16×C53—41no—
177D8×C53—41no—
178C2×C8×C53C22×C411yescharacteristic
179C2×C8×C53C22×C411yescharacteristic
180C4⋊C4×C53C22×C411yescharacteristic
181C22⋊C4×C53—21no—
182C22⋊Dic53—21no—
183C22⋊Dic53—21no—
184C22⋊Dic53—21no—
185C22⋊Dic53—21no—
186C22⋊Dic53—21no—
187C22⋊Dic53—21no—
188C22⋊Dic53—21no—
189#11719925—41no—
190#11719923—41no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53C22×C411yescharacteristic
195C4⋊Dic53C22×C411yescharacteristic
196C4⋊Dic53C22×C411yescharacteristic
197#11719919—21no—
198#11719919—21no—
199#11719919—21no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53C22×C411yescharacteristic
205C4×Dic53C2×D411yescharacteristic
206C4×Dic53#4111yescharacteristic
207C4×Dic53C2411yescharacteristic
208C2×#88322C22×C411yescharacteristic
209C2×#88322C22×C411yescharacteristic
210C23×C53—21no—
211C23×C53—21no—
212C23×C53C4×D411yescharacteristic
213Q8×C53—21no—
214Q8×C53—21no—
215D4×C53—21no—
216D4×C53—21no—
217D4×C53—21no—
218D4×C53—21no—
219D4×C53—42no—
220D4×C53—81no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—42no—
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53C23×C411yescharacteristic
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—81no—
240C2×Dic53C4×D411yescharacteristic
241C2×Dic53C4×D411yescharacteristic
242C2×Dic53C22×D411yescharacteristic
243C2×Dic53C2×#4111yescharacteristic
244Dic106—21no—
245Dic106—21no—
246#88324—22no—
247#88324—21no—
248#88324—21no—
249#88324—21no—
250#88324—21no—
251#88324—41no—
252#88324—41no—
253#88324—41no—
254#88324—41no—
255#88324—41no—
256C8×C53—21no—
257C8×C53—21no—
258#88322—21no—
259#88322—21no—
260#17955—531nomaximal, Sylow
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—21no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53C2×C4×D411yescharacteristic
271C4×C53—21no—
272C4×C53—41no—
273C4×C53—41no—
274C4×C53#56811yescharacteristic
275C4×C53#56811yescharacteristic
276Dic53—21no—
277Dic53—21no—
278Dic53—21no—
279Dic53—41no—
280Dic53—41no—
281Dic53—41no—
282Dic53—41no—
283Dic53#57711yescharacteristic
284Dic53#57711yescharacteristic
285#3102—531no—
286#2993—531no—
287#2989—531no—
288#1594—531no—
289#1593—531no—
290#1590—531no—
291#1590—531no—
292#1587—531no—
293#1586—531no—
294#1583—531no—
295#1582—531no—
296#1570—531no—
297#1489—531no—
298#1483—531no—
299#1453—531no—
300C2×C53—41no—
301C2×C53—81no—
302C2×C53—81no—
303C2×C53#293011yescharacteristic
304C2×C53#295211yescharacteristic
305C2×C53#295611yescharacteristic
306#529—531no—
307C4×#41—531no—
308#485—531no—
309#482—531no—
310#479—531no—
311#478—531no—
312#464—531no—
313#463—531no—
314#452—531no—
315#449#615711yescharacteristic
316#446—531no—
317#446—531no—
318#446—531no—
319C8×Q8—531no—
320C4×Q16—531no—
321C4×SD16—531no—
322C4×SD16—531no—
323C4×D8—531no—
324#436—531no—
325#436—531no—
326#436—531no—
327#436—1061no—
328#435—1061no—
329#435—1061no—
330#435—1061no—
331C8×D4—531no—
332#433—1061no—
333#432—1061no—
334#431—531no—
335#431—531no—
336#424—1061no—
337#423—531no—
338#423—1061no—
339C2×C4⋊C8—531no—
340C4×M16—531no—
341#334—531no—
342#334—531no—
343#334—531no—
344#334—531no—
345Q8⋊C8—531no—
346Q8⋊C8—531no—
347Q8⋊C8—531no—
348Q8⋊C8—531no—
349D4⋊C8—531no—
350D4⋊C8—531no—
351D4⋊C8—531no—
352D4⋊C8—531no—
353#322—531no—
354#322—531no—
355#322—531no—
356C82—531no—
357C53#1795511yescharacteristic, complemented, Sylow
358C2×#41C2×#615711yescharacteristic
359C2×Q16—531no—
360C2×SD16—531no—
361C2×SD16C2×#615711yescharacteristic
362C2×D8C2×#615711yescharacteristic
363C2×M16—1061no—
364C2×M16—1061no—
365C2×M16—1061no—
366C2×M16—1061no—
367C2×M16—1061no—
368C22×C8—1061no—
369C22×C8—1061no—
370#126—1061no—
371#125—531no—
372#124—531no—
373#123—1061no—
374C22⋊Q8—531no—
375#121C2×#615711yescharacteristic
376C4×Q8—531no—
377C4×Q8—531no—
378C4×Q8—531no—
379C4×D4—531no—
380C4×D4—531no—
381C4×D4—531no—
382C4×D4—1061no—
383C4×D4—1061no—
384C4×D4—1061no—
385#117—531no—
386#117—1061no—
387#117—1061no—
388C2×C42—531no—
389C2×C42—1061no—
390#107—531no—
391#107—531no—
392#106—531no—
393#106—531no—
394C4⋊C8—531no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—531no—
401C4⋊C8—531no—
402C4⋊C8—1061no—
403C4⋊C8—1061no—
404C4⋊C8—1061no—
405C4⋊C8—1061no—
406C4⋊C8—1061no—
407C4⋊C8—1061no—
408#103—531no—
409#103—531no—
410#103—531no—
411#103C2×#615711yescharacteristic
412#102—531no—
413#102—531no—
414#102—531no—
415#102C2×#615711yescharacteristic
416C22⋊C8—531no—
417C22⋊C8—531no—
418C22⋊C8—531no—
419C22⋊C8—1061no—
420C22⋊C8—1061no—
421C22⋊C8—1061no—
422C22⋊C8—1061no—
423C22⋊C8C2×#615711yescharacteristic
424#97—1061no—
425#97—1061no—
426#97—1061no—
427#97—1061no—
428C4×C8—531no—
429C4×C8—531no—
430C4×C8—531no—
431C4×C8—531no—
432C4×C8—531no—
433C4×C8—531no—
434C4×C8—1061no—
435#41—41no—
436#41—41no—
437C2×Q8—531no—
438C2×Q8C22×#615711yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4C22×#615711yescharacteristic
442C2×D4C22×#615711yescharacteristic
443C22×C4—21no—
444C22×C4—531no—
445C22×C4—531no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4C22×#615711yescharacteristic
450Q16—2121no—
451SD16—41no—
452SD16—2121no—
453D8—41no—
454M16—4241no—
455M16—4241no—
456M16—4241no—
457M16—4241no—
458M16—2122no—
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×C8C22×#615711yescharacteristic
474C2×C8C22×#615711yescharacteristic
475C4⋊C4—531no—
476C4⋊C4—531no—
477C4⋊C4—531no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4C22×#615711yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—1061no—
487C22⋊C4—1061no—
488C22⋊C4—1061no—
489C22⋊C4—1061no—
490C22⋊C4—1061no—
491C22⋊C4—1061no—
492C22⋊C4—1061no—
493C42—531no—
494C42—531no—
495C42—531no—
496C42—531no—
497C42—1061no—
498C42—1061no—
499C42—1061no—
500C42—1061no—
501C23—21no—
502C23—21no—
503C23D4×#615711yescharacteristic
504Q8—21no—
505Q8—21no—
506Q8—1061no—
507Q8—1061no—
508D4—21no—
509D4—21no—
510D4—21no—
511D4—21no—
512D4—42no—
513D4—81no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—42no—
519C2×C4—531no—
520C2×C4—531no—
521C2×C4—531no—
522C2×C4—531no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—4241no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4D4×#615711yescharacteristic
536C2×C4C23×#615711yescharacteristic
537C8—21no—
538C8—21no—
539C8—1061no—
540C8—1061no—
541C8—1061no—
542C8—1061no—
543C8—1062no—
544C8—1061no—
545C8—1061no—
546C8—2121no—
547C8—2121no—
548C8—2121no—
549C8—2121no—
550C8—2121no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22C2×D4×#615711yescharacteristic
561C4—21no—
562C4—41no—
563C4—41no—
564C4—531no—
565C4—531no—
566C4—1061no—
567C4—1061no—
568C4—1061no—
569C4—2121no—
570C4—2121no—
571C4—2121no—
572C4—2121no—
573C4#1681170111yescharacteristic
574C4#1681170111yescharacteristic
575C2—41no—
576C2—81no—
577C2—81no—
578C2#2029143411yescharacteristic
579C2#2029147811yescharacteristic
580C2#2029148911yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
Fitting, prime radicalC53×#449C4
socleC22×C53C2×C4×D4
cosocleC4×Dic53C24
hypercenter#449#6157
hypocenterC53#17955
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #20291657C2semidirect
2. #20291634C2semidirect
3. #20290365C2semidirect
4. #20290360C2semidirect
5. #20290352C2semidirect
6. #20290351C2semidirect
7. #20290341C2semidirect
8. #20290336C2semidirect
9. #20290325C2semidirect
10. #20290320C2semidirect
11. #20290282C2semidirect
12. #20290050C2semidirect
13. #20290030C2semidirect
14. #20289932C2semidirect
357. C53#17955semidirect

Maximal subgroups

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

subgroupindex
#202916572
#202916342
#202903652
#202903602
#202903522
#202903512
#202903412
#202903362
#202903252
#202903202
#202902822
#202900502
#202900302
#202899322
#202847262
#1795553