Finite Group Databasebeta

← #28317778

Subgroups — #28317778

Order 13568.

588 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291546C211yesmaximal, characteristic
2#20291510C211yesmaximal, characteristic, complemented
3#20290356C211yesmaximal, characteristic, complemented
4#20290353C211yesmaximal, characteristic, complemented
5#20290344C211yesmaximal, characteristic, complemented
6#20290339C211yesmaximal, characteristic, complemented
7#20290327C211yesmaximal, characteristic, complemented
8#20290320C211yesmaximal, characteristic, complemented
9#20290309C211yesmaximal, characteristic, complemented
10#20290298C211yesmaximal, characteristic, complemented
11#20290268C211yesmaximal, characteristic, complemented
12#20290037C211yesmaximal, characteristic, complemented
13#20290030C211yesmaximal, characteristic
14#20289938C211yesmaximal, characteristic, complemented
15#20285314C211yesmaximal, characteristic
16#16811675—21no—
17#16811674C2211yescharacteristic
18#16811673—21no—
19#16811673C2211yescharacteristic
20#16811673C2211yescharacteristic
21#16811672C2211yescharacteristic
22#16811672C2211yescharacteristic
23Q8×#88324C2211yescharacteristic
24#16811658—21no—
25#16811658C2211yescharacteristic
26#16811654—21no—
27D4×#88324C2211yescharacteristic
28#16811647—21no—
29#16811639C2211yescharacteristic
30#16811629—21no—
31C2×C4⋊#88324C2211yescharacteristic
32#16811587—21no—
33#16811587—21no—
34C4×#11719940C2211yescharacteristic
35C4×#11719940C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811487C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811485C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42#16811473C2211yescharacteristic
43#16811473C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811175C2211yescharacteristic
53#16811154C2211yescharacteristic
54#16811031C411yescharacteristic
55#16811014C411yescharacteristic
56#16810990C411yescharacteristic
57#16810981C411yescharacteristic
58#16810881C411yescharacteristic
59#16810879C411yescharacteristic
60#16810867C411yescharacteristic
61#16810717C2211yescharacteristic
62#16810715C2211yescharacteristic
63#16810714C2211yescharacteristic
64SD16×Dic53C2211yescharacteristic
65M16×Dic53C2211yescharacteristic
66C53×#468C411yescharacteristic
67C2×#11719940—21no—
68C2×#11719940—21no—
69C2×#11719940D411yescharacteristic
70C2×#11719940D411yescharacteristic
71C22×#88324—21no—
72C22×#88324—21no—
73#13133284—21no—
74#13133284—21no—
75#13133284—21no—
76C4⋊#88324—21no—
77C4⋊#88324—21no—
78C4⋊#88324—21no—
79C4⋊#88324—21no—
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84C22⋊#88324—21no—
85C22⋊#88324—21no—
86C22⋊#88324—21no—
87#13133272—21no—
88#13133272—21no—
89#13133272—21no—
90#13133272—21no—
91#13133272D411yescharacteristic
92#13133272D411yescharacteristic
93C4×#88324—21no—
94C4×#88324—21no—
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×Dic212C2×C411yescharacteristic
102C2×#11719925C2×C411yescharacteristic
103C2×#11719923C2×C411yescharacteristic
104C2×#11719917C2×C411yescharacteristic
105#13133216C2×C411yescharacteristic
106#13133215C2×C411yescharacteristic
107#13133210—21no—
108#13133208C2×C411yescharacteristic
109#13133207—21no—
110#13133204C2×C411yescharacteristic
111#13133198—21no—
112#13133196C2×C411yescharacteristic
113#13133196C2×C411yescharacteristic
114#13133194—21no—
115Q8×Dic53C2311yescharacteristic
116#13133182—21no—
117#13133182C2311yescharacteristic
118#13133182C2311yescharacteristic
119D4×Dic53—21no—
120D4×Dic53C2311yescharacteristic
121#13133173—21no—
122#13133173—21no—
123C2×C4×Dic53C2311yescharacteristic
124#13133150C2×C411yescharacteristic
125#13133149C2×C411yescharacteristic
126#13133148C2×C411yescharacteristic
127#13133142C2×C411yescharacteristic
128#13133142C2×C411yescharacteristic
129#13133141C2×C411yescharacteristic
130#13133140C2×C411yescharacteristic
131#13133140C2×C411yescharacteristic
132#13133138C2×C411yescharacteristic
133#13133127C2×C411yescharacteristic
134#13133127C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C2×SD16×C53C2×C411yescharacteristic
138C2×M16×C53C2×C411yescharacteristic
139C22⋊Q8×C53C2×C411yescharacteristic
140#121×C53C2×C411yescharacteristic
141#107×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143#102×C53C2×C411yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×D4×C53—21no—
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53C22×C411yescharacteristic
148C22×Dic53—21no—
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53#4111yescharacteristic
151C2×Dic106—21no—
152C2×Dic106C22×C411yescharacteristic
153C2×Dic106C22×C411yescharacteristic
154#11719940—22no—
155#11719940—41no—
156#11719940—42no—
157#11719940—42no—
158C2×#88324—21no—
159C2×#88324—21no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—42no—
165C2×#88324—42no—
166C2×#88324C2×D411yescharacteristic
167C2×#88324C2×D411yescharacteristic
168C2×#88324C2×D411yescharacteristic
169C2×#88324C2×D411yescharacteristic
170SD16×C53—41no—
171M16×C53—41no—
172C2×C8×C53C22×C411yescharacteristic
173C2×C8×C53C22×C411yescharacteristic
174C4⋊C4×C53—21no—
175C4⋊C4×C53C22×C411yescharacteristic
176C4⋊C4×C53C22×C411yescharacteristic
177C22⋊C4×C53—21no—
178C22⋊C4×C53—21no—
179C22⋊Dic53—21no—
180C22⋊Dic53—21no—
181C22⋊Dic53—21no—
182C22⋊Dic53—21no—
183C22⋊Dic53—21no—
184#11719925—22no—
185#11719923—22no—
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—
196#11719919—21no—
197#11719919—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53C22×C411yescharacteristic
202C4×Dic53C2×D411yescharacteristic
203C4×Dic53#4111yescharacteristic
204C4×Dic53C2411yescharacteristic
205#11719917—22no—
206C2×#88322C22×C411yescharacteristic
207C2×#88322C22×C411yescharacteristic
208Dic212—41no—
209C23×C53—21no—
210C23×C53C4×D411yescharacteristic
211Q8×C53—21no—
212Q8×C53—21no—
213D4×C53—21no—
214D4×C53—21no—
215D4×C53—81no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53C4×D411yescharacteristic
222C2×C4×C53C23×C411yescharacteristic
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53C4×D411yescharacteristic
233C2×Dic53C4×D411yescharacteristic
234C2×Dic53C22×D411yescharacteristic
235C2×Dic53C2×#4111yescharacteristic
236Dic106—21no—
237Dic106—21no—
238Dic106—21no—
239Dic106—21no—
240Dic106—81no—
241#88324—22no—
242#88324—21no—
243#88324—21no—
244#88324—21no—
245#88324—41no—
246#88324—41no—
247#88324—41no—
248#88324—41no—
249#88324—41no—
250#88324#13712yes—
251C8×C53—21no—
252C8×C53—21no—
253#88322—21no—
254#88322#13112yes—
255#18077—531nomaximal, Sylow
256C22×C53—21no—
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53C2×C4×D411yescharacteristic
263C4×C53—21no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53—41no—
267C4×C53C2×#13111yescharacteristic
268C4×C53#56811yescharacteristic
269Dic53—21no—
270Dic53—21no—
271Dic53—21no—
272Dic53—41no—
273Dic53—41no—
274Dic53—41no—
275Dic53—41no—
276Dic53C2×#13711yescharacteristic
277Dic53#57511yescharacteristic
278#3172—531no—
279#2969—531no—
280#2962—531no—
281#1592—531no—
282#1591—531no—
283#1588—531no—
284#1587—531no—
285#1584—531no—
286#1582—531no—
287#1579—531no—
288C8×SD16—531no—
289#1565—531no—
290#1485—531no—
291#1483—531no—
292#1455—531no—
293C2×C53—41no—
294C2×C53—81no—
295C2×C53#293511yescharacteristic
296C2×C53C4×#13711yescharacteristic
297C2×C53#295611yescharacteristic
298#533—531no—
299#529—531no—
300#501—531no—
301#497—531no—
302#491—531no—
303#489—531no—
304#470—531no—
305#470—531no—
306#468—531no—
307#468#615711yescharacteristic
308#446—531no—
309#446—531no—
310#446—531no—
311#446—1061no—
312#446—1061no—
313C8×Q8—531no—
314C8×Q8—531no—
315C8×Q8—531no—
316C4×Q16—531no—
317C4×Q16—531no—
318C4×SD16—531no—
319C4×SD16—531no—
320#436—531no—
321#436—1061no—
322#435—1061no—
323C8×D4—531no—
324#433—1061no—
325#431—531no—
326#424—1061no—
327C2×C4⋊C8—531no—
328#405—1061no—
329#405—1061no—
330C4×M16—531no—
331C4×M16—531no—
332C4×M16—531no—
333#335—531no—
334#335—531no—
335#334—531no—
336#334—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8—531no—
342Q8⋊C8—531no—
343D4⋊C8—531no—
344D4⋊C8—531no—
345#322—531no—
346#322—531no—
347#322—531no—
348C82—531no—
349C53#1807711yescharacteristic, complemented, Sylow
350C2×Q16—531no—
351C2×Q16—531no—
352C2×SD16—531no—
353C2×SD16C2×#615711yescharacteristic
354C2×M16—531no—
355C2×M16—531no—
356C2×M16—531no—
357C2×M16—1061no—
358C2×M16—1061no—
359C2×M16C2×#615711yescharacteristic
360C22×C8—1061no—
361C22×C8—1061no—
362C4⋊Q8—531no—
363C4⋊Q8—531no—
364#126—1061no—
365#125—531no—
366#125—1061no—
367#124—531no—
368#123—1061no—
369C22⋊Q8—531no—
370C22⋊Q8—531no—
371C22⋊Q8—1061no—
372C22⋊Q8C2×#615711yescharacteristic
373#121C2×#615711yescharacteristic
374C4×Q8—531no—
375C4×Q8—531no—
376C4×Q8—531no—
377C4×Q8—1061no—
378C4×D4—531no—
379C4×D4—1061no—
380#117—1061no—
381#117—1061no—
382C2×C42—531no—
383#107—531no—
384#107C2×#615711yescharacteristic
385#106—531no—
386#106—531no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—531no—
390C4⋊C8—531no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397C4⋊C8—1061no—
398#103—531no—
399#103—531no—
400#103—531no—
401#103—531no—
402#103—531no—
403#103C2×#615711yescharacteristic
404#102—531no—
405#102C2×#615711yescharacteristic
406C22⋊C8—1061no—
407C22⋊C8—1061no—
408C22⋊C8—1061no—
409#97—531no—
410#97—531no—
411#97—531no—
412#97—531no—
413#97—1061no—
414#97—1061no—
415#97—1061no—
416#97—1061no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—531no—
423C4×C8—1061no—
424C4×C8—1061no—
425C4×C8—1061no—
426C4×C8—1061no—
427C2×Q8—531no—
428C2×Q8—531no—
429C2×Q8—1061no—
430C2×Q8C22×#615711yescharacteristic
431C2×D4—21no—
432C2×D4C22×#615711yescharacteristic
433C22×C4—531no—
434C22×C4—531no—
435C22×C4—1061no—
436C22×C4C22×#615711yescharacteristic
437Q16—1062no—
438Q16—2121no—
439SD16—41no—
440SD16—1062no—
441M16—41no—
442M16—1062no—
443M16—1062no—
444M16—2121no—
445M16—2122no—
446M16—2122no—
447C2×C8—531no—
448C2×C8—531no—
449C2×C8—531no—
450C2×C8—531no—
451C2×C8—531no—
452C2×C8—531no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—2122no—
460C2×C8—2122no—
461C2×C8C22×#615711yescharacteristic
462C2×C8C22×#615711yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—531no—
465C4⋊C4—531no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4C22×#615711yescharacteristic
477C4⋊C4C22×#615711yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—1061no—
481C22⋊C4—1061no—
482C22⋊C4—1061no—
483C22⋊C4—1061no—
484C22⋊C4—1061no—
485C42—531no—
486C42—531no—
487C42—531no—
488C42—531no—
489C42—1061no—
490C42—1061no—
491C42—1061no—
492C23—21no—
493C23D4×#615711yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—1061no—
497Q8—1061no—
498Q8—1061no—
499Q8—1061no—
500Q8—4241no—
501D4—21no—
502D4—21no—
503D4—81no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—531no—
510C2×C4—531no—
511C2×C4—531no—
512C2×C4—531no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4D4×#615711yescharacteristic
523C2×C4C23×#615711yescharacteristic
524C8—21no—
525C8—21no—
526C8—532no—
527C8—532no—
528C8—1061no—
529C8—1061no—
530C8—1061no—
531C8—1062no—
532C8—1061no—
533C8—2121no—
534C8—2121no—
535C8—2121no—
536C8—2121no—
537C8—2121no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22C2×D4×#615711yescharacteristic
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—531no—
550C4—531no—
551C4—1061no—
552C4—1061no—
553C4—1061no—
554C4—2121no—
555C4—2121no—
556C4—2121no—
557C4—2121no—
558C4D53⋊#13111yescharacteristic
559C4#1681170111yescharacteristic
560C2—41no—
561C2—81no—
562C2#2029145911yescharacteristic
563C2#137×#615711yescharacteristic
564C2#2029148911yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#468C4
socleC22×C53C2×C4×D4
prime radical#121×C53C2×C4
cosocleC4×Dic53C24
hypercenter#468#6157
hypocenterC53#18077
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #20291510C2semidirect
3. #20290356C2semidirect
4. #20290353C2semidirect
5. #20290344C2semidirect
6. #20290339C2semidirect
7. #20290327C2semidirect
8. #20290320C2semidirect
9. #20290309C2semidirect
10. #20290298C2semidirect
11. #20290268C2semidirect
12. #20290037C2semidirect
14. #20289938C2semidirect
349. C53#18077semidirect

Maximal subgroups

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

subgroupindex
#202915462
#202915102
#202903562
#202903532
#202903442
#202903392
#202903272
#202903202
#202903092
#202902982
#202902682
#202900372
#202900302
#202899382
#202853142
#1807753