Finite Group Databasebeta

← #28317339

Subgroups — #28317339

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291560C211yesmaximal, characteristic
2#20291509C211yesmaximal, characteristic, complemented
3#20290358C211yesmaximal, characteristic, complemented
4#20290351C211yesmaximal, characteristic, complemented
5#20290344C211yesmaximal, characteristic, complemented
6#20290341C211yesmaximal, characteristic, complemented
7#20290329C211yesmaximal, characteristic, complemented
8#20290324C211yesmaximal, characteristic, complemented
9#20290309C211yesmaximal, characteristic, complemented
10#20290302C211yesmaximal, characteristic, complemented
11#20290289C211yesmaximal, characteristic, complemented
12#20290078C211yesmaximal, characteristic, complemented
13#20290011C211yesmaximal, characteristic
14#20289933C211yesmaximal, characteristic, complemented
15#20284783C211yesmaximal, characteristic
16#16811675—21no—
17#16811675—21no—
18#16811674C2211yescharacteristic
19#16811674C2211yescharacteristic
20#16811672C2211yescharacteristic
21#16811672C2211yescharacteristic
22Q8×#88324C2211yescharacteristic
23Q8×#88324C2211yescharacteristic
24#16811658C2211yescharacteristic
25#16811658C2211yescharacteristic
26#16811656—21no—
27#16811656—21no—
28#16811645C2211yescharacteristic
29#16811643—21no—
30#16811641C2211yescharacteristic
31#16811630—21no—
32D53⋊(C4⋊C8)C2211yescharacteristic
33#16811588—21no—
34C4×#11719936C2211yescharacteristic
35C4×#11719940—21no—
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811178C2211yescharacteristic
53C4×#11719947C2211yescharacteristic
54#16810962C411yescharacteristic
55#16810962C411yescharacteristic
56#16810909C411yescharacteristic
57#16810903C411yescharacteristic
58#16810830C411yescharacteristic
59#16810782C411yescharacteristic
60#16810782C411yescharacteristic
61#16810715C2211yescharacteristic
62#16810715C2211yescharacteristic
63SD16×Dic53C2211yescharacteristic
64SD16×Dic53C2211yescharacteristic
65#16810675C2211yescharacteristic
66C53×#461C411yescharacteristic
67C2×#11719936D411yescharacteristic
68C2×#11719936D411yescharacteristic
69C2×#11719940—21no—
70C2×#11719940—21no—
71C2×D53⋊C8—21no—
72#13133284—21no—
73#13133284—21no—
74#13133284—21no—
75#13133284—21no—
76#13133284—21no—
77#13133284—21no—
78C4⋊#88324—21no—
79C4⋊#88324—21no—
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84#13133274—21no—
85#13133274—21no—
86C22⋊#88324—21no—
87C22⋊#88324—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×#11719947C2×C411yescharacteristic
102C2×#11719925C2×C411yescharacteristic
103C2×#11719925C2×C411yescharacteristic
104#13133217C2×C411yescharacteristic
105#13133208—21no—
106#13133197—21no—
107#13133193C2×C411yescharacteristic
108#13133193C2×C411yescharacteristic
109Q8×Dic53C2311yescharacteristic
110Q8×Dic53C2311yescharacteristic
111C4×Dic106C2×C411yescharacteristic
112#13133182—21no—
113#13133182C2311yescharacteristic
114D4×Dic53C2311yescharacteristic
115C4×C22⋊D53—21no—
116#13133177—21no—
117#13133175—21no—
118#13133175C2×C411yescharacteristic
119#13133173—21no—
120C2×C4×Dic53—21no—
121C42×D53C2311yescharacteristic
122#13133150C2×C411yescharacteristic
123#13133150C2×C411yescharacteristic
124#13133149C2×C411yescharacteristic
125#13133149C2×C411yescharacteristic
126C4⋊#88322C2×C411yescharacteristic
127#13133142C2×C411yescharacteristic
128#13133142C2×C411yescharacteristic
129#13133141C2×C411yescharacteristic
130#13133141C2×C411yescharacteristic
131#13133138C2×C411yescharacteristic
132#13133138C2×C411yescharacteristic
133#13133129C2×C411yescharacteristic
134#13133129C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C2×SD16×C53C2×C411yescharacteristic
138C2×SD16×C53C2×C411yescharacteristic
139C4⋊Q8×C53C2×C411yescharacteristic
140C4×D4×C53C2×C411yescharacteristic
141C4⋊C8×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143#103×C53C2×C411yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53—21no—
148C2×C22⋊D53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151#11719947—22no—
152#11719947—41no—
153C2×C4×D53C22×C411yescharacteristic
154C2×C4×D53C2×D411yescharacteristic
155C2×C4×D53#4111yescharacteristic
156C2×Dic106—21no—
157C2×Dic106C22×C411yescharacteristic
158#11719940—42no—
159#11719940—42no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—21no—
166C2×#88324C2×D411yescharacteristic
167C2×#88324C2×D411yescharacteristic
168C2×#88324C2×D411yescharacteristic
169C2×#88324C2×D411yescharacteristic
170#11719936—22no—
171#11719936—41no—
172D53⋊C8—42no—
173SD16×C53—41no—
174SD16×C53—41no—
175C2×C8×C53C22×C411yescharacteristic
176C2×C8×C53C22×C411yescharacteristic
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53C22×C411yescharacteristic
180C22⋊C4×C53—21no—
181C42×C53C22×C411yescharacteristic
182C22⋊Dic53—21no—
183#11719925—22no—
184#11719925—22no—
185#11719921—21no—
186#11719921—21no—
187#11719921—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53C22×C411yescharacteristic
192C4⋊Dic53C22×C411yescharacteristic
193C4⋊Dic53C22×C411yescharacteristic
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×Dic53C2×D411yescharacteristic
204C4×Dic53#4111yescharacteristic
205C4×Dic53C2411yescharacteristic
206C2×#88322C22×C411yescharacteristic
207C2×#88322C22×C411yescharacteristic
208C23×C53—21no—
209C22×D53C4×D411yescharacteristic
210Q8×C53—21no—
211Q8×C53—21no—
212Q8×C53—21no—
213Q8×C53—21no—
214D4×C53—21no—
215D4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—81no—
220C2×C4×C53C4×D411yescharacteristic
221C2×C4×C53C4×D411yescharacteristic
222C2×C4×C53C23×C411yescharacteristic
223C22⋊D53—42no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—42no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53C4×D411yescharacteristic
232C2×Dic53C22×D411yescharacteristic
233C2×Dic53C2×#4111yescharacteristic
234C4×D53—21no—
235C4×D53—21no—
236C4×D53—21no—
237C4×D53—21no—
238C4×D53—41no—
239Dic106—21no—
240Dic106—81no—
241Dic106#13112yes—
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—41no—
251#88324#13712yes—
252C8×C53—21no—
253C8×C53—21no—
254#88322—21no—
255#88322—21no—
256#18013—531nomaximal, Sylow
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53C2×C4×D411yescharacteristic
261C2×D53—21no—
262C2×D53—21no—
263C2×D53—21no—
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—41no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53C2×#13111yescharacteristic
270C4×C53#56811yescharacteristic
271Dic53—22no—
272Dic53—21no—
273Dic53—21no—
274Dic53—41no—
275Dic53—41no—
276Dic53—41no—
277Dic53C2×#13711yescharacteristic
278Dic53#57511yescharacteristic
279#3107—531no—
280#2971—531no—
281#2962—531no—
282#1592—531no—
283#1590—531no—
284#1588—531no—
285#1587—531no—
286#1584—531no—
287#1583—531no—
288#1579—531no—
289C8×Q16—531no—
290#1573—531no—
291#1497—531no—
292#1477—531no—
293#1453—531no—
294C2×C53—81no—
295C2×C53#293311yescharacteristic
296C2×C53C4×#13711yescharacteristic
297C2×C53#295711yescharacteristic
298D53—41no—
299#533—531no—
300C4×#41—531no—
301#485—531no—
302#485—531no—
303#475—531no—
304#474—531no—
305#462—531no—
306#461#615711yescharacteristic
307#452—531no—
308#452—531no—
309#446—531no—
310#446—531no—
311#446—1061no—
312#446—1061no—
313C8×Q8—531no—
314C8×Q8—531no—
315C8×Q8—531no—
316C8×Q8—531no—
317C4×Q16—531no—
318C4×Q16—531no—
319C4×SD16—531no—
320C4×SD16—531no—
321#436—531no—
322#436—531no—
323#436—1061no—
324#436—1061no—
325#432—531no—
326#432—1061no—
327#431—531no—
328#431—531no—
329#424—1061no—
330C2×C4⋊C8—531no—
331#405—1061no—
332C4×M16—531no—
333C4×M16—1061no—
334#335—531no—
335#335—531no—
336#334—531no—
337#334—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8—531no—
342Q8⋊C8—531no—
343Q8⋊C8—531no—
344D4⋊C8—531no—
345D4⋊C8—531no—
346#322—531no—
347#322—531no—
348#322—531no—
349C82—531no—
350C53#1801311yescharacteristic, complemented, Sylow
351C2×#41—531no—
352C2×Q16—531no—
353C2×Q16—531no—
354C2×SD16C2×#615711yescharacteristic
355C2×SD16C2×#615711yescharacteristic
356C2×M16—531no—
357C2×M16—531no—
358C2×M16—1061no—
359C2×M16—1061no—
360C22×C8—1061no—
361C4⋊Q8—531no—
362C4⋊Q8C2×#615711yescharacteristic
363#125—1061no—
364C22⋊Q8—531no—
365C22⋊Q8—531no—
366C22⋊Q8—1061no—
367C4×Q8—531no—
368C4×Q8—531no—
369C4×Q8—531no—
370C4×Q8—531no—
371C4×Q8—1061no—
372C4×D4—531no—
373C4×D4—1061no—
374C4×D4—1061no—
375C4×D4C2×#615711yescharacteristic
376#117—531no—
377#117—1061no—
378#117—1061no—
379C2×C42—531no—
380C2×C42—1061no—
381#107—531no—
382#107—531no—
383#106—531no—
384#106—531no—
385C4⋊C8—531no—
386C4⋊C8—531no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—531no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397C4⋊C8—1061no—
398C4⋊C8C2×#615711yescharacteristic
399#103—531no—
400#103—531no—
401#103—531no—
402#103—531no—
403#103C2×#615711yescharacteristic
404#103C2×#615711yescharacteristic
405#102—531no—
406#102—531no—
407C22⋊C8—531no—
408C22⋊C8—531no—
409C22⋊C8—1061no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413#97—531no—
414#97—531no—
415#97—1061no—
416#97—1061no—
417#97—1061no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—531no—
423C4×C8—531no—
424C4×C8—1061no—
425C4×C8—1061no—
426C4×C8—1061no—
427C4×C8—1061no—
428#41—1062no—
429#41—2121no—
430C2×Q8—531no—
431C2×Q8—1061no—
432C2×Q8C22×#615711yescharacteristic
433C2×Q8C22×#615711yescharacteristic
434C2×D4—1061no—
435C2×D4C22×#615711yescharacteristic
436C22×C4—21no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—1061no—
441C22×C4—1061no—
442Q16—1062no—
443Q16—1062no—
444SD16—41no—
445SD16—41no—
446M16—1062no—
447M16—2122no—
448M16—2121no—
449M16—2122no—
450C2×C8—531no—
451C2×C8—531no—
452C2×C8—531no—
453C2×C8—531no—
454C2×C8—531no—
455C2×C8—531no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—2122no—
463C2×C8C22×#615711yescharacteristic
464C2×C8C22×#615711yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—531no—
468C4⋊C4—531no—
469C4⋊C4—531no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4C22×#615711yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—1061no—
480C22⋊C4—1061no—
481C22⋊C4—1061no—
482C22⋊C4—1061no—
483C42—531no—
484C42—531no—
485C42—531no—
486C42—1061no—
487C42—1061no—
488C42—1061no—
489C42—1061no—
490C42—1061no—
491C42C22×#615711yescharacteristic
492C23—21no—
493C23—531no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—532no—
499Q8—1061no—
500Q8—4241no—
501D4—21no—
502D4—21no—
503D4—2122no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—81no—
508C2×C4—531no—
509C2×C4—531no—
510C2×C4—531no—
511C2×C4—1061no—
512C2×C4—1061no—
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×C4—2122no—
523C2×C4D4×#615711yescharacteristic
524C2×C4D4×#615711yescharacteristic
525C2×C4C23×#615711yescharacteristic
526C8—21no—
527C8—21no—
528C8—532no—
529C8—1061no—
530C8—1061no—
531C8—1061no—
532C8—1061no—
533C8—1061no—
534C8—2121no—
535C8—2121no—
536C8—2121no—
537C8—2121no—
538C8—2121no—
539C8—2121no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—1061no—
544C22—1061no—
545C22—1061no—
546C22C2×D4×#615711yescharacteristic
547C4—21no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—531no—
553C4—531no—
554C4—1061no—
555C4—1062no—
556C4—1061no—
557C4—2121no—
558C4—2121no—
559C4—2121no—
560C4D53⋊#13111yescharacteristic
561C4#1681170111yescharacteristic
562C2—81no—
563C2—2121no—
564C2#2029144611yescharacteristic
565C2#137×#615711yescharacteristic
566C2#2029149311yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#461C4
socleC22×C53C2×C4×D4
prime radicalC2×#11719947C2×C4
cosocleC4×Dic53C24
hypercenter#461#6157
hypocenterC53#18013
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #20291509C2semidirect
3. #20290358C2semidirect
4. #20290351C2semidirect
5. #20290344C2semidirect
6. #20290341C2semidirect
7. #20290329C2semidirect
8. #20290324C2semidirect
9. #20290309C2semidirect
10. #20290302C2semidirect
11. #20290289C2semidirect
12. #20290078C2semidirect
14. #20289933C2semidirect
350. C53#18013semidirect

Maximal subgroups

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

subgroupindex
#202915602
#202915092
#202903582
#202903512
#202903442
#202903412
#202903292
#202903242
#202903092
#202903022
#202902892
#202900782
#202900112
#202899332
#202847832
#1801353