Finite Group Databasebeta

← #28317366

Subgroups — #28317366

Order 13568.

600 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#20291662C211yesmaximal, characteristic
2#20291615C211yesmaximal, characteristic, complemented
3#20290354C211yesmaximal, characteristic, complemented
4#20290352C211yesmaximal, characteristic, complemented
5#20290345C211yesmaximal, characteristic, complemented
6#20290339C211yesmaximal, characteristic, complemented
7#20290339C211yesmaximal, characteristic, complemented
8#20290330C211yesmaximal, characteristic, complemented
9#20290303C211yesmaximal, characteristic, complemented
10#20290298C211yesmaximal, characteristic, complemented
11#20290290C211yesmaximal, characteristic, complemented
12#20290066C211yesmaximal, characteristic
13C2×#16811473C211yesmaximal, characteristic, complemented
14#20289920C211yesmaximal, characteristic, complemented
15#20284792C211yesmaximal, characteristic
16#16811675—21no—
17#16811675—21no—
18#16811673C2211yescharacteristic
19#16811673C2211yescharacteristic
20#16811673C2211yescharacteristic
21#16811672C2211yescharacteristic
22#16811672C2211yescharacteristic
23#16811672C2211yescharacteristic
24#16811658C2211yescharacteristic
25#16811654—21no—
26D4×#88324—21no—
27D4×#88324C2211yescharacteristic
28#16811647—21no—
29#16811644C2211yescharacteristic
30#16811643—21no—
31#16811640C2211yescharacteristic
32#16811639—21no—
33#16811629—21no—
34C2×C4⋊#88324C2211yescharacteristic
35C2×C4×#88324C2211yescharacteristic
36#16811486C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811485C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42#16811473C2211yescharacteristic
43#16811473C2211yescharacteristic
44#16811473C2211yescharacteristic
45#16811473C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50C8×#88324C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811154C2211yescharacteristic
53C2×#13133182C2211yescharacteristic
54#16810953C411yescharacteristic
55#16810929C411yescharacteristic
56#16810929C411yescharacteristic
57#16810844C411yescharacteristic
58#16810844C411yescharacteristic
59#16810781C411yescharacteristic
60#16810761C411yescharacteristic
61#16810717C2211yescharacteristic
62#16810717C2211yescharacteristic
63#16810714C2211yescharacteristic
64#16810714C2211yescharacteristic
65#16810673C2211yescharacteristic
66C53×#480C411yescharacteristic
67C2×#11719940—21no—
68C22×#88324—21no—
69C22×#88324—21no—
70C22×#88324—21no—
71#13133284—21no—
72#13133284—21no—
73#13133284—21no—
74#13133284D411yescharacteristic
75#13133284D411yescharacteristic
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—
87C22⋊#88324D411yescharacteristic
88C22⋊#88324D411yescharacteristic
89#13133272—21no—
90#13133272—21no—
91#13133272—21no—
92#13133272—21no—
93C4×#88324—21no—
94C4×#88324—21no—
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C22×Dic106C2×C411yescharacteristic
102C2×Dic212C2×C411yescharacteristic
103C2×Dic212C2×C411yescharacteristic
104C2×#11719923C2×C411yescharacteristic
105C2×#11719923C2×C411yescharacteristic
106#13133210—21no—
107#13133208C2×C411yescharacteristic
108#13133204C2×C411yescharacteristic
109#13133198—21no—
110#13133196C2×C411yescharacteristic
111#13133182—21no—
112#13133182—21no—
113#13133182C2×C411yescharacteristic
114#13133182C2×C411yescharacteristic
115#13133182C2311yescharacteristic
116#13133182C2311yescharacteristic
117#13133182C2311yescharacteristic
118D4×Dic53—21no—
119D4×Dic53C2311yescharacteristic
120#13133173—21no—
121C2×#11719919—21no—
122C2×C4×Dic53—21no—
123C2×C4×Dic53C2311yescharacteristic
124#13133148C2×C411yescharacteristic
125#13133148C2×C411yescharacteristic
126#13133146C2×C411yescharacteristic
127#13133146C2×C411yescharacteristic
128#13133142C2×C411yescharacteristic
129#13133142C2×C411yescharacteristic
130#13133140C2×C411yescharacteristic
131#13133140C2×C411yescharacteristic
132#13133140C2×C411yescharacteristic
133#13133140C2×C411yescharacteristic
134C22⋊#88322C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137#121×C53C2×C411yescharacteristic
138C2×C4⋊C4×C53C2×C411yescharacteristic
139#107×C53C2×C411yescharacteristic
140#107×C53C2×C411yescharacteristic
141#102×C53C2×C411yescharacteristic
142#102×C53C2×C411yescharacteristic
143C22⋊C8×C53C2×C411yescharacteristic
144C2×D4×C53—21no—
145C2×D4×C53C22×C411yescharacteristic
146C22×C4×C53—21no—
147C22×C4×C53C22×C411yescharacteristic
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53#4111yescharacteristic
152C2×Dic106—22no—
153C2×Dic106—21no—
154C2×Dic106—41no—
155C2×Dic106C22×C411yescharacteristic
156C2×Dic106C22×C411yescharacteristic
157C2×Dic106C22×C411yescharacteristic
158#11719940—81no—
159C2×#88324—24no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—21no—
166C2×#88324—81no—
167C2×#88324—81no—
168C2×#88324C2×D411yescharacteristic
169C2×#88324C2×D411yescharacteristic
170C2×#88324C2×D411yescharacteristic
171C2×#88324C2×D411yescharacteristic
172C2×C8×C53C22×C411yescharacteristic
173C2×C8×C53C22×C411yescharacteristic
174C4⋊C4×C53—21no—
175C4⋊C4×C53C22×C411yescharacteristic
176C4⋊C4×C53C22×C411yescharacteristic
177C4⋊C4×C53C22×C411yescharacteristic
178C22⋊C4×C53—21no—
179C22⋊Dic53—21no—
180C22⋊Dic53—21no—
181C22⋊Dic53—21no—
182C22⋊Dic53—21no—
183#11719923—22no—
184#11719923—41no—
185C4⋊Dic53—21no—
186C4⋊Dic53C22×C411yescharacteristic
187#11719919—21no—
188#11719919—21no—
189#11719919—21no—
190#11719919—21no—
191#11719919—21no—
192#11719919—21no—
193#11719919—21no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53C22×C411yescharacteristic
200C4×Dic53C2×D411yescharacteristic
201C4×Dic53#4111yescharacteristic
202C4×Dic53C2411yescharacteristic
203C2×#88322C22×C411yescharacteristic
204C2×#88322C22×C411yescharacteristic
205Dic212—22no—
206Dic212—41no—
207C23×C53—21no—
208C23×C53C4×D411yescharacteristic
209D4×C53—21no—
210D4×C53—21no—
211D4×C53—42no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—42no—
218C2×C4×C53C4×D411yescharacteristic
219C2×C4×C53C23×C411yescharacteristic
220C2×Dic53—21no—
221C2×Dic53—22no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—42no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53C4×D411yescharacteristic
231C2×Dic53C4×D411yescharacteristic
232C2×Dic53C22×D411yescharacteristic
233C2×Dic53C2×#4111yescharacteristic
234Dic106—21no—
235Dic106—21no—
236Dic106—21no—
237Dic106—21no—
238Dic106—21no—
239Dic106—42no—
240Dic106#13112yes—
241#88324—22no—
242#88324—22no—
243#88324—21no—
244#88324—21no—
245#88324—21no—
246#88324—21no—
247#88324—41no—
248#88324—41no—
249#88324—41no—
250#88324—41no—
251C8×C53—21no—
252C8×C53—21no—
253#88322—21no—
254#88322—21no—
255#18017—531nomaximal, Sylow
256C22×C53—21no—
257C22×C53—21no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53#44312yes—
262C22×C53C2×C4×D411yescharacteristic
263C4×C53—21no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53—41no—
267C4×C53C2×#13111yescharacteristic
268C4×C53#56811yescharacteristic
269Dic53—22no—
270Dic53—21no—
271Dic53—21no—
272Dic53—21no—
273Dic53—41no—
274Dic53—41no—
275Dic53—41no—
276Dic53#57711yescharacteristic
277Dic53#57811yescharacteristic
278#3108—531no—
279#2994—531no—
280#2984—531no—
281#1591—531no—
282#1590—531no—
283#1588—531no—
284#1587—531no—
285#1587—531no—
286#1585—531no—
287C8×Q16—531no—
288C8×SD16—531no—
289#1573—531no—
290#1494—531no—
291C2×Q8⋊C8—531no—
292#1450—531no—
293C2×C53—22no—
294C2×C53—81no—
295C2×C53#293311yescharacteristic
296C2×C53#294711yescharacteristic
297C2×C53C2×#44311yescharacteristic
298#529—531no—
299C2×C4×Q8—531no—
300#483—531no—
301#480#615711yescharacteristic
302#479—531no—
303#479—531no—
304#464—531no—
305#464—531no—
306#451—531no—
307#448—531no—
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#435—1061no—
322C8×D4—531no—
323C8×D4—1061no—
324#433—1061no—
325#432—531no—
326#432—1061no—
327#431—531no—
328#431—531no—
329#431—1061no—
330#424—1061no—
331C2×C4⋊C8—531no—
332C2×C4×C8—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—
347C82—531no—
348C82—531no—
349C53#1801711yescharacteristic, complemented, Sylow
350C22×Q8—531no—
351C2×Q16—531no—
352C2×Q16—531no—
353C2×SD16—531no—
354C2×SD16—531no—
355C2×M16—1061no—
356C22×C8—1061no—
357C22×C8—1061no—
358C22×C8—1061no—
359#126—1061no—
360#125—531no—
361#124—531no—
362#123—1061no—
363C22⋊Q8—531no—
364#121C2×#615711yescharacteristic
365C4×Q8—531no—
366C4×Q8—531no—
367C4×Q8—531no—
368C4×Q8—531no—
369C4×Q8—531no—
370C4×Q8—1061no—
371C4×Q8—1061no—
372C4×D4—531no—
373C4×D4—1061no—
374#117—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4C2×#615711yescharacteristic
377C2×C42—531no—
378C2×C42—1061no—
379#107C2×#615711yescharacteristic
380#107C2×#615711yescharacteristic
381#106—531no—
382#106—531no—
383C4⋊C8—531no—
384C4⋊C8—531no—
385C4⋊C8—531no—
386C4⋊C8—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#103—531no—
404#102C2×#615711yescharacteristic
405#102C2×#615711yescharacteristic
406C22⋊C8—531no—
407C22⋊C8—531no—
408C22⋊C8—531no—
409C22⋊C8—1061no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8C2×#615711yescharacteristic
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—531no—
430C2×Q8—1062no—
431C2×Q8—1061no—
432C2×Q8—2121no—
433C2×D4—21no—
434C2×D4C22×#615711yescharacteristic
435C22×C4—21no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4C22×#615711yescharacteristic
441Q16—1062no—
442Q16—2121no—
443SD16—1062no—
444SD16—2121no—
445M16—4241no—
446C2×C8—531no—
447C2×C8—531no—
448C2×C8—531no—
449C2×C8—531no—
450C2×C8—531no—
451C2×C8—531no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1064no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—4241no—
460C2×C8—4241no—
461C2×C8C22×#615711yescharacteristic
462C2×C8C22×#615711yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—531no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4C22×#615711yescharacteristic
474C4⋊C4C22×#615711yescharacteristic
475C4⋊C4C22×#615711yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—1061no—
478C22⋊C4—1061no—
479C22⋊C4—1061no—
480C22⋊C4—1061no—
481C42—531no—
482C42—531no—
483C42—531no—
484C42—531no—
485C42—1061no—
486C42—1061no—
487C42—1061no—
488C42—1061no—
489C42—1061no—
490C23—21no—
491C23D4×#615711yescharacteristic
492Q8—532no—
493Q8—1061no—
494Q8—1061no—
495Q8—1061no—
496Q8—1061no—
497Q8—1061no—
498Q8—2122no—
499D4—21no—
500D4—21no—
501D4—42no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—42no—
508C2×C4—531no—
509C2×C4—531no—
510C2×C4—531no—
511C2×C4—531no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1062no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2122no—
522C2×C4D4×#615711yescharacteristic
523C2×C4C23×#615711yescharacteristic
524C8—21no—
525C8—21no—
526C8—1061no—
527C8—1061no—
528C8—1061no—
529C8—1061no—
530C8—1062no—
531C8—1062no—
532C8—1061no—
533C8—1061no—
534C8—2121no—
535C8—2121no—
536C8—2121no—
537C8—2121no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22#1681166512yes—
544C22C2×D4×#615711yescharacteristic
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—531no—
550C4—531no—
551C4—1062no—
552C4—1061no—
553C4—1061no—
554C4—1061no—
555C4—2121no—
556C4—2121no—
557C4—2121no—
558C4D53⋊#13111yescharacteristic
559C4#1681170211yescharacteristic
560C2—22no—
561C2—81no—
562C2#2029144911yescharacteristic
563C2#2029147311yescharacteristic
564C2C2×#1681166511yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#480C4
socleC22×C53C2×C4×D4
prime radical#121×C53C2×C4
cosocleC4×Dic53C24
hypercenter#480#6157
hypocenterC53#18017
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #20291615C2semidirect
3. #20290354C2semidirect
4. #20290352C2semidirect
5. #20290345C2semidirect
6. #20290339C2semidirect
7. #20290339C2semidirect
8. #20290330C2semidirect
9. #20290303C2semidirect
10. #20290298C2semidirect
11. #20290290C2semidirect
13. C2×#16811473C2semidirect
14. #20289920C2semidirect
349. C53#18017semidirect

Maximal subgroups

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

subgroupindex
#202916622
#202916152
#202903542
#202903522
#202903452
#202903392
#202903302
#202903032
#202902982
#202902902
#202900662
C2×#168114732
#202899202
#202847922
#1801753