Finite Group Databasebeta

← #28316774

Subgroups — #28316774

Order 13568.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291657C211yesmaximal, characteristic
2#20291610C211yesmaximal, characteristic, complemented
3#20290345C211yesmaximal, characteristic, complemented
4#20290339C211yesmaximal, characteristic, complemented
5#20290327C211yesmaximal, characteristic, complemented
6#20290325C211yesmaximal, characteristic, complemented
7#20290310C211yesmaximal, characteristic, complemented
8#20290306C211yesmaximal, characteristic, complemented
9#20290303C211yesmaximal, characteristic, complemented
10#20290298C211yesmaximal, characteristic, complemented
11#20290291C211yesmaximal, characteristic, complemented
12#20290045C211yesmaximal, characteristic, complemented
13#20290026C211yesmaximal, characteristic
14#20289936C211yesmaximal, characteristic, complemented
15#20284792C211yesmaximal, characteristic
16#16811673C2211yescharacteristic
17#16811672—21no—
18#16811672—21no—
19#16811672C2211yescharacteristic
20#16811672C2211yescharacteristic
21#16811672C2211yescharacteristic
22#16811672C2211yescharacteristic
23#16811672C2211yescharacteristic
24#16811658—21no—
25#16811658C2211yescharacteristic
26#16811654—21no—
27D4×#88324C2211yescharacteristic
28#16811647—21no—
29#16811640—21no—
30#16811640—21no—
31#16811640C2211yescharacteristic
32#16811639C2211yescharacteristic
33#16811639C2211yescharacteristic
34#16811629—21no—
35#16811626C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42#16811473C2211yescharacteristic
43#16811473C2211yescharacteristic
44#16811473C2211yescharacteristic
45#16811473C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
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—
68C2×#11719940—21no—
69C22×#88324—21no—
70#13133284—21no—
71#13133284—21no—
72#13133284—21no—
73#13133284—21no—
74#13133284—21no—
75#13133284—21no—
76#13133284—21no—
77C4⋊#88324—21no—
78C4⋊#88324D411yescharacteristic
79C4⋊#88324D411yescharacteristic
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84C22⋊#88324—21no—
85C22⋊#88324—21no—
86C22⋊#88324—21no—
87C22⋊#88324—21no—
88C22⋊#88324D411yescharacteristic
89C22⋊#88324D411yescharacteristic
90#13133272—21no—
91C4×#88324—21no—
92C4×#88324—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—21no—
153C2×Dic106—22no—
154C2×Dic106—22no—
155C2×Dic106C22×C411yescharacteristic
156C2×Dic106C22×C411yescharacteristic
157C2×Dic106C22×C411yescharacteristic
158#11719940—81no—
159#11719940—81no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—21no—
166C2×#88324—81no—
167C2×#88324C2×D411yescharacteristic
168C2×#88324C2×D411yescharacteristic
169C2×#88324C2×D411yescharacteristic
170C2×#88324C2×D411yescharacteristic
171C2×C8×C53C22×C411yescharacteristic
172C2×C8×C53C22×C411yescharacteristic
173C4⋊C4×C53—21no—
174C4⋊C4×C53C22×C411yescharacteristic
175C4⋊C4×C53C22×C411yescharacteristic
176C4⋊C4×C53C22×C411yescharacteristic
177C22⋊C4×C53—21no—
178C22⋊Dic53—21no—
179C22⋊Dic53—21no—
180C22⋊Dic53—21no—
181C22⋊Dic53—21no—
182#11719923—22no—
183#11719923—22no—
184C4⋊Dic53—21no—
185C4⋊Dic53C22×C411yescharacteristic
186#11719919—21no—
187#11719919—21no—
188#11719919—21no—
189#11719919—21no—
190#11719919—21no—
191#11719919—21no—
192#11719919—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53C22×C411yescharacteristic
199C4×Dic53C2×D411yescharacteristic
200C4×Dic53#4111yescharacteristic
201C4×Dic53C2411yescharacteristic
202C2×#88322C22×C411yescharacteristic
203C2×#88322C22×C411yescharacteristic
204Dic212—22no—
205Dic212—22no—
206C23×C53—21no—
207C23×C53C4×D411yescharacteristic
208D4×C53—21no—
209D4×C53—21no—
210D4×C53—81no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—81no—
217C2×C4×C53C4×D411yescharacteristic
218C2×C4×C53C23×C411yescharacteristic
219C2×Dic53—24no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53C4×D411yescharacteristic
230C2×Dic53C4×D411yescharacteristic
231C2×Dic53C22×D411yescharacteristic
232C2×Dic53C2×#4111yescharacteristic
233Dic106—24no—
234Dic106—21no—
235Dic106—21no—
236Dic106—21no—
237Dic106—21no—
238Dic106#13112yes—
239Dic106#13112yes—
240#88324—21no—
241#88324—21no—
242#88324—21no—
243#88324—21no—
244#88324—41no—
245#88324—41no—
246#88324—41no—
247#88324—41no—
248#88324—41no—
249#88324—41no—
250C8×C53—21no—
251C8×C53—21no—
252#88322—21no—
253#88322—21no—
254#17919—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—21no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53C2×C4×D411yescharacteristic
262C4×C53—21no—
263C4×C53—41no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53C2×#13111yescharacteristic
267C4×C53C2×#13111yescharacteristic
268Dic53—22no—
269Dic53—22no—
270Dic53—21no—
271Dic53—21no—
272Dic53—21no—
273Dic53—41no—
274Dic53—41no—
275Dic53#57711yescharacteristic
276Dic53#57811yescharacteristic
277#3108—531no—
278#2993—531no—
279#2982—531no—
280#1588—531no—
281#1587—531no—
282#1584—531no—
283#1583—531no—
284#1579—531no—
285SD16⋊C8—531no—
286C8×Q16—531no—
287C8×SD16—531no—
288#1574—531no—
289#1487—531no—
290#1482—531no—
291#1454—531no—
292C2×C53—41no—
293C2×C53—81no—
294C2×C53#292911yescharacteristic
295C2×C53#294711yescharacteristic
296C2×C53#295711yescharacteristic
297#529—531no—
298C2×C4×Q8—531no—
299#483—531no—
300#480#615711yescharacteristic
301#479—531no—
302#479—531no—
303#464—531no—
304#464—531no—
305#451—531no—
306#448—531no—
307#446—531no—
308C8×Q8—531no—
309C8×Q8—531no—
310C8×Q8—531no—
311C8×Q8—531no—
312C8×Q8—531no—
313C8×Q8—1061no—
314C8×Q8—1061no—
315C4×Q16—531no—
316C4×Q16—531no—
317C4×SD16—531no—
318C4×SD16—531no—
319#436—531no—
320#436—1061no—
321#435—1061no—
322C8×D4—531no—
323#433—1061no—
324#431—531no—
325#431—531no—
326#431—531no—
327#431—531no—
328#431—1061no—
329#431—1061no—
330#424—1061no—
331#423—531no—
332#335—531no—
333#335—531no—
334#334—531no—
335#334—531no—
336Q8⋊C8—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8—531no—
342D4⋊C8—531no—
343D4⋊C8—531no—
344#322—531no—
345#322—531no—
346#322—531no—
347C82—531no—
348C53#1791911yescharacteristic, complemented, Sylow
349C22×Q8—531no—
350C2×Q16—531no—
351C2×Q16—531no—
352C2×SD16—531no—
353C2×SD16—531no—
354C2×M16—1061no—
355C2×M16—1061no—
356C22×C8—1061no—
357#126—1061no—
358#125—531no—
359#124—531no—
360#123—1061no—
361C22⋊Q8—531no—
362#121C2×#615711yescharacteristic
363C4×Q8—531no—
364C4×Q8—531no—
365C4×Q8—531no—
366C4×Q8—531no—
367C4×Q8—531no—
368C4×Q8—1061no—
369C4×Q8—1061no—
370C4×D4—531no—
371C4×D4—1061no—
372#117—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4C2×#615711yescharacteristic
375C2×C42—531no—
376C2×C42—1061no—
377#107C2×#615711yescharacteristic
378#107C2×#615711yescharacteristic
379#106—531no—
380#106—531no—
381C4⋊C8—531no—
382C4⋊C8—531no—
383C4⋊C8—531no—
384C4⋊C8—531no—
385C4⋊C8—531no—
386C4⋊C8—531no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397#103—531no—
398#103—531no—
399#103—531no—
400#103—531no—
401#103—531no—
402#103—531no—
403#102C2×#615711yescharacteristic
404#102C2×#615711yescharacteristic
405C22⋊C8—531no—
406C22⋊C8—531no—
407C22⋊C8—531no—
408C22⋊C8—1061no—
409C22⋊C8—1061no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8C2×#615711yescharacteristic
413#97—1061no—
414C4×C8—531no—
415C4×C8—531no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—1061no—
421C4×C8—1061no—
422C4×C8—1061no—
423C4×C8—1061no—
424C4×C8—1061no—
425C4×C8—1061no—
426C2×Q8—531no—
427C2×Q8—531no—
428C2×Q8—531no—
429C2×Q8—1062no—
430C2×Q8—1062no—
431C2×Q8—1061no—
432C2×D4—21no—
433C2×D4C22×#615711yescharacteristic
434C22×C4—21no—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4C22×#615711yescharacteristic
440Q16—1062no—
441Q16—1062no—
442SD16—1062no—
443SD16—1062no—
444M16—4241no—
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—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—4241no—
459C2×C8C22×#615711yescharacteristic
460C2×C8C22×#615711yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—531no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4C22×#615711yescharacteristic
472C4⋊C4C22×#615711yescharacteristic
473C4⋊C4C22×#615711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—1061no—
476C22⋊C4—1061no—
477C22⋊C4—1061no—
478C22⋊C4—1061no—
479C42—531no—
480C42—531no—
481C42—531no—
482C42—531no—
483C42—1061no—
484C42—1061no—
485C42—1061no—
486C42—1061no—
487C42—1061no—
488C23—21no—
489C23D4×#615711yescharacteristic
490Q8—532no—
491Q8—532no—
492Q8—1064no—
493Q8—1061no—
494Q8—1061no—
495Q8—1061no—
496Q8—1061no—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—81no—
506C2×C4—531no—
507C2×C4—531no—
508C2×C4—531no—
509C2×C4—531no—
510C2×C4—1061no—
511C2×C4—1061no—
512C2×C4—1064no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4D4×#615711yescharacteristic
521C2×C4C23×#615711yescharacteristic
522C8—21no—
523C8—21no—
524C8—1061no—
525C8—1061no—
526C8—1061no—
527C8—1061no—
528C8—1061no—
529C8—1061no—
530C8—2121no—
531C8—2121no—
532C8—2121no—
533C8—2121no—
534C8—2121no—
535C8—2121no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22C2×D4×#615711yescharacteristic
543C4—21no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—531no—
548C4—531no—
549C4—1062no—
550C4—1062no—
551C4—1061no—
552C4—1061no—
553C4—1061no—
554C4—2121no—
555C4—2121no—
556C4D53⋊#13111yescharacteristic
557C4D53⋊#13111yescharacteristic
558C2—41no—
559C2—81no—
560C2#2029143211yescharacteristic
561C2#2029147311yescharacteristic
562C2#2029149111yescharacteristic
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#17919
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. #20291610C2semidirect
3. #20290345C2semidirect
4. #20290339C2semidirect
5. #20290327C2semidirect
6. #20290325C2semidirect
7. #20290310C2semidirect
8. #20290306C2semidirect
9. #20290303C2semidirect
10. #20290298C2semidirect
11. #20290291C2semidirect
12. #20290045C2semidirect
14. #20289936C2semidirect
348. C53#17919semidirect

Maximal subgroups

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

subgroupindex
#202916572
#202916102
#202903452
#202903392
#202903272
#202903252
#202903102
#202903062
#202903032
#202902982
#202902912
#202900452
#202900262
#202899362
#202847922
#1791953