Finite Group Databasebeta

← #27541063

Subgroups — #27541063

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287063C211yesmaximal, characteristic, complemented
2#20276951C211yesmaximal, characteristic, complemented
3#20276910C211yesmaximal, characteristic, complemented
4#20276898C211yesmaximal, characteristic, complemented
5#20276862C211yesmaximal, characteristic, complemented
6#20276816C211yesmaximal, characteristic, complemented
7#20276813C211yesmaximal, characteristic, complemented
8#20276767C211yesmaximal, characteristic, complemented
9#20276638C211yesmaximal, characteristic, complemented
10#20276627C211yesmaximal, characteristic, complemented
11#20276589C211yesmaximal, characteristic, complemented
12#20276541C211yesmaximal, characteristic, complemented
13#20276427C211yesmaximal, characteristic, complemented
14C2×#16810424C211yesmaximal, characteristic, complemented
15C53×#3280C211yesmaximal, characteristic, complemented
16#16811277—21no—
17C2×#13133160C2211yescharacteristic, complemented
18C2×#13133158C2211yescharacteristic
19C2×#13133158C2211yescharacteristic
20C2×#13133157C2211yescharacteristic, complemented
21C22×#11719908C2211yescharacteristic
22#16811028C2211yescharacteristic, complemented
23#16810937—21no—
24#16810937—21no—
25#16810904—21no—
26#16810859C2211yescharacteristic
27#16810859C2211yescharacteristic, complemented
28C4×D424C2211yescharacteristic, complemented
29C2×#13133151C2211yescharacteristic
30C2×#13133151C2211yescharacteristic, complemented
31#16810450C2211yescharacteristic, complemented
32#16810450C2211yescharacteristic, complemented
33#16810450C2211yescharacteristic, complemented
34C4⋊#11719908C2211yescharacteristic, complemented
35#16810427C2211yescharacteristic, complemented
36#16810427C2211yescharacteristic, complemented
37#16810427C2211yescharacteristic, complemented
38#16810424C2211yescharacteristic
39#16810424C2211yescharacteristic
40#16810424C2211yescharacteristic
41#16810424C2211yescharacteristic
42#16810424C2211yescharacteristic
43#16810424C2211yescharacteristic
44#16810423C2211yescharacteristic, complemented
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic, complemented
47#16810399C2211yescharacteristic, complemented
48C2×#135×C53C2211yescharacteristic
49C22×D8×C53C2211yescharacteristic
50C53×#546—21no—
51C53×#495C2211yescharacteristic
52C53×#467C2211yescharacteristic
53C53×#466C2211yescharacteristic
54C53×#463—21no—
55C53×#450—21no—
56C53×#449—21no—
57C4×D8×C53C2211yescharacteristic
58C8×D4×C53C2211yescharacteristic
59C2×#11719947—41no—
60C2×D424C2311yescharacteristic
61#13133217—21no—
62#13133201—41no—
63#13133201—41no—
64#13133197—41no—
65#13133191—21no—
66#13133191—21no—
67C4×D212—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70#13133160—41no—
71#13133158—22no—
72#13133158—41no—
73#13133157—41no—
74C2×#11719908—41no—
75C2×#11719908C2311yescharacteristic
76C2×#11719908C2311yescharacteristic
77C2×#11719908C2311yescharacteristic
78C2×#11719908C2311yescharacteristic
79#13133151—21no—
80#13133151—21no—
81#13133151—21no—
82#13133151C2311yescharacteristic
83#13133151C2311yescharacteristic
84#13133151C2311yescharacteristic
85#13133139—21no—
86#13133139—21no—
87#13133139—21no—
88C2×#41×C53—21no—
89C22×D4×C53—21no—
90#135×C53—22no—
91#135×C53—41no—
92C2×Q16×C53C2311yescharacteristic
93C2×SD16×C53—21no—
94C2×SD16×C53—21no—
95C2×D8×C53—21no—
96C2×D8×C53—22no—
97C2×D8×C53—41no—
98C2×D8×C53C2311yescharacteristic
99C2×D8×C53C2311yescharacteristic
100C22×C8×C53C2311yescharacteristic
101C22×C8×C53C2311yescharacteristic
102#124×C53—21no—
103#123×C53—41no—
104C22⋊Q8×C53—21no—
105#121×C53—21no—
106#121×C53—41no—
107#120×C53—41no—
108C4×D4×C53—21no—
109C4×D4×C53D411yescharacteristic
110C2×C22⋊C4×C53—41no—
111#107×C53C2311yescharacteristic
112C4⋊C8×C53D411yescharacteristic
113#103×C53—21no—
114#102×C53—21no—
115#102×C53—21no—
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C24×C53—41no—
120#41×C53—42no—
121#41×C53—81no—
122C2×Q8×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—42no—
126C2×D4×C53—41no—
127C2×D4×C53—41no—
128C2×D4×C53—41no—
129C2×D4×C53—81no—
130C2×D4×C53D811yescharacteristic
131C22×C4×C53—41no—
132C22×C4×C53—41no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C2×C22⋊D53—41no—
136C2×C22⋊D53—41no—
137C22×Dic53—41no—
138C22×Dic53—41no—
139#11719947—84no—
140C2×D212—21no—
141C2×C4×D53—41no—
142C2×Dic106—41no—
143Q16×C53—21no—
144Q16×C53—21no—
145SD16×C53—42no—
146SD16×C53—81no—
147D8×C53—21no—
148D8×C53—21no—
149D8×C53—21no—
150D8×C53—42no—
151D8×C53#4112yescomplemented
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53C2×D411yescharacteristic
157C2×C8×C53C2×D411yescharacteristic
158C2×C8×C53C2×D411yescharacteristic
159C2×C8×C53C2411yescharacteristic
160C4⋊C4×C53—21no—
161C4⋊C4×C53—41no—
162C4⋊C4×C53D811yescharacteristic
163C22⋊C4×C53—41no—
164C22⋊C4×C53—41no—
165C22⋊C4×C53—41no—
166C22⋊C4×C53—41no—
167C22⋊C4×C53—41no—
168C22⋊C4×C53D811yescharacteristic
169C22⋊C4×C53D811yescharacteristic
170C42×C53C2×D411yescharacteristic
171#11719921—41no—
172#11719921—41no—
173#11719921—41no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—41no—
178C4⋊Dic53—41no—
179C4⋊Dic53—41no—
180D424—21no—
181D424—21no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185#11719908—21no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53—82no—
189C23×C53—82no—
190C23×C53—161no—
191C23×C53C2×D811yescharacteristic
192C23×C53C2×D811yescharacteristic
193C22×D53—41no—
194Q8×C53—41no—
195Q8×C53—41no—
196D4×C53—22no—
197D4×C53—22no—
198D4×C53—41no—
199D4×C53—41no—
200D4×C53—41no—
201D4×C53—82no—
202D4×C53—82no—
203D4×C53—161no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—82no—
211C2×C4×C53—161no—
212C2×C4×C53C2×D811yescharacteristic
213C2×C4×C53C2×D811yescharacteristic
214C2×C4×C53C2×D811yescharacteristic
215C2×C4×C53C2×D811yescharacteristic
216C2×C4×C53C22×D411yescharacteristic
217C22⋊D53—82no—
218C22⋊D53—82no—
219C2×Dic53—41no—
220C2×Dic53—41no—
221C2×Dic53—41no—
222C2×Dic53—82no—
223C2×Dic53—82no—
224D212—41no—
225D212—41no—
226C4×D53—82no—
227Dic106—82no—
228C8×C53—21no—
229C8×C53—21no—
230C8×C53—21no—
231C8×C53C2×#4111yescharacteristic
232C8×C53H(5,𝔽2)11yescharacteristic
233#33873—531nomaximal, Sylow
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—42no—
240C22×C53—81no—
241C22×C53—81no—
242C22×C53—81no—
243C22×C53—81no—
244C22×C53—81no—
245C22×C53—164no—
246C22×C53#50912yes—
247C22×C53C22×D811yescharacteristic
248C2×D53—81no—
249C2×D53—81no—
250C2×D53—81no—
251C4×C53—21no—
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—81no—
256C4×C53—81no—
257C4×C53#54611yescharacteristic
258C4×C53#57711yescharacteristic
259Dic53—81no—
260Dic53—81no—
261Dic53—81no—
262#3334—531no—
263#3280D5311yescharacteristic, complemented
264#2243—531no—
265#2235—531no—
266#2233—531no—
267#2226—531no—
268#2218—531no—
269#2217—531no—
270#2210—531no—
271#2186—531no—
272#2184—531no—
273#2177—531no—
274#2168—531no—
275#2139—531no—
276C2×#357—531no—
277C2×C53—22no—
278C2×C53—41no—
279C2×C53—82no—
280C2×C53—161no—
281C2×C53#329411yescharacteristic
282C2×C53C2×#50911yescharacteristic
283C2×C53#341611yescharacteristic
284D53—161no—
285C2×#135C2×D5311yescharacteristic
286C22×D8C2×D5311yescharacteristic
287#547—1061no—
288#546—21no—
289C2×SD32—531no—
290C2×D16—531no—
291C2×D16—531no—
292C2×M32—531no—
293C22×C16—531no—
294#500—531no—
295#495C2×D5311yescharacteristic
296#480—1061no—
297#480—1061no—
298#474—1061no—
299#467C2×D5311yescharacteristic
300#466—531no—
301#466—531no—
302#466C2×D5311yescharacteristic
303#463—21no—
304#450—21no—
305#449—21no—
306C4×D8—531no—
307C4×D8C2×D5311yescharacteristic
308C8×D4C2×D5311yescharacteristic
309C2×#107—531no—
310C2×#107—531no—
311#367—531no—
312#367—531no—
313#367—531no—
314C4⋊C16—531no—
315#358—531no—
316#358—531no—
317#358—531no—
318#357—531no—
319#357—531no—
320#357—531no—
321#357—531no—
322#357—531no—
323#357—531no—
324#357—531no—
325C22⋊C16—531no—
326C22⋊C16—531no—
327#346—531no—
328C53#3387311yescharacteristic, complemented, Sylow
329C2×#41—21no—
330C2×#41—2121no—
331C22×D4—21no—
332#135—22no—
333#135—41no—
334C2×Q16C22×D5311yescharacteristic
335C2×SD16—21no—
336C2×SD16—21no—
337C2×D8—22no—
338C2×D8—21no—
339C2×D8—41no—
340C2×D8—531no—
341C2×D8C22×D5311yescharacteristic
342C2×D8C22×D5311yescharacteristic
343C22×C8C22×D5311yescharacteristic
344C22×C8C22×D5311yescharacteristic
345C4⋊Q8—1061no—
346#124—21no—
347#123—41no—
348#123—2121no—
349#123—2121no—
350C22⋊Q8—21no—
351C22⋊Q8—2121no—
352#121—21no—
353#121—41no—
354#121—1061no—
355#121—1061no—
356#120—41no—
357C4×D4—21no—
358C4×D4—1061no—
359C4×D4C22⋊D5311yescharacteristic
360C2×C4⋊C4—1061no—
361C2×C4⋊C4—1061no—
362C2×C22⋊C4—41no—
363SD32—2121no—
364D16—1062no—
365D16—2121no—
366M32—2121no—
367C2×C16—531no—
368C2×C16—531no—
369C2×C16—531no—
370C2×C16—531no—
371C2×C16—2121no—
372#107—531no—
373#107—531no—
374#107—531no—
375#107—1061no—
376#107—1061no—
377#107—1061no—
378#107C22×D5311yescharacteristic
379C4⋊C8C22⋊D5311yescharacteristic
380#103—21no—
381#102—21no—
382#102—21no—
383#102—1061no—
384#102—1061no—
385#102—1061no—
386C22⋊C8C22⋊D5311yescharacteristic
387C22⋊C8C22⋊D5311yescharacteristic
388C4×C8C22×D5311yescharacteristic
389C24—41no—
390#41—42no—
391#41—81no—
392#41—4244no—
393C2×Q8—21no—
394C2×Q8—2121no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—42no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—81no—
402C2×D4—1061no—
403C2×D4—2121no—
404C2×D4—2121no—
405C2×D4#1171992211yescharacteristic
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—2121no—
409C22×C4—2121no—
410C22×C4—2121no—
411C22×C4C2×C22⋊D5311yescharacteristic
412C22×C4C2×C22⋊D5311yescharacteristic
413Q16—21no—
414Q16—21no—
415SD16—42no—
416SD16—81no—
417D8—21no—
418D8—21no—
419D8—21no—
420D8—42no—
421D8—1061no—
422D8—1061no—
423D8#1171994712yescomplemented
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8C2×C22⋊D5311yescharacteristic
429C2×C8C2×C22⋊D5311yescharacteristic
430C2×C8C2×C22⋊D5311yescharacteristic
431C2×C8C23×D5311yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—41no—
434C4⋊C4—1061no—
435C4⋊C4—1061no—
436C4⋊C4—1061no—
437C4⋊C4—2121no—
438C4⋊C4—2121no—
439C4⋊C4—2121no—
440C4⋊C4#1171992211yescharacteristic
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—2121no—
447C22⋊C4—2121no—
448C22⋊C4—2121no—
449C22⋊C4#1171992211yescharacteristic
450C22⋊C4#1171992211yescharacteristic
451C42C2×C22⋊D5311yescharacteristic
452C16—1061no—
453C16—1061no—
454C16—1061no—
455C16—1061no—
456C23—41no—
457C23—41no—
458C23—82no—
459C23—82no—
460C23—161no—
461C23—2121no—
462C23C2×#1171992211yescharacteristic
463C23C2×#1171992211yescharacteristic
464Q8—41no—
465Q8—41no—
466Q8—4242no—
467D4—22no—
468D4—22no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4—82no—
473D4—82no—
474D4—161no—
475D4—4242no—
476D4—4242no—
477D4—2121no—
478D4—2121no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—82no—
486C2×C4—161no—
487C2×C4—4242no—
488C2×C4—4242no—
489C2×C4—4242no—
490C2×C4—2121no—
491C2×C4—2121no—
492C2×C4—2121no—
493C2×C4C2×#1171992211yescharacteristic
494C2×C4C2×#1171992211yescharacteristic
495C2×C4C2×#1171992211yescharacteristic
496C2×C4C2×#1171992211yescharacteristic
497C2×C4C22×C22⋊D5311yescharacteristic
498C8—21no—
499C8—21no—
500C8—21no—
501C8C2×#1171994711yescharacteristic
502C8#1313326211yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—42no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—164no—
515C22—4241no—
516C22—4241no—
517C22—4241no—
518C22#1681105812yes—
519C22C22×#1171992211yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—81no—
525C4—81no—
526C4—4241no—
527C4—4241no—
528C4—4241no—
529C4#1681126711yescharacteristic
530C4#1681143411yescharacteristic
531C2—22no—
532C2—41no—
533C2—82no—
534C2—161no—
535C2—8481no—
536C2#2028652811yescharacteristic
537C2C2×#1681105811yescharacteristic
538C2#2028786211yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3280C2
socleC22×C53C22×D8
hypercenter#3280D53
hypocenterC53#33873
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. #20287063C2semidirect
2. #20276951C2semidirect
3. #20276910C2semidirect
4. #20276898C2semidirect
5. #20276862C2semidirect
6. #20276816C2semidirect
7. #20276813C2semidirect
8. #20276767C2semidirect
9. #20276638C2semidirect
10. #20276627C2semidirect
11. #20276589C2semidirect
12. #20276541C2semidirect
13. #20276427C2semidirect
14. C2×#16810424C2semidirect
15. C53×#3280C2semidirect
17. C2×#13133160C22semidirect
20. C2×#13133157C22semidirect
22. #16811028C22semidirect
27. #16810859C22semidirect
28. C4×D424C22semidirect
30. C2×#13133151C22semidirect
31. #16810450C22semidirect
32. #16810450C22semidirect
33. #16810450C22semidirect
34. C4⋊#11719908C22semidirect
35. #16810427C22semidirect
36. #16810427C22semidirect
37. #16810427C22semidirect
44. #16810423C22semidirect
46. C22⋊#11719908C22semidirect
47. #16810399C22semidirect
151. D8×C53#41semidirect
263. #3280D53semidirect
328. C53#33873semidirect
423. D8#11719947semidirect

Maximal subgroups

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

subgroupindex
#202870632
#202769512
#202769102
#202768982
#202768622
#202768162
#202768132
#202767672
#202766382
#202766272
#202765892
#202765412
#202764272
C2×#168104242
C53×#32802
#3387353