Finite Group Databasebeta

← #27541237

Subgroups — #27541237

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D424C211yesmaximal, characteristic, complemented
2#20276958C211yesmaximal, characteristic, complemented
3#20276906C211yesmaximal, characteristic, complemented
4#20276900C211yesmaximal, characteristic, complemented
5#20276869C211yesmaximal, characteristic, complemented
6#20276816C211yesmaximal, characteristic, complemented
7#20276799C211yesmaximal, characteristic, complemented
8#20276750C211yesmaximal, characteristic, complemented
9#20276626C211yesmaximal, characteristic, complemented
10#20276626C211yesmaximal, characteristic, complemented
11#20276589C211yesmaximal, characteristic, complemented
12#20276541C211yesmaximal, characteristic, complemented
13#20276430C211yesmaximal, characteristic, complemented
14C2×#16810423C211yesmaximal, characteristic, complemented
15C53×#3339C211yesmaximal, characteristic, complemented
16C22×D424C2211yescharacteristic
17C22×D424C2211yescharacteristic
18D4×D212—21no—
19C2×#13133158C2211yescharacteristic
20C2×#13133158C2211yescharacteristic, complemented
21C2×#13133158C2211yescharacteristic, complemented
22C2×#13133157C2211yescharacteristic, complemented
23C22×#11719908C2211yescharacteristic, complemented
24#16811000C2211yescharacteristic
25#16810859C2211yescharacteristic
26#16810859C2211yescharacteristic
27#16810819—21no—
28#16810759—21no—
29#16810759—21no—
30C4×D424C2211yescharacteristic
31#16810450C2211yescharacteristic, complemented
32#16810450C2211yescharacteristic, complemented
33#16810448C2211yescharacteristic, complemented
34C4⋊#11719908C2211yescharacteristic, complemented
35#16810424C2211yescharacteristic, complemented
36#16810423C2211yescharacteristic
37#16810423C2211yescharacteristic
38#16810423C2211yescharacteristic
39#16810423C2211yescharacteristic, complemented
40#16810423C2211yescharacteristic, complemented
41#16810423C2211yescharacteristic, complemented
42#16810423C2211yescharacteristic, complemented
43#16810423C2211yescharacteristic, complemented
44#16810423C2211yescharacteristic, complemented
45C22⋊#11719908C2211yescharacteristic, complemented
46C22⋊#11719908C2211yescharacteristic, complemented
47#16810399C2211yescharacteristic, complemented
48C53×#553—21no—
49C53×#499C2211yescharacteristic, complemented
50C53×#482—21no—
51C53×#481—21no—
52C53×#478—21no—
53C53×#466C2211yescharacteristic, complemented
54C53×#466C2211yescharacteristic, complemented
55C4×D8×C53C2211yescharacteristic, complemented
56C8×D4×C53C2211yescharacteristic
57C53×#427C2211yescharacteristic, complemented
58C2×#107×C53C2211yescharacteristic, complemented
59C22×D212—21no—
60C22×D212—21no—
61C2×D4×D53—41no—
62C2×D424—21no—
63C2×D424—21no—
64C2×D424—21no—
65C2×D424—22no—
66C2×D424—22no—
67C2×D424—41no—
68C2×D424—41no—
69C2×D424C2311yescharacteristic
70C2×D424C2311yescharacteristic
71C2×D424C2311yescharacteristic
72#13133214—21no—
73#13133192—41no—
74#13133191—21no—
75#13133191—21no—
76#13133187—41no—
77#13133187—41no—
78C4×D212—21no—
79#13133158—22no—
80#13133158—41no—
81#13133158—41no—
82#13133157—41no—
83C2×#11719908—41no—
84C2×#11719908C2311yescharacteristic
85C2×#11719908C2311yescharacteristic
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88#13133151C2311yescharacteristic
89#13133139—21no—
90#13133139—21no—
91#13133139—21no—
92C2×D8×C53C2311yescharacteristic
93C22×C8×C53C2311yescharacteristic
94C22×C8×C53C2311yescharacteristic
95#126×C53—41no—
96#125×C53—21no—
97#123×C53—41no—
98#121×C53—21no—
99#121×C53—21no—
100#121×C53—41no—
101C4×D4×C53—21no—
102C4×D4×C53—41no—
103C4×D4×C53D411yescharacteristic
104#117×C53—21no—
105C2×C4⋊C4×C53—21no—
106#107×C53—21no—
107#107×C53C2311yescharacteristic
108#107×C53C2311yescharacteristic
109#107×C53C2311yescharacteristic
110#106×C53—21no—
111#106×C53—21no—
112C4⋊C8×C53D411yescharacteristic
113#102×C53—21no—
114#102×C53—21no—
115#102×C53—21no—
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C23×D53—41no—
120C23×D53—41no—
121C2×D4×C53—21no—
122C2×D4×C53—41no—
123C2×D4×C53—41no—
124C2×D4×C53D811yescharacteristic
125C22×C4×C53—41no—
126C22×C4×C53—41no—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C2×C22⋊D53—41no—
130C2×C22⋊D53—41no—
131D4×D53—84no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212—21no—
135C2×D212—42no—
136C2×D212—42no—
137C2×D212—41no—
138C2×D212—41no—
139C2×D212—41no—
140C2×D212—41no—
141C2×D212—81no—
142C2×D212—81no—
143C2×C4×D53—41no—
144D8×C53—21no—
145D8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53C2×D411yescharacteristic
151C2×C8×C53C2×D411yescharacteristic
152C2×C8×C53C2×D411yescharacteristic
153C2×C8×C53C2411yescharacteristic
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—41no—
158C4⋊C4×C53—41no—
159C4⋊C4×C53—41no—
160C4⋊C4×C53D811yescharacteristic
161C22⋊C4×C53—41no—
162C22⋊C4×C53—41no—
163C22⋊C4×C53—41no—
164C22⋊C4×C53—41no—
165C22⋊C4×C53D811yescharacteristic
166C22⋊C4×C53D811yescharacteristic
167C42×C53—41no—
168C42×C53C2×D411yescharacteristic
169#11719921—41no—
170#11719921—41no—
171#11719921—41no—
172C4⋊Dic53—21no—
173D424—21no—
174D424—21no—
175D424—21no—
176D424—21no—
177D424—21no—
178D424—42no—
179D424—42no—
180D424—81no—
181D424#4112yes—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185#11719908—21no—
186C23×C53—41no—
187C23×C53C2×D811yescharacteristic
188C23×C53C2×D811yescharacteristic
189C22×D53—41no—
190C22×D53—41no—
191C22×D53—41no—
192C22×D53—82no—
193C22×D53—82no—
194C22×D53—82no—
195C22×D53—82no—
196C22×D53—161no—
197C22×D53—161no—
198D4×C53—22no—
199D4×C53—41no—
200D4×C53—41no—
201D4×C53—82no—
202D4×C53—161no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—161no—
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—
220D212—22no—
221D212—41no—
222D212—41no—
223D212—41no—
224D212—41no—
225D212—41no—
226D212—82no—
227D212—82no—
228D212—82no—
229D212—161no—
230C4×D53—82no—
231C8×C53—21no—
232C8×C53—21no—
233C8×C53—21no—
234C8×C53C2×#4111yescharacteristic
235C8×C53H(5,𝔽2)11yescharacteristic
236#33885—531nomaximal, Sylow
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—81no—
243C22×C53—81no—
244C22×C53—81no—
245C22×C53#50912yes—
246C22×C53C22×D811yescharacteristic
247C2×D53—42no—
248C2×D53—81no—
249C2×D53—81no—
250C2×D53—81no—
251C2×D53—81no—
252C2×D53—81no—
253C2×D53—81no—
254C2×D53—81no—
255C2×D53—81no—
256C2×D53—164no—
257C2×D53—164no—
258C4×C53—21no—
259C4×C53—21no—
260C4×C53—21no—
261C4×C53—21no—
262C4×C53—81no—
263C4×C53—81no—
264C4×C53—81no—
265C4×C53#54611yescharacteristic
266C4×C53#57711yescharacteristic
267Dic53—81no—
268#3339D5311yescharacteristic, complemented
269D4×D8—531no—
270#2244—531no—
271#2234—531no—
272#2233—531no—
273#2228—531no—
274#2218—531no—
275#2215—531no—
276#2206—531no—
277#2184—531no—
278#2184—531no—
279#2177—531no—
280#2168—531no—
281#2140—531no—
282C2×#357—531no—
283C2×C53—22no—
284C2×C53—41no—
285C2×C53—161no—
286C2×C53#329411yescharacteristic
287C2×C53C2×#50911yescharacteristic
288C2×C53#341511yescharacteristic
289D53—82no—
290D53—161no—
291D53—161no—
292C22×D8—531no—
293C22×D8—531no—
294#553—21no—
295D42—1061no—
296C2×D16—531no—
297C2×D16—531no—
298C2×D16—531no—
299C2×M32—531no—
300C22×C16—531no—
301#499C2×D5311yescharacteristic, complemented
302#493—531no—
303#482—21no—
304#481—21no—
305#478—21no—
306#466—531no—
307#466—531no—
308#466C2×D5311yescharacteristic, complemented
309#466C2×D5311yescharacteristic, complemented
310#459—1061no—
311#447—1061no—
312#447—1061no—
313C4×D8—531no—
314C4×D8C2×D5311yescharacteristic, complemented
315C8×D4C2×D5311yescharacteristic
316#427C2×D5311yescharacteristic, complemented
317C2×#107C2×D5311yescharacteristic, complemented
318#367—531no—
319#367—531no—
320#366—531no—
321C4⋊C16—531no—
322#357—531no—
323#357—531no—
324#357—531no—
325#357—531no—
326#357—531no—
327#357—531no—
328#357—531no—
329#357—531no—
330#357—531no—
331#357—531no—
332C22⋊C16—531no—
333C22⋊C16—531no—
334#346—531no—
335C53#3388511yescharacteristic, complemented, Sylow
336C22×D4—1061no—
337C22×D4—1061no—
338C22×D4—2121no—
339C2×D8—531no—
340C2×D8—531no—
341C2×D8—531no—
342C2×D8—1062no—
343C2×D8—1062no—
344C2×D8—1061no—
345C2×D8—1061no—
346C2×D8—1061no—
347C2×D8—2121no—
348C2×D8—2121no—
349C2×D8C22×D5311yescharacteristic
350C22×C8C22×D5311yescharacteristic
351C22×C8C22×D5311yescharacteristic
352#127—1061no—
353#126—41no—
354#125—21no—
355#123—41no—
356#121—21no—
357#121—21no—
358#121—41no—
359#121—1061no—
360#121—1061no—
361#121—2121no—
362#120—2121no—
363#120—2121no—
364C4×D4—21no—
365C4×D4—41no—
366C4×D4—1061no—
367C4×D4C22⋊D5311yescharacteristic
368#117—21no—
369C2×C4⋊C4—21no—
370D16—1062no—
371D16—2121no—
372D16—2121no—
373M32—2121no—
374C2×C16—531no—
375C2×C16—531no—
376C2×C16—531no—
377C2×C16—531no—
378C2×C16—2121no—
379#107—21no—
380#107—531no—
381#107C22×D5311yescharacteristic
382#107C22×D5311yescharacteristic
383#107C22×D5311yescharacteristic
384#106—21no—
385#106—21no—
386C4⋊C8C22⋊D5311yescharacteristic
387#102—21no—
388#102—21no—
389#102—21no—
390#102—1061no—
391#102—1061no—
392#102—1061no—
393C22⋊C8C22⋊D5311yescharacteristic
394C22⋊C8C22⋊D5311yescharacteristic
395C4×C8C22×D5311yescharacteristic
396C24—2121no—
397C24—2121no—
398C2×D4—21no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—1061no—
402C2×D4—1061no—
403C2×D4—1061no—
404C2×D4—4241no—
405C2×D4—4241no—
406C2×D4—4244no—
407C2×D4—2122no—
408C2×D4—2122no—
409C2×D4—2121no—
410C2×D4—2121no—
411C2×D4—2121no—
412C2×D4—2121no—
413C2×D4—2121no—
414C2×D4—2121no—
415C2×D4#1171992211yescharacteristic
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—2121no—
419C22×C4C2×C22⋊D5311yescharacteristic
420C22×C4C2×C22⋊D5311yescharacteristic
421D8—21no—
422D8—21no—
423D8—532no—
424D8—1061no—
425D8—1061no—
426D8—1061no—
427D8—1061no—
428D8—1061no—
429D8—4241no—
430D8—2122no—
431D8—2122no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8C2×C22⋊D5311yescharacteristic
437C2×C8C2×C22⋊D5311yescharacteristic
438C2×C8C2×C22⋊D5311yescharacteristic
439C2×C8C23×D5311yescharacteristic
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—1061no—
447C4⋊C4#1171992211yescharacteristic
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—2121no—
453C22⋊C4—2121no—
454C22⋊C4—2121no—
455C22⋊C4#1171992211yescharacteristic
456C22⋊C4#1171992211yescharacteristic
457C42—41no—
458C42C2×C22⋊D5311yescharacteristic
459C16—1061no—
460C16—1061no—
461C16—1061no—
462C16—1061no—
463C23—41no—
464C23—8481no—
465C23—8481no—
466C23—4242no—
467C23—4242no—
468C23—4242no—
469C23—4242no—
470C23—2121no—
471C23—2121no—
472C23—2121no—
473C23C2×#1171992211yescharacteristic
474C23C2×#1171992211yescharacteristic
475D4—22no—
476D4—41no—
477D4—41no—
478D4—82no—
479D4—161no—
480D4—8481no—
481D4—1062no—
482D4—4242no—
483D4—4242no—
484D4—4242no—
485D4—4242no—
486D4—4242no—
487D4—2121no—
488D4—2121no—
489D4—2121no—
490D4—2121no—
491D4—2121no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—161no—
500C2×C4—161no—
501C2×C4—4242no—
502C2×C4—2121no—
503C2×C4C2×#1171992211yescharacteristic
504C2×C4C2×#1171992211yescharacteristic
505C2×C4C2×#1171992211yescharacteristic
506C2×C4C2×#1171992211yescharacteristic
507C2×C4C22×C22⋊D5311yescharacteristic
508C8—21no—
509C8—21no—
510C8—21no—
511C8#41×D5311yescharacteristic
512C8#1313326311yescharacteristic
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—8484no—
522C22—8484no—
523C22—4241no—
524C22—4241no—
525C22—4241no—
526C22—4241no—
527C22—4241no—
528C22—4241no—
529C22—4241no—
530C22—4241no—
531C22—2122no—
532C22#1681105812yes—
533C22C22×#1171992211yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—4241no—
542C4#1681126111yescharacteristic
543C4#1681143111yescharacteristic
544C2—22no—
545C2—41no—
546C2—161no—
547C2—8481no—
548C2—8481no—
549C2—4242no—
550C2#2028652211yescharacteristic
551C2C2×#1681105811yescharacteristic
552C2#2028785411yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3339C2
socleC22×C53C22×D8
hypercenter#3339D53
hypocenterC53#33885
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3339

Complements

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

NH ≅ G/Ntype
1. D4×D424C2semidirect
2. #20276958C2semidirect
3. #20276906C2semidirect
4. #20276900C2semidirect
5. #20276869C2semidirect
6. #20276816C2semidirect
7. #20276799C2semidirect
8. #20276750C2semidirect
9. #20276626C2semidirect
10. #20276626C2semidirect
11. #20276589C2semidirect
12. #20276541C2semidirect
13. #20276430C2semidirect
14. C2×#16810423C2semidirect
15. C53×#3339C2semidirect
20. C2×#13133158C22semidirect
21. C2×#13133158C22semidirect
22. C2×#13133157C22semidirect
23. C22×#11719908C22semidirect
31. #16810450C22semidirect
32. #16810450C22semidirect
33. #16810448C22semidirect
34. C4⋊#11719908C22semidirect
35. #16810424C22semidirect
39. #16810423C22semidirect
40. #16810423C22semidirect
41. #16810423C22semidirect
42. #16810423C22semidirect
43. #16810423C22semidirect
44. #16810423C22semidirect
45. C22⋊#11719908C22semidirect
46. C22⋊#11719908C22semidirect
47. #16810399C22semidirect
49. C53×#499C22semidirect
53. C53×#466C22semidirect
54. C53×#466C22semidirect
55. C4×D8×C53C22semidirect
57. C53×#427C22semidirect
58. C2×#107×C53C22semidirect
268. #3339D53semidirect
301. #499C2×D53semidirect
308. #466C2×D53semidirect
309. #466C2×D53semidirect
314. C4×D8C2×D53semidirect
316. #427C2×D53semidirect
317. C2×#107C2×D53semidirect
335. C53#33885semidirect

Maximal subgroups

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

subgroupindex
D4×D4242
#202769582
#202769062
#202769002
#202768692
#202768162
#202767992
#202767502
#202766262
#202765892
#202765412
#202764302
C2×#168104232
C53×#33392
#3388553