Finite Group Databasebeta

← #27539194

Subgroups — #27539194

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283568C211yesmaximal, characteristic, complemented
2#20276920C211yesmaximal, characteristic, complemented
3#20276903C211yesmaximal, characteristic, complemented
4#20276859C211yesmaximal, characteristic, complemented
5#20276849C211yesmaximal, characteristic, complemented
6#20276767C211yesmaximal, characteristic, complemented
7#20276767C211yesmaximal, characteristic, complemented
8#20276643C211yesmaximal, characteristic, complemented
9#20276626C211yesmaximal, characteristic
10#20276589C211yesmaximal, characteristic, complemented
11#20276577C211yesmaximal, characteristic, complemented
12C4×#13133160C211yesmaximal, characteristic, complemented
13C4×#13133158C211yesmaximal, characteristic, complemented
14#20276522C211yesmaximal, characteristic, complemented
15C53×#3175C211yesmaximal, characteristic, complemented
16C22×D424C2211yescharacteristic
17#16811097—21no—
18C2×#13133160C2211yescharacteristic
19C2×#13133160C2211yescharacteristic
20C2×#13133158C2211yescharacteristic
21C2×#13133158C2211yescharacteristic
22#16810736—21no—
23#16810736—21no—
24C4×D424C2211yescharacteristic
25C4×D424C2211yescharacteristic
26C4×D424C2211yescharacteristic
27C4×D424C2211yescharacteristic
28#16810661C2211yescharacteristic
29#16810629—21no—
30#16810450C2211yescharacteristic, complemented
31#16810450C2211yescharacteristic, complemented
32#16810448C2211yescharacteristic, complemented
33#16810448C2211yescharacteristic, complemented
34C4⋊#11719908C2211yescharacteristic, complemented
35C4⋊#11719908C2211yescharacteristic, complemented
36#16810427C2211yescharacteristic, complemented
37#16810427C2211yescharacteristic, complemented
38#16810424C2211yescharacteristic, complemented
39#16810424C2211yescharacteristic, complemented
40#16810423C2211yescharacteristic
41#16810423C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44C22⋊#11719908C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46#16810399C2211yescharacteristic, complemented
47C4×#11719908C2211yescharacteristic, complemented
48C53×#542—21no—
49C53×#500C2211yescharacteristic, complemented
50C53×#495C2211yescharacteristic, complemented
51C53×#489—21no—
52C53×#468—21no—
53C53×#467C2211yescharacteristic
54C53×#466C2211yescharacteristic
55C53×#440—21no—
56C4×Q16×C53C2211yescharacteristic, complemented
57C4×D8×C53C2211yescharacteristic, complemented
58C53×#405C2211yescharacteristic
59C22×D212—21no—
60C2×D424—21no—
61C2×D424—22no—
62C2×D424—22no—
63C2×D424C2311yescharacteristic
64C2×D424C2311yescharacteristic
65C4×D212—21no—
66C4×D212—21no—
67C4×D212—21no—
68C4×D212—21no—
69#13133179—41no—
70#13133179—41no—
71#13133176—21no—
72C22⋊C4×D53—41no—
73C22⋊C4×D53—41no—
74#13133160—22no—
75#13133160—22no—
76#13133158—22no—
77#13133158—22no—
78C2×#11719908C2311yescharacteristic
79C2×#11719908C2311yescharacteristic
80C2×#11719908C2311yescharacteristic
81C2×#11719908C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133151C2311yescharacteristic
84#13133149—21no—
85#13133139—21no—
86#13133139—21no—
87#13133139—21no—
88#13133139—21no—
89C2×Q16×C53C2311yescharacteristic
90C2×SD16×C53—21no—
91C2×D8×C53C2311yescharacteristic
92C2×M16×C53D411yescharacteristic
93C22×C8×C53C2311yescharacteristic
94C4⋊Q8×C53—21no—
95#126×C53—41no—
96#124×C53—21no—
97#124×C53—41no—
98#123×C53—41no—
99C22⋊Q8×C53—21no—
100C22⋊Q8×C53—41no—
101#121×C53—21no—
102C4×Q8×C53—21no—
103C4×D4×C53—21no—
104#117×C53D411yescharacteristic
105#107×C53—21no—
106#107×C53C2311yescharacteristic
107#107×C53C2311yescharacteristic
108#103×C53—21no—
109#103×C53—21no—
110#102×C53—21no—
111#102×C53—21no—
112#97×C53D411yescharacteristic
113#97×C53D411yescharacteristic
114C4×C8×C53C2311yescharacteristic
115C4×C8×C53C2311yescharacteristic
116C23×D53—41no—
117C2×Q8×C53—21no—
118C2×Q8×C53—41no—
119C2×D4×C53—21no—
120C2×D4×C53—41no—
121C22×C4×C53—41no—
122C22×C4×C53C2×D411yescharacteristic
123C2×D212—24no—
124C2×D212—21no—
125C2×D212—21no—
126C2×D212—42no—
127C2×D212—41no—
128C2×C4×D53—41no—
129C2×C4×D53—41no—
130C2×C4×D53—41no—
131C2×C4×D53—41no—
132Q16×C53—21no—
133Q16×C53—21no—
134SD16×C53—42no—
135D8×C53—21no—
136D8×C53—21no—
137M16×C53—41no—
138C2×C8×C53—21no—
139C2×C8×C53—21no—
140C2×C8×C53C2×D411yescharacteristic
141C2×C8×C53C2×D411yescharacteristic
142C2×C8×C53C2×D411yescharacteristic
143C2×C8×C53C2411yescharacteristic
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—41no—
147C4⋊C4×C53—41no—
148C4⋊C4×C53—41no—
149C4⋊C4×C53D811yescharacteristic
150C4⋊C4×C53D811yescharacteristic
151C22⋊C4×C53—41no—
152C22⋊C4×C53—41no—
153C22⋊C4×C53—41no—
154C22⋊C4×C53—41no—
155C22⋊C4×C53—41no—
156C22⋊C4×C53D811yescharacteristic
157C22⋊C4×C53D811yescharacteristic
158C42×C53—41no—
159C42×C53C2×D411yescharacteristic
160C42×C53C2×D411yescharacteristic
161C22⋊Dic53—41no—
162#11719921—41no—
163#11719921—41no—
164#11719921—41no—
165#11719921—41no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4×Dic53—41no—
169D424—24no—
170D424—21no—
171D424—21no—
172D424#4112yes—
173D424#4112yes—
174#11719908—21no—
175#11719908—21no—
176#11719908—21no—
177#11719908—21no—
178C23×C53—41no—
179C23×C53C2×D811yescharacteristic
180C22×D53—41no—
181C22×D53—41no—
182C22×D53—44no—
183C22×D53—161no—
184C22×D53—161no—
185Q8×C53—41no—
186Q8×C53—41no—
187Q8×C53—161no—
188D4×C53—41no—
189D4×C53—41no—
190D4×C53—161no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—161no—
197C2×C4×C53C2×D811yescharacteristic
198C2×C4×C53C2×D811yescharacteristic
199C2×C4×C53C2×D811yescharacteristic
200C2×C4×C53C2×D811yescharacteristic
201C2×C4×C53C2×D811yescharacteristic
202C2×C4×C53C22×D411yescharacteristic
203C2×Dic53—41no—
204C2×Dic53—41no—
205D212—22no—
206D212—22no—
207D212—44no—
208D212—41no—
209D212—41no—
210C4×D53—161no—
211C4×D53—161no—
212C4×D53—161no—
213C4×D53—161no—
214C8×C53—21no—
215C8×C53—21no—
216C8×C53—21no—
217C8×C53C2×#4111yescharacteristic
218C8×C53C2×#4111yescharacteristic
219#33678—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—81no—
224C22×C53—81no—
225C22×C53—81no—
226C22×C53C22×D811yescharacteristic
227C2×D53—42no—
228C2×D53—42no—
229C2×D53—81no—
230C2×D53—81no—
231C2×D53—81no—
232C2×D53—81no—
233C2×D53—164no—
234C4×C53—21no—
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—21no—
239C4×C53—81no—
240C4×C53—81no—
241C4×C53—81no—
242C4×C53#52511yescharacteristic
243C4×C53#57511yescharacteristic
244Dic53—81no—
245Dic53—81no—
246#3175D5311yescharacteristic, complemented
247#2948—531no—
248#2237—531no—
249#2234—531no—
250#2226—531no—
251#2224—531no—
252#2210—531no—
253#2210—531no—
254#2187—531no—
255#2184—531no—
256#2177—531no—
257#2175—531no—
258C4×SD32—531no—
259C4×D16—531no—
260#2164—531no—
261C2×C53—41no—
262C2×C53—161no—
263C2×C53#310011yescharacteristic
264C2×C53#341511yescharacteristic
265C2×C53#341611yescharacteristic
266D53—82no—
267D53—82no—
268C22×D8—531no—
269#542—21no—
270#518—1061no—
271C2×SD32—531no—
272C2×SD32—531no—
273C2×D16—531no—
274C2×D16—531no—
275#500C2×D5311yescharacteristic, complemented
276#495C2×D5311yescharacteristic, complemented
277#489—21no—
278#468—21no—
279#467C2×D5311yescharacteristic
280#466C2×D5311yescharacteristic
281#442—1061no—
282#442—1061no—
283#440—21no—
284C4×Q16C2×D5311yescharacteristic, complemented
285C4×D8—531no—
286C4×D8—531no—
287C4×D8—531no—
288C4×D8—531no—
289C4×D8C2×D5311yescharacteristic, complemented
290#427—531no—
291#418—1061no—
292#405C2×D5311yescharacteristic
293#367—531no—
294#367—531no—
295#366—531no—
296#366—531no—
297C4⋊C16—531no—
298C4⋊C16—531no—
299#358—531no—
300#358—531no—
301#357—531no—
302#357—531no—
303#357—531no—
304#357—531no—
305#357—531no—
306#357—531no—
307C22⋊C16—531no—
308C22⋊C16—531no—
309#346—531no—
310C4×C16—531no—
311C53#3367811yescharacteristic, complemented, Sylow
312C22×D4—1061no—
313C2×Q16C22×D5311yescharacteristic
314C2×SD16—21no—
315C2×D8—531no—
316C2×D8—531no—
317C2×D8—1062no—
318C2×D8—1062no—
319C2×D8—1061no—
320C2×D8C22×D5311yescharacteristic
321C2×M16C22⋊D5311yescharacteristic
322C22×C8C22×D5311yescharacteristic
323C4⋊Q8—21no—
324#126—41no—
325#124—21no—
326#124—41no—
327#123—41no—
328C22⋊Q8—21no—
329C22⋊Q8—41no—
330#121—21no—
331C4×Q8—21no—
332C4×D4—21no—
333C4×D4—1061no—
334C4×D4—1061no—
335C4×D4—1061no—
336C4×D4—1061no—
337C4×D4—2121no—
338C4×D4—2121no—
339#117—1061no—
340#117C22⋊D5311yescharacteristic
341C2×C22⋊C4—2121no—
342C2×C22⋊C4—2121no—
343SD32—1062no—
344SD32—1062no—
345D16—1062no—
346D16—1062no—
347C2×C16—531no—
348C2×C16—531no—
349C2×C16—531no—
350C2×C16—531no—
351#107—21no—
352#107—531no—
353#107—531no—
354#107C22×D5311yescharacteristic
355#107C22×D5311yescharacteristic
356#106—1061no—
357#103—21no—
358#103—21no—
359#102—21no—
360#102—21no—
361#102—1061no—
362#102—1061no—
363#102—1061no—
364#102—1061no—
365#97C22⋊D5311yescharacteristic
366#97C22⋊D5311yescharacteristic
367C4×C8C22×D5311yescharacteristic
368C4×C8C22×D5311yescharacteristic
369C24—2121no—
370C2×Q8—21no—
371C2×Q8—41no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—1064no—
375C2×D4—1061no—
376C2×D4—1061no—
377C2×D4—2122no—
378C2×D4—2121no—
379C22×C4—41no—
380C22×C4—2121no—
381C22×C4—2121no—
382C22×C4—2121no—
383C22×C4—2121no—
384C22×C4C2×C22⋊D5311yescharacteristic
385Q16—21no—
386Q16—21no—
387SD16—42no—
388D8—21no—
389D8—21no—
390D8—532no—
391D8—532no—
392D8—1064no—
393D8—1061no—
394D8—1061no—
395M16—41no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8C2×C22⋊D5311yescharacteristic
399C2×C8C2×C22⋊D5311yescharacteristic
400C2×C8C2×C22⋊D5311yescharacteristic
401C2×C8C23×D5311yescharacteristic
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—1061no—
408C4⋊C4—1061no—
409C4⋊C4#1171992211yescharacteristic
410C4⋊C4#1171992211yescharacteristic
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—2121no—
417C22⋊C4—2121no—
418C22⋊C4—2121no—
419C22⋊C4—2121no—
420C22⋊C4—2121no—
421C22⋊C4#1171992211yescharacteristic
422C22⋊C4#1171992211yescharacteristic
423C42—41no—
424C42—2121no—
425C42C2×C22⋊D5311yescharacteristic
426C42C2×C22⋊D5311yescharacteristic
427C16—1061no—
428C16—1061no—
429C16—1061no—
430C16—1061no—
431C23—41no—
432C23—8481no—
433C23—8481no—
434C23—2124no—
435C23—2121no—
436C23—2121no—
437C23C2×#1171992211yescharacteristic
438Q8—41no—
439Q8—41no—
440Q8—161no—
441D4—41no—
442D4—41no—
443D4—161no—
444D4—1062no—
445D4—1062no—
446D4—2124no—
447D4—2121no—
448D4—2121no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—161no—
455C2×C4—8481no—
456C2×C4—8481no—
457C2×C4—8481no—
458C2×C4—8481no—
459C2×C4—2121no—
460C2×C4—2121no—
461C2×C4C2×#1171992211yescharacteristic
462C2×C4C2×#1171992211yescharacteristic
463C2×C4C2×#1171992211yescharacteristic
464C2×C4C2×#1171992211yescharacteristic
465C2×C4C2×#1171992211yescharacteristic
466C2×C4C22×C22⋊D5311yescharacteristic
467C8—21no—
468C8—21no—
469C8—21no—
470C8#41×D5311yescharacteristic
471C8#41×D5311yescharacteristic
472C22—21no—
473C22—21no—
474C22—21no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—8484no—
479C22—4241no—
480C22—4241no—
481C22—4241no—
482C22—4241no—
483C22—2122no—
484C22—2122no—
485C22C22×#1171992211yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—4241no—
495C4—4241no—
496C4#1681113011yescharacteristic
497C4#1681142111yescharacteristic
498C2—41no—
499C2—161no—
500C2—4242no—
501C2—4242no—
502C2#2028471111yescharacteristic
503C2#2028785411yescharacteristic
504C2#2028785911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3175C2
socleC22×C53C22×D8
prime radical#20276626C2
hypercenter#3175D53
hypocenterC53#33678
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3175

Complements

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

NH ≅ G/Ntype
1. #20283568C2semidirect
2. #20276920C2semidirect
3. #20276903C2semidirect
4. #20276859C2semidirect
5. #20276849C2semidirect
6. #20276767C2semidirect
7. #20276767C2semidirect
8. #20276643C2semidirect
10. #20276589C2semidirect
11. #20276577C2semidirect
12. C4×#13133160C2semidirect
13. C4×#13133158C2semidirect
14. #20276522C2semidirect
15. C53×#3175C2semidirect
30. #16810450C22semidirect
31. #16810450C22semidirect
32. #16810448C22semidirect
33. #16810448C22semidirect
34. C4⋊#11719908C22semidirect
35. C4⋊#11719908C22semidirect
36. #16810427C22semidirect
37. #16810427C22semidirect
38. #16810424C22semidirect
39. #16810424C22semidirect
46. #16810399C22semidirect
47. C4×#11719908C22semidirect
49. C53×#500C22semidirect
50. C53×#495C22semidirect
56. C4×Q16×C53C22semidirect
57. C4×D8×C53C22semidirect
246. #3175D53semidirect
275. #500C2×D53semidirect
276. #495C2×D53semidirect
284. C4×Q16C2×D53semidirect
289. C4×D8C2×D53semidirect
311. C53#33678semidirect

Maximal subgroups

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

subgroupindex
#202835682
#202769202
#202769032
#202768592
#202768492
#202767672
#202766432
#202766262
#202765892
#202765772
C4×#131331602
C4×#131331582
#202765222
C53×#31752
#3367853