Finite Group Databasebeta

← #27539201

Subgroups — #27539201

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#20285355C211yesmaximal, characteristic, complemented
2#20283567C211yesmaximal, characteristic, complemented
3#20276920C211yesmaximal, characteristic, complemented
4#20276907C211yesmaximal, characteristic, complemented
5#20276860C211yesmaximal, characteristic, complemented
6#20276850C211yesmaximal, characteristic, complemented
7#20276772C211yesmaximal, characteristic, complemented
8#20276772C211yesmaximal, characteristic, complemented
9#20276627C211yesmaximal, characteristic
10#20276588C211yesmaximal, characteristic, complemented
11#20276580C211yesmaximal, characteristic, complemented
12SD32×Dic53C211yesmaximal, characteristic, complemented
13#20276559C211yesmaximal, characteristic, complemented
14#20276519C211yesmaximal, characteristic, complemented
15C53×#2187C211yesmaximal, characteristic, complemented
16#16811246—21no—
17#16811098—21no—
18C2×#13133158C2211yescharacteristic
19C2×#13133158C2211yescharacteristic
20#16811027C2211yescharacteristic, complemented
21#16811008C2211yescharacteristic, complemented
22#16810984—21no—
23#16810871—21no—
24#16810859C2211yescharacteristic
25#16810735—21no—
26#16810735—21no—
27#16810726—21no—
28Q16×Dic53C2211yescharacteristic, complemented
29D8×Dic53C2211yescharacteristic
30D8×Dic53C2211yescharacteristic
31D8×Dic53C2211yescharacteristic
32D8×Dic53C2211yescharacteristic
33#16810705C2211yescharacteristic, complemented
34#16810661C2211yescharacteristic
35#16810628—21no—
36#16810578C2211yescharacteristic
37#16810451C2211yescharacteristic, complemented
38#16810450C2211yescharacteristic, complemented
39#16810449C2211yescharacteristic, complemented
40#16810448C2211yescharacteristic, complemented
41#16810441C2211yescharacteristic, complemented
42#16810441C2211yescharacteristic, complemented
43#16810427C2211yescharacteristic, complemented
44#16810427C2211yescharacteristic, complemented
45#16810425C2211yescharacteristic, complemented
46#16810425C2211yescharacteristic, complemented
47#16810424C2211yescharacteristic
48#16810424C2211yescharacteristic
49C22⋊#11719908C2211yescharacteristic
50#16810398C2211yescharacteristic, complemented
51C16×Dic53C2211yescharacteristic, complemented
52C22×D8×C53C2211yescharacteristic
53C2×SD32×C53C2211yescharacteristic
54C2×SD32×C53C2211yescharacteristic
55C53×#467C2211yescharacteristic
56C53×#357C2211yescharacteristic
57C53×#357C2211yescharacteristic
58C53×C22⋊C16C2211yescharacteristic
59C2×#11719924—21no—
60C2×D424C2311yescharacteristic
61C2×#11719917D411yescharacteristic
62#13133216—21no—
63#13133212—41no—
64#13133206—21no—
65#13133203—41no—
66#13133199—41no—
67#13133197—41no—
68#13133191—21no—
69Q8×Dic53—21no—
70D4×Dic53—21no—
71D4×Dic53—21no—
72D4×Dic53—21no—
73D4×Dic53—21no—
74D4×Dic53—41no—
75D4×Dic53—41no—
76#13133179—21no—
77#13133176—21no—
78#13133176D411yescharacteristic
79C2×C22⋊Dic53—41no—
80C2×C22⋊Dic53—41no—
81#13133158—22no—
82#13133158—22no—
83C2×#11719908C2311yescharacteristic
84C2×#11719908C2311yescharacteristic
85#13133151C2311yescharacteristic
86#13133151C2311yescharacteristic
87#13133151C2311yescharacteristic
88#13133150—21no—
89#13133149—21no—
90#13133141—21no—
91#13133139—21no—
92#13133138—21no—
93#13133138—21no—
94#13133138—21no—
95#13133138—21no—
96#13133137—21no—
97#13133127D411yescharacteristic
98#13133127D411yescharacteristic
99C8×Dic53C2311yescharacteristic
100C8×Dic53C2311yescharacteristic
101C22×D4×C53—21no—
102C2×Q16×C53C2311yescharacteristic
103C2×D8×C53—22no—
104C2×D8×C53—22no—
105C2×D8×C53—21no—
106C2×D8×C53C2311yescharacteristic
107C2×D8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22⋊Q8×C53—21no—
110SD32×C53—22no—
111SD32×C53—22no—
112C2×C16×C53C2311yescharacteristic
113C2×C16×C53C2311yescharacteristic
114#107×C53C2311yescharacteristic
115#103×C53—21no—
116C24×C53—41no—
117C2×Q8×C53—21no—
118C2×D4×C53—24no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—42no—
122C2×D4×C53—41no—
123C22×C4×C53C2×D411yescharacteristic
124C2×C22⋊D53—41no—
125C22×Dic53—41no—
126C22×Dic53—41no—
127C22×Dic53—41no—
128C22×Dic53—41no—
129C2×D212—21no—
130C2×C4×D53—41no—
131C2×Dic106—41no—
132Q16×C53—21no—
133Q16×C53—21no—
134D8×C53—24no—
135D8×C53—21no—
136D8×C53—21no—
137D8×C53#4112yes—
138D8×C53#4112yes—
139C2×C8×C53—21no—
140C2×C8×C53—21no—
141C2×C8×C53C2×D411yescharacteristic
142C2×C8×C53C2411yescharacteristic
143C4⋊C4×C53—21no—
144C4⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C22⋊Dic53—41no—
147C22⋊Dic53—41no—
148C22⋊Dic53—41no—
149C22⋊Dic53—41no—
150C22⋊Dic53—41no—
151C22⋊Dic53D811yescharacteristic
152C22⋊Dic53D811yescharacteristic
153#11719924—42no—
154#11719921—41no—
155#11719921—41no—
156#11719921—41no—
157#11719921—41no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—41no—
162C4⋊Dic53D811yescharacteristic
163C4⋊Dic53D811yescharacteristic
164#11719919—41no—
165C4×Dic53—41no—
166C4×Dic53—41no—
167C4×Dic53C2×D411yescharacteristic
168C4×Dic53C2×D411yescharacteristic
169#11719917—41no—
170C2×#88322C2×D411yescharacteristic
171C2×#88322C2×D411yescharacteristic
172D424—21no—
173D424—21no—
174C16×C53—21no—
175C16×C53—21no—
176#11719908—21no—
177#11719908—21no—
178C23×C53—44no—
179C23×C53—41no—
180C23×C53—41no—
181C23×C53—161no—
182C23×C53—161no—
183C23×C53C2×D811yescharacteristic
184C22×D53—41no—
185Q8×C53—41no—
186Q8×C53—41no—
187D4×C53—22no—
188D4×C53—22no—
189D4×C53—44no—
190D4×C53—41no—
191D4×C53—41no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53C2×D811yescharacteristic
197C2×C4×C53C22×D411yescharacteristic
198C22⋊D53—161no—
199C2×Dic53—41no—
200C2×Dic53—41no—
201C2×Dic53—41no—
202C2×Dic53—161no—
203C2×Dic53—161no—
204C2×Dic53—161no—
205C2×Dic53—161no—
206C2×Dic53C2×D811yescharacteristic
207C2×Dic53C2×D811yescharacteristic
208C2×Dic53C2×D811yescharacteristic
209C2×Dic53C2×D811yescharacteristic
210D212—41no—
211D212—41no—
212C4×D53—161no—
213Dic106—161no—
214C8×C53—21no—
215C8×C53C2×#4111yescharacteristic
216C8×C53C2×#4111yescharacteristic
217#88322—21no—
218#88322—21no—
219#33678—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—42no—
224C22×C53—42no—
225C22×C53—81no—
226C22×C53—81no—
227C22×C53—81no—
228C22×C53—81no—
229C22×C53—164no—
230C22×C53C22×D811yescharacteristic
231C2×D53—81no—
232C2×D53—81no—
233C2×D53—81no—
234C4×C53—21no—
235C4×C53—81no—
236C4×C53—81no—
237C4×C53#52511yescharacteristic
238C4×C53#57511yescharacteristic
239Dic53—21no—
240Dic53—21no—
241Dic53—21no—
242Dic53—21no—
243Dic53—81no—
244Dic53—81no—
245Dic53—81no—
246#3175—531no—
247#2948—531no—
248#2237—531no—
249#2234—531no—
250#2226—531no—
251#2224—531no—
252#2210—531no—
253#2210—531no—
254#2187D5311yescharacteristic, complemented
255#2184—531no—
256#2177—531no—
257#2175—531no—
258C4×SD32—531no—
259C4×D16—531no—
260#2164—531no—
261C2×C53—41no—
262C2×C53—82no—
263C2×C53—82no—
264C2×C53#310011yescharacteristic
265C2×C53#341511yescharacteristic
266C2×C53#341611yescharacteristic
267D53—161no—
268C22×D8C2×D5311yescharacteristic
269#542—1061no—
270#518—1061no—
271C2×SD32C2×D5311yescharacteristic
272C2×SD32C2×D5311yescharacteristic
273C2×D16—531no—
274C2×D16—531no—
275#500—531no—
276#495—531no—
277#489—1061no—
278#468—1061no—
279#467C2×D5311yescharacteristic
280#466—531no—
281#442—1061no—
282#442—1061no—
283#440—1061no—
284C4×Q16—531no—
285C4×D8—531no—
286C4×D8—531no—
287C4×D8—531no—
288C4×D8—531no—
289C4×D8—531no—
290#427—531no—
291#418—1061no—
292#405—531no—
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#357C2×D5311yescharacteristic
306#357C2×D5311yescharacteristic
307C22⋊C16—531no—
308C22⋊C16C2×D5311yescharacteristic
309#346—531no—
310C4×C16—531no—
311C53#3367811yescharacteristic, complemented, Sylow
312C22×D4—21no—
313C2×Q16C22×D5311yescharacteristic
314C2×SD16—1061no—
315C2×D8—22no—
316C2×D8—22no—
317C2×D8—21no—
318C2×D8—531no—
319C2×D8C22×D5311yescharacteristic
320C2×D8C22×D5311yescharacteristic
321C2×M16—531no—
322C22×C8C22×D5311yescharacteristic
323C4⋊Q8—1061no—
324#126—2121no—
325#124—1061no—
326#124—2121no—
327#123—2121no—
328C22⋊Q8—21no—
329C22⋊Q8—2121no—
330#121—1061no—
331C4×Q8—1061no—
332C4×D4—1061no—
333C4×D4—1061no—
334C4×D4—1061no—
335C4×D4—1061no—
336C4×D4—1061no—
337C4×D4—2121no—
338C4×D4—2121no—
339#117—531no—
340#117—1061no—
341C2×C22⋊C4—2121no—
342C2×C22⋊C4—2121no—
343SD32—22no—
344SD32—22no—
345D16—1062no—
346D16—1062no—
347C2×C16—531no—
348C2×C16—531no—
349C2×C16C22×D5311yescharacteristic
350C2×C16C22×D5311yescharacteristic
351#107—531no—
352#107—531no—
353#107—531no—
354#107—1061no—
355#107C22×D5311yescharacteristic
356#106—1061no—
357#103—21no—
358#103—1061no—
359#102—1061no—
360#102—1061no—
361#102—1061no—
362#102—1061no—
363#102—1061no—
364#102—1061no—
365#97—531no—
366#97—531no—
367C4×C8—531no—
368C4×C8—531no—
369C24—41no—
370C2×Q8—21no—
371C2×Q8—2121no—
372C2×D4—24no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—42no—
376C2×D4—41no—
377C2×D4—1061no—
378C2×D4—2121no—
379C22×C4—2121no—
380C22×C4—2121no—
381C22×C4—2121no—
382C22×C4—2121no—
383C22×C4—2121no—
384C22×C4D4×D5311yescharacteristic
385Q16—21no—
386Q16—21no—
387SD16—2122no—
388D8—24no—
389D8—21no—
390D8—21no—
391D8—1061no—
392D8—1061no—
393D8#1171994712yes—
394D8#1171994712yes—
395M16—2121no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—531no—
399C2×C8—531no—
400C2×C8D4×D5311yescharacteristic
401C2×C8C23×D5311yescharacteristic
402C4⋊C4—21no—
403C4⋊C4—41no—
404C4⋊C4—531no—
405C4⋊C4—531no—
406C4⋊C4—1061no—
407C4⋊C4—1061no—
408C4⋊C4—1061no—
409C4⋊C4—2121no—
410C4⋊C4—2121no—
411C22⋊C4—41no—
412C22⋊C4—531no—
413C22⋊C4—531no—
414C22⋊C4—2121no—
415C22⋊C4—2121no—
416C22⋊C4—2121no—
417C22⋊C4—2121no—
418C22⋊C4—2121no—
419C22⋊C4—2121no—
420C22⋊C4—2121no—
421C22⋊C4—2121no—
422C22⋊C4—2121no—
423C42—531no—
424C42—531no—
425C42—2121no—
426C42—2121no—
427C16—21no—
428C16—21no—
429C16—1061no—
430C16—1061no—
431C23—44no—
432C23—41no—
433C23—41no—
434C23—161no—
435C23—161no—
436C23—2121no—
437C23D8×D5311yescharacteristic
438Q8—41no—
439Q8—41no—
440Q8—8481no—
441D4—22no—
442D4—22no—
443D4—44no—
444D4—41no—
445D4—41no—
446D4—8481no—
447D4—2121no—
448D4—2121no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—531no—
454C2×C4—531no—
455C2×C4—531no—
456C2×C4—531no—
457C2×C4—8481no—
458C2×C4—8481no—
459C2×C4—8481no—
460C2×C4—8481no—
461C2×C4—8481no—
462C2×C4—2121no—
463C2×C4—2121no—
464C2×C4—2121no—
465C2×C4D8×D5311yescharacteristic
466C2×C4C2×D4×D5311yescharacteristic
467C8—21no—
468C8—1061no—
469C8—1061no—
470C8C2×#1171994711yescharacteristic
471C8C2×#1171994711yescharacteristic
472C22—21no—
473C22—21no—
474C22—21no—
475C22—42no—
476C22—42no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—164no—
482C22—4241no—
483C22—4241no—
484C22—4241no—
485C22C2×D8×D5311yescharacteristic
486C4—21no—
487C4—81no—
488C4—81no—
489C4—1061no—
490C4—1061no—
491C4—1061no—
492C4—1061no—
493C4—4241no—
494C4—4241no—
495C4—4241no—
496C4#1681112911yescharacteristic
497C4#1681141911yescharacteristic
498C2—41no—
499C2—82no—
500C2—82no—
501C2—8481no—
502C2#2028471011yescharacteristic
503C2#2028785511yescharacteristic
504C2#2028785711yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2187C2
socleC22×C53C22×D8
prime radical#20276627C2
hypercenter#2187D53
hypocenterC53#33678
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
1. #20285355C2semidirect
2. #20283567C2semidirect
3. #20276920C2semidirect
4. #20276907C2semidirect
5. #20276860C2semidirect
6. #20276850C2semidirect
7. #20276772C2semidirect
8. #20276772C2semidirect
10. #20276588C2semidirect
11. #20276580C2semidirect
12. SD32×Dic53C2semidirect
13. #20276559C2semidirect
14. #20276519C2semidirect
15. C53×#2187C2semidirect
20. #16811027C22semidirect
21. #16811008C22semidirect
28. Q16×Dic53C22semidirect
33. #16810705C22semidirect
37. #16810451C22semidirect
38. #16810450C22semidirect
39. #16810449C22semidirect
40. #16810448C22semidirect
41. #16810441C22semidirect
42. #16810441C22semidirect
43. #16810427C22semidirect
44. #16810427C22semidirect
45. #16810425C22semidirect
46. #16810425C22semidirect
50. #16810398C22semidirect
51. C16×Dic53C22semidirect
254. #2187D53semidirect
311. C53#33678semidirect

Maximal subgroups

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

subgroupindex
#202853552
#202835672
#202769202
#202769072
#202768602
#202768502
#202767722
#202766272
#202765882
#202765802
SD32×Dic532
#202765592
#202765192
C53×#21872
#3367853