Finite Group Databasebeta

← #27539200

Subgroups — #27539200

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#20285354C211yesmaximal, characteristic, complemented
2#20276923C211yesmaximal, characteristic, complemented
3#20276906C211yesmaximal, characteristic, complemented
4#20276862C211yesmaximal, characteristic, complemented
5#20276852C211yesmaximal, characteristic, complemented
6#20276770C211yesmaximal, characteristic, complemented
7#20276770C211yesmaximal, characteristic, complemented
8#20276644C211yesmaximal, characteristic, complemented
9#20276627C211yesmaximal, characteristic
10#20276589C211yesmaximal, characteristic, complemented
11#20276579C211yesmaximal, characteristic, complemented
12C4×#13133161C211yesmaximal, characteristic, complemented
13C4×#13133158C211yesmaximal, characteristic, complemented
14#20276522C211yesmaximal, characteristic, complemented
15C53×#2948C211yesmaximal, characteristic, complemented
16#16811244—21no—
17C2×#13133161C2211yescharacteristic
18C2×#13133161C2211yescharacteristic
19C2×#13133158C2211yescharacteristic
20C2×#13133158C2211yescharacteristic
21#16811028C2211yescharacteristic, complemented
22#16811007C2211yescharacteristic, complemented
23#16810983—21no—
24#16810869—21no—
25#16810864C2211yescharacteristic
26#16810859C2211yescharacteristic
27#16810725—21no—
28C4×Dic212C2211yescharacteristic, complemented
29C4×D424C2211yescharacteristic, complemented
30#16810450C2211yescharacteristic, complemented
31#16810450C2211yescharacteristic, complemented
32#16810448C2211yescharacteristic, complemented
33#16810448C2211yescharacteristic, complemented
34C4⋊#11719908C2211yescharacteristic, complemented
35C4⋊#11719908C2211yescharacteristic, complemented
36#16810426C2211yescharacteristic, complemented
37#16810426C2211yescharacteristic, complemented
38#16810424C2211yescharacteristic
39#16810424C2211yescharacteristic
40#16810424C2211yescharacteristic
41#16810424C2211yescharacteristic
42#16810423C2211yescharacteristic, complemented
43#16810423C2211yescharacteristic, complemented
44C22⋊#11719908C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46#16810399C2211yescharacteristic, complemented
47C4×#11719908C2211yescharacteristic, complemented
48C22×D8×C53C2211yescharacteristic
49C53×#518—21no—
50C53×#442—21no—
51C53×#442—21no—
52C4×D8×C53C2211yescharacteristic
53C4×D8×C53C2211yescharacteristic
54C4×D8×C53C2211yescharacteristic
55C4×D8×C53C2211yescharacteristic
56C53×#427C2211yescharacteristic
57C53×#418—21no—
58C53×#405C2211yescharacteristic
59C2×Dic212C2311yescharacteristic
60C2×#11719913—21no—
61C2×D424C2311yescharacteristic
62#13133217—21no—
63#13133212—41no—
64#13133205—21no—
65#13133203—41no—
66#13133199—41no—
67#13133196—21no—
68#13133195—41no—
69#13133191—21no—
70C4×Dic106—21no—
71C4×D212—21no—
72#13133161—22no—
73#13133161—22no—
74#13133158—22no—
75#13133158—22no—
76C2×#11719908C2311yescharacteristic
77C2×#11719908C2311yescharacteristic
78C2×#11719908C2311yescharacteristic
79C2×#11719908C2311yescharacteristic
80#13133151—21no—
81#13133151C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133142—21no—
84#13133142—21no—
85#13133139—21no—
86#13133139—21no—
87C22×D4×C53—21no—
88C2×D8×C53—22no—
89C2×D8×C53—22no—
90C2×D8×C53—21no—
91C2×D8×C53C2311yescharacteristic
92C2×D8×C53C2311yescharacteristic
93C2×M16×C53D411yescharacteristic
94C22×C8×C53C2311yescharacteristic
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53—41no—
100C4×D4×C53—41no—
101#117×C53—21no—
102#117×C53D411yescharacteristic
103C2×C22⋊C4×C53—41no—
104C2×C22⋊C4×C53—41no—
105#107×C53C2311yescharacteristic
106#107×C53C2311yescharacteristic
107#106×C53—21no—
108#102×C53—21no—
109#102×C53—21no—
110#102×C53—21no—
111#102×C53—21no—
112#97×C53D411yescharacteristic
113#97×C53D411yescharacteristic
114C4×C8×C53C2311yescharacteristic
115C4×C8×C53C2311yescharacteristic
116C24×C53—41no—
117C2×D4×C53—24no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—42no—
121C2×D4×C53—41no—
122C22×C4×C53—41no—
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53—41no—
126C22×C4×C53C2×D411yescharacteristic
127C2×C22⋊D53—41no—
128C2×D212—21no—
129C2×C4×D53—41no—
130C2×Dic106—21no—
131C2×Dic106—41no—
132D8×C53—24no—
133D8×C53—21no—
134D8×C53—21no—
135D8×C53#4112yes—
136D8×C53#4112yes—
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×C53D811yescharacteristic
147C4⋊C4×C53D811yescharacteristic
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C22⋊C4×C53—41no—
151C22⋊C4×C53—41no—
152C22⋊C4×C53—41no—
153C22⋊C4×C53D811yescharacteristic
154C22⋊C4×C53D811yescharacteristic
155C42×C53—41no—
156C42×C53C2×D411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C22⋊Dic53—41no—
159#11719921—41no—
160#11719921—41no—
161#11719921—41no—
162#11719921—41no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—41no—
166#11719919—41no—
167#11719919—41no—
168C4×Dic53—41no—
169Dic212—21no—
170Dic212—21no—
171D424—21no—
172D424—21no—
173#11719913—42no—
174#11719908—21no—
175#11719908—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—
185D4×C53—22no—
186D4×C53—22no—
187D4×C53—44no—
188D4×C53—41no—
189D4×C53—41no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—161no—
195C2×C4×C53—161no—
196C2×C4×C53—161no—
197C2×C4×C53—161no—
198C2×C4×C53C2×D811yescharacteristic
199C2×C4×C53C2×D811yescharacteristic
200C2×C4×C53C2×D811yescharacteristic
201C2×C4×C53C2×D811yescharacteristic
202C2×C4×C53C2×D811yescharacteristic
203C2×C4×C53C22×D411yescharacteristic
204C22⋊D53—161no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208D212—41no—
209D212—41no—
210C4×D53—161no—
211Dic106—41no—
212Dic106—41no—
213Dic106—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—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—21no—
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—21no—
239C4×C53—81no—
240C4×C53—81no—
241C4×C53#52511yescharacteristic
242C4×C53#57511yescharacteristic
243Dic53—81no—
244Dic53—81no—
245Dic53—81no—
246#3175—531no—
247#2948D5311yescharacteristic, complemented
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—82no—
263C2×C53—82no—
264C2×C53#310011yescharacteristic
265C2×C53#341511yescharacteristic
266C2×C53#341611yescharacteristic
267D53—161no—
268C22×D8C2×D5311yescharacteristic
269#542—1061no—
270#518—21no—
271C2×SD32—531no—
272C2×SD32—531no—
273C2×D16—531no—
274C2×D16—531no—
275#500—531no—
276#495—531no—
277#489—1061no—
278#468—1061no—
279#467—531no—
280#466—531no—
281#442—21no—
282#442—21no—
283#440—1061no—
284C4×Q16—531no—
285C4×D8—531no—
286C4×D8C2×D5311yescharacteristic
287C4×D8C2×D5311yescharacteristic
288C4×D8C2×D5311yescharacteristic
289C4×D8C2×D5311yescharacteristic
290#427C2×D5311yescharacteristic
291#418—21no—
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—21no—
313C2×Q16—531no—
314C2×SD16—1061no—
315C2×D8—22no—
316C2×D8—22no—
317C2×D8—21no—
318C2×D8—531no—
319C2×D8C22×D5311yescharacteristic
320C2×D8C22×D5311yescharacteristic
321C2×M16C22⋊D5311yescharacteristic
322C22×C8C22×D5311yescharacteristic
323C4⋊Q8—1061no—
324#126—2121no—
325#124—1061no—
326#124—2121no—
327#123—2121no—
328C22⋊Q8—1061no—
329C22⋊Q8—2121no—
330#121—1061no—
331C4×Q8—1061no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—41no—
337C4×D4—41no—
338C4×D4—1061no—
339#117—21no—
340#117C22⋊D5311yescharacteristic
341C2×C22⋊C4—41no—
342C2×C22⋊C4—41no—
343SD32—1062no—
344SD32—1062no—
345D16—1062no—
346D16—1062no—
347C2×C16—531no—
348C2×C16—531no—
349C2×C16—531no—
350C2×C16—531no—
351#107—531no—
352#107—531no—
353#107—1061no—
354#107C22×D5311yescharacteristic
355#107C22×D5311yescharacteristic
356#106—21no—
357#103—1061no—
358#103—1061no—
359#102—21no—
360#102—21no—
361#102—21no—
362#102—21no—
363#102—1061no—
364#102—1061no—
365#97C22⋊D5311yescharacteristic
366#97C22⋊D5311yescharacteristic
367C4×C8C22×D5311yescharacteristic
368C4×C8C22×D5311yescharacteristic
369C24—41no—
370C2×Q8—1061no—
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—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—2121no—
384C22×C4C2×C22⋊D5311yescharacteristic
385Q16—1061no—
386Q16—1061no—
387SD16—2122no—
388D8—24no—
389D8—21no—
390D8—21no—
391D8—1061no—
392D8—1061no—
393D8#1171994512yes—
394D8#1171994512yes—
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—1061no—
405C4⋊C4—1061no—
406C4⋊C4—2121no—
407C4⋊C4—2121no—
408C4⋊C4—2121no—
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—44no—
432C23—41no—
433C23—41no—
434C23—161no—
435C23—161no—
436C23—2121no—
437C23C2×#1171992211yescharacteristic
438Q8—8481no—
439Q8—2121no—
440Q8—2121no—
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—161no—
454C2×C4—161no—
455C2×C4—161no—
456C2×C4—161no—
457C2×C4—8481no—
458C2×C4—2121no—
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—
470C8C2×#1171994511yescharacteristic
471C8C2×#1171994511yescharacteristic
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—
485C22C22×#1171992211yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—81no—
492C4—81no—
493C4—4241no—
494C4—4241no—
495C4—4241no—
496C4#1681113411yescharacteristic
497C4#1681142111yescharacteristic
498C2—41no—
499C2—82no—
500C2—82no—
501C2—8481no—
502C2#2028471511yescharacteristic
503C2#2028785411yescharacteristic
504C2#2028786211yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2948C2
socleC22×C53C22×D8
prime radical#20276627C2
hypercenter#2948D53
hypocenterC53#33678
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2948

Complements

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

NH ≅ G/Ntype
1. #20285354C2semidirect
2. #20276923C2semidirect
3. #20276906C2semidirect
4. #20276862C2semidirect
5. #20276852C2semidirect
6. #20276770C2semidirect
7. #20276770C2semidirect
8. #20276644C2semidirect
10. #20276589C2semidirect
11. #20276579C2semidirect
12. C4×#13133161C2semidirect
13. C4×#13133158C2semidirect
14. #20276522C2semidirect
15. C53×#2948C2semidirect
21. #16811028C22semidirect
22. #16811007C22semidirect
28. C4×Dic212C22semidirect
29. C4×D424C22semidirect
30. #16810450C22semidirect
31. #16810450C22semidirect
32. #16810448C22semidirect
33. #16810448C22semidirect
34. C4⋊#11719908C22semidirect
35. C4⋊#11719908C22semidirect
36. #16810426C22semidirect
37. #16810426C22semidirect
42. #16810423C22semidirect
43. #16810423C22semidirect
46. #16810399C22semidirect
47. C4×#11719908C22semidirect
247. #2948D53semidirect
311. C53#33678semidirect

Maximal subgroups

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

subgroupindex
#202853542
#202769232
#202769062
#202768622
#202768522
#202767702
#202766442
#202766272
#202765892
#202765792
C4×#131331612
C4×#131331582
#202765222
C53×#29482
#3367853