Finite Group Databasebeta

← #27541864

Subgroups — #27541864

Order 13568.

480 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287607C211yesmaximal, characteristic, complemented
2#20276956C211yesmaximal, characteristic
3#20276951C211yesmaximal, characteristic, complemented
4#20276946C211yesmaximal, characteristic, complemented
5#20276879C211yesmaximal, characteristic, complemented
6#20276872C211yesmaximal, characteristic, complemented
7#20276862C211yesmaximal, characteristic, complemented
8#20276774C211yesmaximal, characteristic, complemented
9#20276774C211yesmaximal, characteristic, complemented
10#20276752C211yesmaximal, characteristic
11#20276624C211yesmaximal, characteristic, complemented
12#20276584C211yesmaximal, characteristic, complemented
13#20276579C211yesmaximal, characteristic, complemented
14C4×#13133161C211yesmaximal, characteristic, complemented
15C53×#3396C211yesmaximal, characteristic
16#16811351—21no—
17C2×#13133163C2211yescharacteristic
18C2×#13133161C2211yescharacteristic
19C2×#13133161C2211yescharacteristic
20#16811028C2211yescharacteristic
21#16811025C2211yescharacteristic
22#16811025C2211yescharacteristic
23#16810931—21no—
24#16810931—21no—
25#16810909—21no—
26C4×Dic212C2211yescharacteristic
27C4×Dic212C2211yescharacteristic
28C4×Dic212C2211yescharacteristic
29#16810450C2211yescharacteristic
30#16810450C2211yescharacteristic
31#16810448C2211yescharacteristic
32C4⋊#11719908C2211yescharacteristic
33C4⋊#11719908C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35#16810427C2211yescharacteristic
36#16810427C2211yescharacteristic
37#16810427C2211yescharacteristic
38#16810426C2211yescharacteristic
39#16810424C2211yescharacteristic
40#16810424C2211yescharacteristic
41#16810424C2211yescharacteristic
42#16810424C2211yescharacteristic
43#16810424C2211yescharacteristic
44#16810424C2211yescharacteristic
45#16810399C2211yescharacteristic
46#16810399C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C53×#559—21no—
49C53×#495C2211yescharacteristic
50C53×#495C2211yescharacteristic
51C53×#493C2211yescharacteristic
52C53×#463—21no—
53C53×#463—21no—
54C53×#460—21no—
55C8×Q8×C53C2211yescharacteristic
56C4×Q16×C53C2211yescharacteristic
57C4×D8×C53C2211yescharacteristic
58C4×D8×C53C2211yescharacteristic
59C2×Dic212C2311yescharacteristic
60#13133217—21no—
61#13133215—41no—
62#13133209—21no—
63#13133209—21no—
64#13133207—41no—
65#13133207—41no—
66C4×Dic106—21no—
67C4×Dic106—21no—
68C4×Dic106—21no—
69#13133182—41no—
70#13133163—22no—
71#13133161—22no—
72#13133161—22no—
73C2×#11719908C2311yescharacteristic
74C2×#11719908C2311yescharacteristic
75C2×#11719908C2311yescharacteristic
76C2×#11719908C2311yescharacteristic
77#13133151—21no—
78#13133151C2311yescharacteristic
79#13133151C2311yescharacteristic
80#13133151C2311yescharacteristic
81#13133149—21no—
82#13133149—21no—
83#13133142—21no—
84#13133142—21no—
85#13133142—21no—
86C2×Q16×C53C2311yescharacteristic
87C2×SD16×C53—21no—
88C2×SD16×C53—21no—
89C2×D8×C53—21no—
90C2×D8×C53C2311yescharacteristic
91C2×D8×C53C2311yescharacteristic
92#127×C53—21no—
93#124×C53—21no—
94#124×C53—21no—
95#124×C53—41no—
96#123×C53—41no—
97#123×C53—41no—
98C4×Q8×C53—21no—
99C4×Q8×C53D411yescharacteristic
100C4×D4×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53—41no—
103#107×C53C2311yescharacteristic
104C4⋊C8×C53D411yescharacteristic
105C4⋊C8×C53D411yescharacteristic
106C4⋊C8×C53D411yescharacteristic
107#103×C53—21no—
108#102×C53—21no—
109#102×C53—21no—
110C4×C8×C53C2311yescharacteristic
111C4×C8×C53C2311yescharacteristic
112C4×C8×C53C2311yescharacteristic
113C2×Q8×C53—21no—
114C2×Q8×C53D811yescharacteristic
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—41no—
118C2×D4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53—41no—
121C2×Dic106—21no—
122C2×Dic106—41no—
123Q16×C53—21no—
124Q16×C53—21no—
125SD16×C53—81no—
126SD16×C53—81no—
127D8×C53—21no—
128D8×C53—21no—
129D8×C53—21no—
130D8×C53—21no—
131D8×C53—81no—
132C2×C8×C53C2×D411yescharacteristic
133C2×C8×C53C2×D411yescharacteristic
134C2×C8×C53C2×D411yescharacteristic
135C2×C8×C53C2411yescharacteristic
136C4⋊C4×C53—21no—
137C4⋊C4×C53—41no—
138C4⋊C4×C53D811yescharacteristic
139C4⋊C4×C53D811yescharacteristic
140C4⋊C4×C53D811yescharacteristic
141C22⋊C4×C53—41no—
142C22⋊C4×C53—41no—
143C22⋊C4×C53—41no—
144C22⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C22⋊C4×C53—41no—
147C42×C53—41no—
148C42×C53C2×D411yescharacteristic
149C42×C53C2×D411yescharacteristic
150C42×C53C2×D411yescharacteristic
151C4⋊Dic53—21no—
152C4⋊Dic53—21no—
153C4⋊Dic53—21no—
154C4⋊Dic53—41no—
155#11719919—41no—
156#11719919—41no—
157#11719919—41no—
158#11719919—41no—
159#11719919—41no—
160#11719919—41no—
161#11719919—41no—
162C4×Dic53—41no—
163C4×Dic53—41no—
164C4×Dic53—41no—
165Dic212—21no—
166Dic212#4112yes—
167#11719908—21no—
168#11719908—21no—
169#11719908—21no—
170#11719908—21no—
171C23×C53—41no—
172C23×C53—41no—
173Q8×C53—41no—
174Q8×C53—41no—
175Q8×C53—41no—
176D4×C53—41no—
177D4×C53—41no—
178D4×C53—41no—
179D4×C53—41no—
180D4×C53—82no—
181D4×C53—161no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—161no—
185C2×C4×C53—161no—
186C2×C4×C53C2×D811yescharacteristic
187C2×C4×C53C2×D811yescharacteristic
188C2×C4×C53C2×D811yescharacteristic
189C2×C4×C53C2×D811yescharacteristic
190C2×C4×C53C2×D811yescharacteristic
191C2×C4×C53C2×D811yescharacteristic
192C2×C4×C53C22×D411yescharacteristic
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53—41no—
197Dic106—22no—
198Dic106—41no—
199Dic106—82no—
200C8×C53—21no—
201C8×C53—21no—
202C8×C53—21no—
203C8×C53C2×#4111yescharacteristic
204C8×C532−1+411yescharacteristic
205#33938—531nomaximal, Sylow
206C22×C53—81no—
207C22×C53—81no—
208C22×C53—81no—
209C22×C53—81no—
210C22×C53—81no—
211C22×C53—81no—
212C22×C53C22×D811yescharacteristic
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—21no—
217C4×C53—21no—
218C4×C53—21no—
219C4×C53—81no—
220C4×C53—81no—
221C4×C53#54811yescharacteristic
222C4×C53#57711yescharacteristic
223Dic53—42no—
224Dic53—81no—
225Dic53—81no—
226Dic53—81no—
227#3396D5311yescharacteristic
228#3383—531no—
229#2244—531no—
230#2243—531no—
231#2242—531no—
232#2229—531no—
233#2228—531no—
234#2226—531no—
235#2211—531no—
236#2211—531no—
237#2207—531no—
238#2183—531no—
239#2176—531no—
240#2175—531no—
241C4×SD32—531no—
242C2×C53—161no—
243C2×C53—161no—
244C2×C53#335811yescharacteristic
245C2×C53#341411yescharacteristic
246C2×C53#341611yescharacteristic
247#559—21no—
248#557—1061no—
249C2×Q32—531no—
250C2×SD32—531no—
251C2×SD32—531no—
252#500—531no—
253#499—531no—
254#499—531no—
255#495C2×D5311yescharacteristic
256#495C2×D5311yescharacteristic
257#493C2×D5311yescharacteristic
258#479—1061no—
259#479—1061no—
260#475—1061no—
261#463—21no—
262#463—21no—
263#460—21no—
264C8×Q8C2×D5311yescharacteristic
265C4×Q16—531no—
266C4×Q16—531no—
267C4×Q16—531no—
268C4×Q16C2×D5311yescharacteristic
269C4×D8C2×D5311yescharacteristic
270C4×D8C2×D5311yescharacteristic
271#367—531no—
272#367—531no—
273#366—531no—
274C4⋊C16—531no—
275C4⋊C16—531no—
276C4⋊C16—531no—
277#358—531no—
278#358—531no—
279#358—531no—
280#358—531no—
281#357—531no—
282#357—531no—
283#357—531no—
284#357—531no—
285#357—531no—
286#357—531no—
287#346—531no—
288#346—531no—
289C4×C16—531no—
290C53#3393811yescharacteristic, complemented, Sylow
291C2×Q16—531no—
292C2×Q16C22×D5311yescharacteristic
293C2×SD16—21no—
294C2×SD16—21no—
295C2×D8—21no—
296C2×D8C22×D5311yescharacteristic
297C2×D8C22×D5311yescharacteristic
298C4⋊Q8—1061no—
299C4⋊Q8—2121no—
300#127—21no—
301#125—1061no—
302#125—1061no—
303#125—2121no—
304#125—2121no—
305#124—21no—
306#124—21no—
307#124—41no—
308#123—41no—
309#123—41no—
310C4×Q8—21no—
311C4×Q8—1061no—
312C4×Q8—1061no—
313C4×Q8—1061no—
314C4×Q8—2121no—
315C4×Q8C22⋊D5311yescharacteristic
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—41no—
319Q32—1062no—
320SD32—1062no—
321SD32—1062no—
322C2×C16—531no—
323C2×C16—531no—
324C2×C16—531no—
325C2×C16—531no—
326#107—531no—
327#107—531no—
328#107—531no—
329#107—1061no—
330#107C22×D5311yescharacteristic
331#106—1061no—
332#106—1061no—
333C4⋊C8C22⋊D5311yescharacteristic
334C4⋊C8C22⋊D5311yescharacteristic
335C4⋊C8C22⋊D5311yescharacteristic
336#103—21no—
337#103—1061no—
338#103—1061no—
339#103—1061no—
340#102—21no—
341#102—21no—
342C4×C8C22×D5311yescharacteristic
343C4×C8C22×D5311yescharacteristic
344C4×C8C22×D5311yescharacteristic
345C2×Q8—21no—
346C2×Q8—1061no—
347C2×Q8—2121no—
348C2×Q8#1171992211yescharacteristic
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—41no—
352C2×D4—41no—
353C22×C4—41no—
354C22×C4—41no—
355Q16—21no—
356Q16—21no—
357Q16—532no—
358Q16—1061no—
359SD16—81no—
360SD16—81no—
361D8—21no—
362D8—21no—
363D8—21no—
364D8—21no—
365D8—81no—
366C2×C8C2×C22⋊D5311yescharacteristic
367C2×C8C2×C22⋊D5311yescharacteristic
368C2×C8C2×C22⋊D5311yescharacteristic
369C2×C8C23×D5311yescharacteristic
370C4⋊C4—21no—
371C4⋊C4—41no—
372C4⋊C4—1061no—
373C4⋊C4—1061no—
374C4⋊C4—1061no—
375C4⋊C4—2121no—
376C4⋊C4—2121no—
377C4⋊C4—2121no—
378C4⋊C4—2121no—
379C4⋊C4—2121no—
380C4⋊C4—2121no—
381C4⋊C4—2121no—
382C4⋊C4—2121no—
383C4⋊C4#1171992211yescharacteristic
384C4⋊C4#1171992211yescharacteristic
385C4⋊C4#1171992211yescharacteristic
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C42—41no—
393C42—2121no—
394C42—2121no—
395C42—2121no—
396C42C2×C22⋊D5311yescharacteristic
397C42C2×C22⋊D5311yescharacteristic
398C42C2×C22⋊D5311yescharacteristic
399C16—1061no—
400C16—1061no—
401C16—1061no—
402C16—1061no—
403C23—41no—
404C23—41no—
405Q8—41no—
406Q8—41no—
407Q8—41no—
408Q8—1062no—
409Q8—4242no—
410Q8—2121no—
411D4—41no—
412D4—41no—
413D4—41no—
414D4—41no—
415D4—82no—
416D4—161no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—161no—
420C2×C4—161no—
421C2×C4—2121no—
422C2×C4—2121no—
423C2×C4—2121no—
424C2×C4—2121no—
425C2×C4C2×#1171992211yescharacteristic
426C2×C4C2×#1171992211yescharacteristic
427C2×C4C2×#1171992211yescharacteristic
428C2×C4C2×#1171992211yescharacteristic
429C2×C4C2×#1171992211yescharacteristic
430C2×C4C2×#1171992211yescharacteristic
431C2×C4C22×C22⋊D5311yescharacteristic
432C8—21no—
433C8—21no—
434C8—21no—
435C8#41×D5311yescharacteristic
436C8#1313326411yescharacteristic
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22C22×#1171992211yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—81no—
451C4—81no—
452C4—4241no—
453C4—4241no—
454C4—4241no—
455C4—2122no—
456C4#1681128011yescharacteristic
457C4#1681143411yescharacteristic
458C2—161no—
459C2—161no—
460C2#2028734611yescharacteristic
461C2#2028784911yescharacteristic
462C2#2028786211yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3396C2
socleC22×C53C22×D8
prime radicalC53×#493C22
hypercenter#3396D53
hypocenterC53#33938
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3396

Complements

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

NH ≅ G/Ntype
1. #20287607C2semidirect
3. #20276951C2semidirect
4. #20276946C2semidirect
5. #20276879C2semidirect
6. #20276872C2semidirect
7. #20276862C2semidirect
8. #20276774C2semidirect
9. #20276774C2semidirect
11. #20276624C2semidirect
12. #20276584C2semidirect
13. #20276579C2semidirect
14. C4×#13133161C2semidirect
290. C53#33938semidirect

Maximal subgroups

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

subgroupindex
#202876072
#202769562
#202769512
#202769462
#202768792
#202768722
#202768622
#202767742
#202767522
#202766242
#202765842
#202765792
C4×#131331612
C53×#33962
#3393853