Finite Group Databasebeta

← #27541870

Subgroups — #27541870

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#20287728C211yesmaximal, characteristic
2#20276958C211yesmaximal, characteristic
3#20276954C211yesmaximal, characteristic, complemented
4#20276949C211yesmaximal, characteristic, complemented
5#20276876C211yesmaximal, characteristic, complemented
6#20276869C211yesmaximal, characteristic, complemented
7#20276859C211yesmaximal, characteristic, complemented
8#20276777C211yesmaximal, characteristic, complemented
9#20276777C211yesmaximal, characteristic, complemented
10#20276755C211yesmaximal, characteristic
11#20276624C211yesmaximal, characteristic, complemented
12#20276584C211yesmaximal, characteristic, complemented
13#20276577C211yesmaximal, characteristic, complemented
14C4×#13133160C211yesmaximal, characteristic, complemented
15C53×#3383C211yesmaximal, characteristic, complemented
16#16811358—21no—
17C2×#13133163C2211yescharacteristic
18C2×#13133160C2211yescharacteristic
19C2×#13133160C2211yescharacteristic
20#16811007C2211yescharacteristic
21#16811007C2211yescharacteristic
22#16811000C2211yescharacteristic
23#16810839—21no—
24#16810839—21no—
25#16810824—21no—
26C4×Dic212C2211yescharacteristic
27C4×D424C2211yescharacteristic
28C4×D424C2211yescharacteristic
29#16810450C2211yescharacteristic
30#16810450C2211yescharacteristic
31#16810448C2211yescharacteristic
32C4⋊#11719908C2211yescharacteristic
33C4⋊#11719908C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35#16810427C2211yescharacteristic
36#16810426C2211yescharacteristic
37#16810426C2211yescharacteristic
38#16810426C2211yescharacteristic
39#16810423C2211yescharacteristic
40#16810423C2211yescharacteristic
41#16810423C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44#16810423C2211yescharacteristic
45#16810399C2211yescharacteristic
46#16810399C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C53×#557—21no—
49C53×#500C2211yescharacteristic
50C53×#499C2211yescharacteristic
51C53×#499C2211yescharacteristic
52C53×#479—21no—
53C53×#479—21no—
54C53×#475—21no—
55C8×Q8×C53C2211yescharacteristic
56C4×Q16×C53C2211yescharacteristic
57C4×Q16×C53C2211yescharacteristic
58C4×Q16×C53C2211yescharacteristic
59C2×Dic212C2311yescharacteristic
60C2×#11719913—21no—
61C2×#11719913—21no—
62C2×D424—21no—
63C2×D424C2311yescharacteristic
64C2×D424C2311yescharacteristic
65#13133214—21no—
66#13133206—41no—
67#13133205—21no—
68#13133205—21no—
69#13133202—41no—
70#13133202—41no—
71C4×Dic106—21no—
72C4×D212—21no—
73C4×D212—21no—
74#13133179—41no—
75#13133163—22no—
76#13133160—22no—
77#13133160—22no—
78C2×#11719908C2311yescharacteristic
79C2×#11719908C2311yescharacteristic
80C2×#11719908C2311yescharacteristic
81C2×#11719908C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133142—21no—
84#13133139—21no—
85#13133139—21no—
86C2×Q16×C53C2311yescharacteristic
87C4⋊Q8×C53—21no—
88C4⋊Q8×C53—41no—
89#125×C53—21no—
90#125×C53—21no—
91#125×C53—41no—
92#125×C53—41no—
93C4×Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53—21no—
96C4×Q8×C53—41no—
97C4×Q8×C53D411yescharacteristic
98#107×C53—21no—
99#107×C53C2311yescharacteristic
100#107×C53C2311yescharacteristic
101#107×C53C2311yescharacteristic
102#106×C53—21no—
103#106×C53—21no—
104C4⋊C8×C53D411yescharacteristic
105C4⋊C8×C53D411yescharacteristic
106C4⋊C8×C53D411yescharacteristic
107#103×C53—21no—
108#103×C53—21no—
109#103×C53—21no—
110C4×C8×C53C2311yescharacteristic
111C4×C8×C53C2311yescharacteristic
112C4×C8×C53C2311yescharacteristic
113C2×Q8×C53—21no—
114C2×Q8×C53—41no—
115C2×Q8×C53D811yescharacteristic
116C2×D212—21no—
117C2×D212—21no—
118C2×D212—41no—
119C2×D212—41no—
120C2×C4×D53—41no—
121C2×C4×D53—41no—
122C2×Dic106—21no—
123Q16×C53—21no—
124Q16×C53#4112yes—
125C2×C8×C53C2×D411yescharacteristic
126C2×C8×C53C2×D411yescharacteristic
127C2×C8×C53C2×D411yescharacteristic
128C2×C8×C53C2411yescharacteristic
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—41no—
133C4⋊C4×C53—41no—
134C4⋊C4×C53—41no—
135C4⋊C4×C53—41no—
136C4⋊C4×C53—41no—
137C4⋊C4×C53—41no—
138C4⋊C4×C53—41no—
139C4⋊C4×C53—41no—
140C4⋊C4×C53D811yescharacteristic
141C4⋊C4×C53D811yescharacteristic
142C4⋊C4×C53D811yescharacteristic
143C42×C53—41no—
144C42×C53—41no—
145C42×C53—41no—
146C42×C53C2×D411yescharacteristic
147C42×C53C2×D411yescharacteristic
148C42×C53C2×D411yescharacteristic
149#11719921—41no—
150#11719921—41no—
151#11719921—41no—
152#11719921—41no—
153#11719921—41no—
154#11719921—41no—
155C4⋊Dic53—21no—
156#11719919—41no—
157C4×Dic53—41no—
158Dic212—21no—
159Dic212—21no—
160D424—21no—
161D424—21no—
162D424—21no—
163D424—21no—
164D424—81no—
165#11719913—81no—
166#11719913—81no—
167#11719908—21no—
168#11719908—21no—
169#11719908—21no—
170#11719908—21no—
171C22×D53—41no—
172C22×D53—41no—
173Q8×C53—22no—
174Q8×C53—41no—
175Q8×C53—41no—
176Q8×C53—82no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53C2×D811yescharacteristic
182C2×C4×C53C2×D811yescharacteristic
183C2×C4×C53C2×D811yescharacteristic
184C2×C4×C53C2×D811yescharacteristic
185C2×C4×C53C2×D811yescharacteristic
186C2×C4×C53C2×D811yescharacteristic
187C2×C4×C53C22×D411yescharacteristic
188C2×Dic53—41no—
189C2×Dic53—41no—
190D212—41no—
191D212—41no—
192D212—41no—
193D212—41no—
194D212—82no—
195D212—161no—
196C4×D53—161no—
197C4×D53—161no—
198Dic106—41no—
199Dic106—41no—
200C8×C53—21no—
201C8×C53—21no—
202C8×C53—21no—
203C8×C53C2×#4111yescharacteristic
204C8×C532−1+411yescharacteristic
205#33938—531nomaximal, Sylow
206C22×C53C22×D811yescharacteristic
207C2×D53—81no—
208C2×D53—81no—
209C2×D53—81no—
210C2×D53—81no—
211C2×D53—81no—
212C2×D53—81no—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—21no—
217C4×C53—21no—
218C4×C53—21no—
219C4×C53—42no—
220C4×C53—81no—
221C4×C53—81no—
222C4×C53—81no—
223C4×C53#54811yescharacteristic
224C4×C53#57711yescharacteristic
225Dic53—81no—
226Dic53—81no—
227#3396—531no—
228#3383D5311yescharacteristic, complemented
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#335811yescharacteristic
243C2×C53#341411yescharacteristic
244C2×C53#341611yescharacteristic
245D53—161no—
246D53—161no—
247#559—1061no—
248#557—21no—
249C2×Q32—531no—
250C2×SD32—531no—
251C2×SD32—531no—
252#500C2×D5311yescharacteristic
253#499C2×D5311yescharacteristic
254#499C2×D5311yescharacteristic
255#495—531no—
256#495—531no—
257#493—531no—
258#479—21no—
259#479—21no—
260#475—21no—
261#463—1061no—
262#463—1061no—
263#460—1061no—
264C8×Q8C2×D5311yescharacteristic
265C4×Q16—531no—
266C4×Q16C2×D5311yescharacteristic
267C4×Q16C2×D5311yescharacteristic
268C4×Q16C2×D5311yescharacteristic
269C4×D8—531no—
270C4×D8—531no—
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—1061no—
294C2×SD16—1061no—
295C2×D8—531no—
296C2×D8—531no—
297C2×D8—1061no—
298C4⋊Q8—21no—
299C4⋊Q8—41no—
300#127—1061no—
301#125—21no—
302#125—21no—
303#125—41no—
304#125—41no—
305#124—1061no—
306#124—1061no—
307#124—2121no—
308#123—2121no—
309#123—2121no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—41no—
314C4×Q8—1061no—
315C4×Q8C22⋊D5311yescharacteristic
316C4×D4—1061no—
317C4×D4—1061no—
318C4×D4—2121no—
319Q32—1062no—
320SD32—1062no—
321SD32—1062no—
322C2×C16—531no—
323C2×C16—531no—
324C2×C16—531no—
325C2×C16—531no—
326#107—21no—
327#107—531no—
328#107C22×D5311yescharacteristic
329#107C22×D5311yescharacteristic
330#107C22×D5311yescharacteristic
331#106—21no—
332#106—21no—
333C4⋊C8C22⋊D5311yescharacteristic
334C4⋊C8C22⋊D5311yescharacteristic
335C4⋊C8C22⋊D5311yescharacteristic
336#103—21no—
337#103—21no—
338#103—21no—
339#103—1061no—
340#102—1061no—
341#102—1061no—
342C4×C8C22×D5311yescharacteristic
343C4×C8C22×D5311yescharacteristic
344C4×C8C22×D5311yescharacteristic
345C2×Q8—21no—
346C2×Q8—41no—
347C2×Q8—1061no—
348C2×Q8#1171992211yescharacteristic
349C2×D4—1061no—
350C2×D4—1061no—
351C2×D4—2121no—
352C2×D4—2121no—
353C22×C4—2121no—
354C22×C4—2121no—
355Q16—21no—
356Q16—1061no—
357Q16—1061no—
358Q16#1171994912yes—
359SD16—4241no—
360SD16—4241no—
361D8—1061no—
362D8—1061no—
363D8—1061no—
364D8—1061no—
365D8—4241no—
366C2×C8C2×C22⋊D5311yescharacteristic
367C2×C8C2×C22⋊D5311yescharacteristic
368C2×C8C2×C22⋊D5311yescharacteristic
369C2×C8C23×D5311yescharacteristic
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—1061no—
382C4⋊C4—2121no—
383C4⋊C4#1171992211yescharacteristic
384C4⋊C4#1171992211yescharacteristic
385C4⋊C4#1171992211yescharacteristic
386C22⋊C4—2121no—
387C22⋊C4—2121no—
388C22⋊C4—2121no—
389C22⋊C4—2121no—
390C22⋊C4—2121no—
391C22⋊C4—2121no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—2121no—
396C42C2×C22⋊D5311yescharacteristic
397C42C2×C22⋊D5311yescharacteristic
398C42C2×C22⋊D5311yescharacteristic
399C16—1061no—
400C16—1061no—
401C16—1061no—
402C16—1061no—
403C23—2121no—
404C23—2121no—
405Q8—22no—
406Q8—41no—
407Q8—41no—
408Q8—82no—
409Q8—2121no—
410Q8—2121no—
411D4—8481no—
412D4—4242no—
413D4—2121no—
414D4—2121no—
415D4—2121no—
416D4—2121no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—8481no—
422C2×C4—8481no—
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—
435C8C2×#1171994911yescharacteristic
436C8#1313326511yescharacteristic
437C22—4241no—
438C22—4241no—
439C22—4241no—
440C22—4241no—
441C22—4241no—
442C22—4241no—
443C22C22×#1171992211yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—42no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—4241no—
455C4—4241no—
456C4#1681128411yescharacteristic
457C4#1681143111yescharacteristic
458C2—8481no—
459C2—8481no—
460C2#2028735011yescharacteristic
461C2#2028784911yescharacteristic
462C2#2028785911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3383C2
socleC22×C53C22×D8
prime radical#16811000C22
hypercenter#3383D53
hypocenterC53#33938
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3383

Complements

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

NH ≅ G/Ntype
3. #20276954C2semidirect
4. #20276949C2semidirect
5. #20276876C2semidirect
6. #20276869C2semidirect
7. #20276859C2semidirect
8. #20276777C2semidirect
9. #20276777C2semidirect
11. #20276624C2semidirect
12. #20276584C2semidirect
13. #20276577C2semidirect
14. C4×#13133160C2semidirect
15. C53×#3383C2semidirect
228. #3383D53semidirect
290. C53#33938semidirect

Maximal subgroups

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

subgroupindex
#202877282
#202769582
#202769542
#202769492
#202768762
#202768692
#202768592
#202767772
#202767552
#202766242
#202765842
#202765772
C4×#131331602
C53×#33832
#3393853