Finite Group Databasebeta

← #38752536

Subgroups — #38752536

Order 25216.

398 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161706C211yesmaximal, characteristic, complemented
2#27161551C211yesmaximal, characteristic
3#27161382C211yesmaximal, characteristic, complemented
4#27161380C211yesmaximal, characteristic, complemented
5#27161379C211yesmaximal, characteristic, complemented
6#27161377C211yesmaximal, characteristic, complemented
7#27161367C211yesmaximal, characteristic, complemented
8#27161366C211yesmaximal, characteristic, complemented
9#27161358C211yesmaximal, characteristic, complemented
10#27161120C211yesmaximal, characteristic, complemented
11#27161116C211yesmaximal, characteristic, complemented
12#27161104C211yesmaximal, characteristic, complemented
13#27161082C211yesmaximal, characteristic
14#102×D197C211yesmaximal, characteristic
15#474×C197C211yesmaximal, characteristic, complemented
16C2×D4×D197C2211yescharacteristic
17C2×#15691341C2211yescharacteristic
18C2×C8×D197C2211yescharacteristic
19#20263325C2211yescharacteristic, complemented
20#20263324—21no—
21#20263318C2211yescharacteristic, complemented
22#20263317C2211yescharacteristic, complemented
23#20263316—21no—
24#20263304—21no—
25#20263302—21no—
26C4×D788C2211yescharacteristic
27D4×Dic197C2211yescharacteristic
28#20263288C2211yescharacteristic
29C4×C22⋊D197—21no—
30#20263286—21no—
31#20263284C2211yescharacteristic
32#20263284C2211yescharacteristic
33#20263283C2211yescharacteristic
34C4⋊C4×D197C2211yescharacteristic
35C2×#15691335—21no—
36C22⋊C4×D197—21no—
37#20263260C2211yescharacteristic, complemented
38#20263259C2211yescharacteristic, complemented
39#20263258C2211yescharacteristic, complemented
40#20263257C2211yescharacteristic, complemented
41#20263257C2211yescharacteristic, complemented
42#20263257C2211yescharacteristic, complemented
43C4⋊#13096802C2211yescharacteristic, complemented
44#20263255C2211yescharacteristic, complemented
45#20263255C2211yescharacteristic, complemented
46#20263248C2211yescharacteristic
47#20263248C2211yescharacteristic
48#20263247C2211yescharacteristic
49#20263247C2211yescharacteristic
50#20263246C2211yescharacteristic
51#20263246C2211yescharacteristic
52C4⋊Q8×C197C2211yescharacteristic, complemented
53C4×D4×C197C2211yescharacteristic
54#107×C197C2211yescharacteristic, complemented
55#107×C197C2211yescharacteristic, complemented
56C4⋊C8×C197C2211yescharacteristic, complemented
57#102×C197C2211yescharacteristic
58#102×C197C2211yescharacteristic
59C23×D197—21no—
60C2×Dic394—21no—
61C2×D788C2311yescharacteristic
62C2×C22⋊D197—21no—
63D4×D197—22no—
64D4×D197Q814yescomplemented
65C22×Dic197—21no—
66C2×C4×D197—21no—
67C2×C4×D197—21no—
68C2×C4×D197—21no—
69C2×C4×D197C2311yescharacteristic
70#15691341—22no—
71C2×#13096802C2311yescharacteristic
72C2×#13096802C2311yescharacteristic
73C8×D197—22no—
74C4⋊Dic197—21no—
75C4⋊Dic197C2311yescharacteristic
76C4⋊Dic197C2311yescharacteristic
77C4⋊Dic197C2311yescharacteristic
78#15691337—21no—
79#15691337—21no—
80#15691337—21no—
81#15691337—21no—
82#15691337—21no—
83#15691337D411yescharacteristic
84#15691337D411yescharacteristic
85C22⋊Dic197—21no—
86#15691335—21no—
87#15691335—21no—
88#15691335—21no—
89#15691335—21no—
90#15691335D411yescharacteristic
91#15691335D411yescharacteristic
92C4×Dic197—21no—
93C4×Dic197—21no—
94C4×Dic197C2311yescharacteristic
95C2×Q8×C197—21no—
96C2×D4×C197C2311yescharacteristic
97C22×C4×C197—21no—
98C2×C8×C197C2311yescharacteristic
99C2×C8×C197C2311yescharacteristic
100C4⋊C4×C197—21no—
101C4⋊C4×C197C2311yescharacteristic
102C4⋊C4×C197C2311yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C22⋊C4×C197—21no—
105C42×C197C2311yescharacteristic
106C22×D197—21no—
107C22×D197—24no—
108C22×D197—81no—
109C22×D197—81no—
110C22×D197C2×D411yescharacteristic
111Dic394—81no—
112D788—21no—
113D788C2×Q812yes—
114C22⋊D197—24no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—21no—
118C2×Dic197—81no—
119C2×Dic197C2×D411yescharacteristic
120C2×Dic197C2×D411yescharacteristic
121C2×Dic197C2×D411yescharacteristic
122C4×D197—81no—
123C4×D197—81no—
124C4×D197—81no—
125C4×D197C2×Q812yes—
126C4×D197#4112yes—
127#13096802—21no—
128#13096802—21no—
129C23×C197—21no—
130Q8×C197—81no—
131D4×C197—21no—
132D4×C197C2×Q812yes—
133C2×C4×C197—21no—
134C2×C4×C197—21no—
135C2×C4×C197—21no—
136C2×C4×C197—81no—
137C2×C4×C197C2×D411yescharacteristic
138C2×C4×C197C2×D411yescharacteristic
139C2×C4×C197C2411yescharacteristic
140C8×C197—21no—
141C8×C197—21no—
142C22×C197—22no—
143C22×C197—41no—
144C22×C197—41no—
145C22×C197C22×D411yescharacteristic
146C2×D197—21no—
147C2×D197—21no—
148C2×D197—22no—
149C2×D197—41no—
150C2×D197—41no—
151C2×D197—84no—
152C2×D197C22⋊Q812yes—
153C4×C197—21no—
154C4×C197—21no—
155C4×C197—41no—
156C4×C197—41no—
157C4×C197—41no—
158C4×C197C22×Q811yescharacteristic
159C4×C197C2×#4111yescharacteristic
160Dic197—21no—
161Dic197—21no—
162Dic197—41no—
163Dic197—41no—
164Dic197—41no—
165Dic197C22⋊Q812yes—
166C2×C197—42no—
167C2×C197C2×C22⋊Q811yescharacteristic
168C2×C197#57611yescharacteristic
169C2×C197#57711yescharacteristic
170D197—22no—
171D197—42no—
172C197#308111yescharacteristic, complemented, Sylow
173#3081—1971nomaximal, Sylow
174#544—1971no—
175#518—1971no—
176#478—1971no—
177#478—1971no—
178#478—1971no—
179#478—1971no—
180#476—1971no—
181#476—1971no—
182#474—1971no—
183#474D19711yescharacteristic, complemented
184#428—1971no—
185#427—1971no—
186#424—1971no—
187#418—1971no—
188C2×#102—1971no—
189C22×D4—1971no—
190C2×M16—1971no—
191C22×C8—1971no—
192C4⋊Q8—3941no—
193C4⋊Q8—1971no—
194C4⋊Q8C2×D19711yescharacteristic, complemented
195#125—3941no—
196#125—1971no—
197#125—1971no—
198C22⋊Q8—3941no—
199C22⋊Q8—3941no—
200C4×D4—3941no—
201C4×D4—3941no—
202C4×D4—1971no—
203C4×D4—1971no—
204C4×D4—1971no—
205C4×D4C2×D19711yescharacteristic
206#117—1971no—
207#117—1971no—
208#117—1971no—
209C2×C4⋊C4—1971no—
210C2×C22⋊C4—3941no—
211C2×C22⋊C4—3941no—
212#107—1971no—
213#107—1971no—
214#107C2×D19711yescharacteristic, complemented
215#107C2×D19711yescharacteristic, complemented
216#106—1971no—
217#106—1971no—
218#106—1971no—
219#106—1971no—
220C4⋊C8—1971no—
221C4⋊C8—1971no—
222C4⋊C8—1971no—
223C4⋊C8C2×D19711yescharacteristic, complemented
224#102—1971no—
225#102—1971no—
226#102—1971no—
227#102—1971no—
228#102—1971no—
229#102—1971no—
230#102C2×D19711yescharacteristic
231#102C2×D19711yescharacteristic
232C24—3941no—
233C2×Q8—21no—
234C2×Q8—3941no—
235C2×D4—3942no—
236C2×D4—3941no—
237C2×D4—1974no—
238C2×D4—1971no—
239C2×D4C22×D19711yescharacteristic
240C22×C4—21no—
241C22×C4—3941no—
242C22×C4—3941no—
243C22×C4—3941no—
244C22×C4—3941no—
245C22×C4—1971no—
246M16—3942no—
247C2×C8—3942no—
248C2×C8—1971no—
249C2×C8—1971no—
250C2×C8C22×D19711yescharacteristic
251C2×C8C22×D19711yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—3941no—
254C4⋊C4—3941no—
255C4⋊C4—3941no—
256C4⋊C4—3941no—
257C4⋊C4—3941no—
258C4⋊C4—3941no—
259C4⋊C4—1971no—
260C4⋊C4—1971no—
261C4⋊C4—1971no—
262C4⋊C4—1971no—
263C4⋊C4—1971no—
264C4⋊C4C22×D19711yescharacteristic
265C4⋊C4C22×D19711yescharacteristic
266C4⋊C4C22×D19711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—3941no—
269C22⋊C4—3941no—
270C22⋊C4—3941no—
271C22⋊C4—3941no—
272C22⋊C4—3941no—
273C22⋊C4—1971no—
274C22⋊C4—1971no—
275C42—3941no—
276C42—3941no—
277C42—1971no—
278C42C22×D19711yescharacteristic
279C23—21no—
280C23—15761no—
281C23—15761no—
282C23—3944no—
283C23—3941no—
284C23—1971no—
285Q8—81no—
286Q8—15761no—
287D4—21no—
288D4—3944no—
289D4—3941no—
290D4—1972no—
291D4Q8×D19712yes—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—81no—
296C2×C4—15761no—
297C2×C4—15761no—
298C2×C4—15761no—
299C2×C4—15761no—
300C2×C4—3941no—
301C2×C4—3941no—
302C2×C4—3941no—
303C2×C4—1971no—
304C2×C4—1971no—
305C2×C4—1971no—
306C2×C4—1972no—
307C2×C4—1972no—
308C2×C4D4×D19711yescharacteristic
309C2×C4D4×D19711yescharacteristic
310C2×C4C23×D19711yescharacteristic
311C8—21no—
312C8—21no—
313C8—3941no—
314C8—3941no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—7881no—
319C22—7881no—
320C22—15764no—
321C22—3942no—
322C22—3941no—
323C22—3941no—
324C22—1972no—
325C22C2×D4×D19711yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—7881no—
332C4—7881no—
333C4—7881no—
334C4—3941no—
335C4—3941no—
336C4—1972no—
337C4C2×Q8×D19711yescharacteristic
338C4#41×D19711yescharacteristic
339C2—42no—
340C2—7882no—
341C2—3942no—
342C2C22⋊Q8×D19711yescharacteristic
343C2#2716188011yescharacteristic
344C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#474×C197C2
socleC22×C197C22×D4
prime radicalC2×D4×D197C22
hypercenter#474D197
hypocenterC197#3081
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #27161706C2semidirect
3. #27161382C2semidirect
4. #27161380C2semidirect
5. #27161379C2semidirect
6. #27161377C2semidirect
7. #27161367C2semidirect
8. #27161366C2semidirect
9. #27161358C2semidirect
10. #27161120C2semidirect
11. #27161116C2semidirect
12. #27161104C2semidirect
15. #474×C197C2semidirect
19. #20263325C22semidirect
21. #20263318C22semidirect
22. #20263317C22semidirect
37. #20263260C22semidirect
38. #20263259C22semidirect
39. #20263258C22semidirect
40. #20263257C22semidirect
41. #20263257C22semidirect
42. #20263257C22semidirect
43. C4⋊#13096802C22semidirect
44. #20263255C22semidirect
45. #20263255C22semidirect
52. C4⋊Q8×C197C22semidirect
54. #107×C197C22semidirect
55. #107×C197C22semidirect
56. C4⋊C8×C197C22semidirect
64. D4×D197Q8semidirect
172. C197#3081semidirect
183. #474D197semidirect
194. C4⋊Q8C2×D197semidirect
214. #107C2×D197semidirect
215. #107C2×D197semidirect
223. C4⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617062
#271615512
#271613822
#271613802
#271613792
#271613772
#271613672
#271613662
#271613582
#271611202
#271611162
#271611042
#271610822
#102×D1972
#474×C1972
#3081197