Finite Group Databasebeta

← #38752291

Subgroups — #38752291

Order 25216.

520 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263345C211yesmaximal, characteristic, complemented
2C22×C8×D197C211yesmaximal, characteristic, complemented
3#27161589C212yesmaximal, complemented
4#27161321C211yesmaximal, characteristic, complemented
5#27161320C211yesmaximal, characteristic, complemented
6#27161312C211yesmaximal, characteristic, complemented
7#27161307C212yesmaximal, complemented
8#27161306C212yesmaximal, complemented
9#27161116C211yesmaximal, characteristic, complemented
10#27161072C212yesmaximal, complemented
11#466×C197C211yesmaximal, characteristic, complemented
12C2×#15691357C2212yes—
13C22×C4×D197C2211yescharacteristic
14#20263345—24no—
15C2×Dic788C2211yescharacteristic, complemented
16C2×#15691346C2212yescomplemented
17C22×#13096802C2211yescharacteristic
18C2×C8×D197—24no—
19C2×C8×D197C2211yescharacteristic
20C2×C8×D197C2211yescharacteristic
21C2×C8×D197C2211yescharacteristic
22C2×C8×D197C2211yescharacteristic
23#20263308—22no—
24#20263306C2212yes—
25#20263305C2212yes—
26#20263297C2212yes—
27#20263294—22no—
28D4×Dic197—22no—
29#20263286—22no—
30#20263284C2212yes—
31#20263260C2211yescharacteristic, complemented
32#20263257C2212yescomplemented
33#20263251C2212yescomplemented
34#20263249C2212yescomplemented
35#20263247C2212yescomplemented
36C2×D8×C197C2211yescharacteristic, complemented
37C22×C8×C197C2211yescharacteristic
38#121×C197C2212yes—
39#107×C197C2211yescharacteristic, complemented
40#102×C197C2212yescomplemented
41C23×D197D411yescharacteristic
42#15691357—24no—
43#15691357—24no—
44C2×Dic394C2312yes—
45C2×C22⋊D197—22no—
46C2×C22⋊D197—22no—
47C22×Dic197—22no—
48C22×Dic197D411yescharacteristic
49C2×C4×D197—22no—
50C2×C4×D197—22no—
51C2×C4×D197D411yescharacteristic
52C2×C4×D197D411yescharacteristic
53C2×C4×D197C2311yescharacteristic
54C2×C4×D197C2311yescharacteristic
55Dic788—22no—
56#15691346—24no—
57C2×#13096802—22no—
58C2×#13096802C2311yescharacteristic
59C2×#13096802C2311yescharacteristic
60C8×D197—22no—
61C8×D197—22no—
62C8×D197—22no—
63C8×D197D414yescomplemented
64C4⋊Dic197—22no—
65C4⋊Dic197C2312yes—
66#15691337—22no—
67C22⋊Dic197—22no—
68C22⋊Dic197—22no—
69#15691335—22no—
70#15691335—22no—
71C4×Dic197—22no—
72C2×D4×C197—22no—
73C2×D4×C197C2312yes—
74C22×C4×C197C2311yescharacteristic
75D8×C197—22no—
76C2×C8×C197—22no—
77C2×C8×C197C2311yescharacteristic
78C2×C8×C197C2311yescharacteristic
79C4⋊C4×C197C2312yes—
80C22⋊C4×C197—22no—
81C22×D197—22no—
82C22×D197—22no—
83C22×D197—22no—
84C22×D197C2×D411yescharacteristic
85C22×D197C2×D411yescharacteristic
86Dic394—22no—
87Dic394—22no—
88C22⋊D197—44no—
89C22⋊D197—44no—
90C2×Dic197—22no—
91C2×Dic197—22no—
92C2×Dic197—22no—
93C2×Dic197—44no—
94C2×Dic197C2×D411yescharacteristic
95C2×Dic197C2×D411yescharacteristic
96C4×D197—22no—
97C4×D197—21no—
98C4×D197—21no—
99C4×D197—22no—
100C4×D197C2×D412yes—
101C4×D197#4112yes—
102#13096802—21no—
103#13096802C2×D412yes—
104C23×C197—22no—
105C23×C197C2×D411yescharacteristic
106D4×C197—22no—
107D4×C197—22no—
108D4×C197—44no—
109C2×C4×C197—21no—
110C2×C4×C197—21no—
111C2×C4×C197—22no—
112C2×C4×C197C2×D411yescharacteristic
113C2×C4×C197C2411yescharacteristic
114C8×C197—21no—
115C8×C197C2×D412yes—
116C22×C197—21no—
117C22×C197—21no—
118C22×C197—42no—
119C22×C197—42no—
120C22×C197—42no—
121C22×C197#13512yescomplemented
122C22×C197C22×D411yescharacteristic
123C2×D197—28no—
124C2×D197—21no—
125C2×D197—21no—
126C2×D197—21no—
127C2×D197—21no—
128C2×D197#12112yes—
129C2×D197#13512yescomplemented
130C4×C197—21no—
131C4×C197—42no—
132C4×C197C22×D411yescharacteristic
133C4×C197C2×#4111yescharacteristic
134Dic197—21no—
135Dic197—42no—
136Dic197—42no—
137Dic197#12112yes—
138C2×C197—22no—
139C2×C197—82no—
140C2×C197C2×#12111yescharacteristic
141C2×C197C2×#13511yescharacteristic
142C2×C197C2×#13511yescharacteristic
143D197—22no—
144D197—22no—
145C197#305011yescharacteristic, complemented, Sylow
146#3050—1971nomaximal, Sylow
147C2×#135—1971no—
148C23×C8—1971no—
149#525—1972no—
150#467—1971no—
151#467—1971no—
152#466—1971no—
153#466D19711yescharacteristic, complemented
154#465—1972no—
155#465—1972no—
156#427—1971no—
157#416—1972no—
158C2×#41—1972no—
159C23×C4—1971no—
160#135—3944no—
161C2×Q16—1971no—
162C2×SD16—1972no—
163C2×D8C2×D19711yescharacteristic, complemented
164C22×C8—3944no—
165C22×C8—1971no—
166C22×C8—1971no—
167C22×C8—1971no—
168C22×C8—1971no—
169C22×C8—1971no—
170C22×C8C2×D19711yescharacteristic
171#123—3942no—
172C22⋊Q8—1972no—
173C22⋊Q8—1972no—
174#121—1972no—
175#121C2×D19712yes—
176#120—3942no—
177C4×D4—3942no—
178C4×D4—3942no—
179#117—1972no—
180#107—1971no—
181#107C2×D19711yescharacteristic, complemented
182#106—1972no—
183#103—1972no—
184#103—1972no—
185#102—1972no—
186#102C2×D19712yescomplemented
187C24—1971no—
188#41—3944no—
189#41—3944no—
190C2×Q8—1972no—
191C2×D4—22no—
192C2×D4—3942no—
193C2×D4—3942no—
194C2×D4C22×D19712yes—
195C22×C4—3942no—
196C22×C4—3942no—
197C22×C4—3942no—
198C22×C4—1971no—
199C22×C4—1971no—
200C22×C4—1971no—
201C22×C4—1971no—
202C22×C4—1971no—
203C22×C4C22×D19711yescharacteristic
204Q16—3942no—
205SD16—3944no—
206D8—22no—
207C2×C8—22no—
208C2×C8—3942no—
209C2×C8—3942no—
210C2×C8—3942no—
211C2×C8—3942no—
212C2×C8—1971no—
213C2×C8—1971no—
214C2×C8—1974no—
215C2×C8C22×D19711yescharacteristic
216C2×C8C22×D19711yescharacteristic
217C4⋊C4—3942no—
218C4⋊C4—3942no—
219C4⋊C4—1972no—
220C4⋊C4C22×D19712yes—
221C22⋊C4—22no—
222C22⋊C4—3942no—
223C22⋊C4—3942no—
224C22⋊C4—3942no—
225C22⋊C4—3942no—
226C42—3942no—
227C23—22no—
228C23—3942no—
229C23—3942no—
230C23—3942no—
231C23—1971no—
232C23—1971no—
233C23D4×D19711yescharacteristic
234Q8—3942no—
235Q8—3942no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—7884no—
240D4—7884no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—7884no—
245C2×C4—3942no—
246C2×C4—3942no—
247C2×C4—3942no—
248C2×C4—3942no—
249C2×C4—3942no—
250C2×C4—3941no—
251C2×C4—3941no—
252C2×C4—1971no—
253C2×C4—1971no—
254C2×C4—1972no—
255C2×C4—1972no—
256C2×C4D4×D19711yescharacteristic
257C2×C4C23×D19711yescharacteristic
258C8—21no—
259C8—3941no—
260C8—1972no—
261C8D4×D19712yes—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—3948no—
268C22—3941no—
269C22—3941no—
270C22—3941no—
271C22—3941no—
272C22—1972no—
273C22—1972no—
274C22#2026334512yescomplemented
275C22C2×D4×D19711yescharacteristic
276C4—21no—
277C4—42no—
278C4—7882no—
279C4—7882no—
280C4—3941no—
281C4—1972no—
282C4C2×D4×D19711yescharacteristic
283C4#41×D19711yescharacteristic
284C2—22no—
285C2—82no—
286C2—3942no—
287C2—3942no—
288C2#121×D19711yescharacteristic
289C2#135×D19711yescharacteristic
290C2C2×#2026334511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#466×C197C2
socleC22×C197C22×D4
hypercenter#466D197
hypocenterC197#3050
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#20263345C2semidirect
2. C22×C8×D197C2semidirect
3. #27161589C2semidirect
4. #27161321C2semidirect
5. #27161320C2semidirect
6. #27161312C2semidirect
7. #27161307C2semidirect
8. #27161306C2semidirect
9. #27161116C2semidirect
10. #27161072C2semidirect
11. #466×C197C2semidirect
15. C2×Dic788C22semidirect
16. C2×#15691346C22semidirect
31. #20263260C22semidirect
32. #20263257C22semidirect
33. #20263251C22semidirect
34. #20263249C22semidirect
35. #20263247C22semidirect
36. C2×D8×C197C22semidirect
39. #107×C197C22semidirect
40. #102×C197C22semidirect
63. C8×D197D4semidirect
121. C22×C197#135semidirect
129. C2×D197#135semidirect
145. C197#3050semidirect
153. #466D197semidirect
163. C2×D8C2×D197semidirect
181. #107C2×D197semidirect
186. #102C2×D197semidirect
274. C22#20263345semidirect

Maximal subgroups

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

subgroupindex
C2×#202633452
C22×C8×D1972
#271615892
#271613212
#271613202
#271613122
#271613072
#271613062
#271611162
#271610722
#466×C1972
#3050197