Finite Group Databasebeta

← #38751561

Subgroups — #38751561

Order 25216.

528 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 175 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×D8×D197C211yesmaximal, characteristic
2#27161551C212yesmaximal, complemented
3#27161192C211yesmaximal, characteristic, complemented
4#27161191C211yesmaximal, characteristic, complemented
5#27161190C211yesmaximal, characteristic, complemented
6C4×#15691343C212yesmaximal, complemented
7#27161159C212yesmaximal, complemented
8#27161114C211yesmaximal, characteristic, complemented
9#27161082C212yesmaximal, complemented
10#27161036C211yesmaximal, characteristic, complemented
11#442×C197C211yesmaximal, characteristic, complemented
12C2×D4×D197C2212yes—
13C2×D1576C2211yescharacteristic
14C2×#15691343C2212yes—
15D8×D197C418yescomplemented
16C2×#15691341C2211yescharacteristic
17C2×C8×D197C2211yescharacteristic
18C4×D788C2212yescomplemented
19D4×Dic197C2212yescomplemented
20#20263288C2212yescomplemented
21C4×C22⋊D197—22no—
22#20263286—22no—
23#20263284C2212yes—
24#20263283C2211yescharacteristic
25C2×#15691335—22no—
26C22⋊C4×D197—22no—
27#20263259C2212yescomplemented
28#20263258C2211yescharacteristic, complemented
29#20263248C2212yescomplemented
30#20263247C2212yescomplemented
31#20263246C2212yescomplemented
32#20263236C2211yescharacteristic, complemented
33C4×#13096802C2211yescharacteristic, complemented
34C8×Dic197C2211yescharacteristic, complemented
35C2×D8×C197C2211yescharacteristic
36C4×D4×C197C2212yescomplemented
37#106×C197C2211yescharacteristic, complemented
38#102×C197C2212yescomplemented
39#97×C197C2211yescharacteristic, complemented
40C23×D197—22no—
41C2×D788C2312yes—
42C2×C22⋊D197—22no—
43D4×D197—24no—
44D4×D197C2×C418yescomplemented
45C22×Dic197—22no—
46C2×C4×D197—22no—
47C2×C4×D197—22no—
48C2×C4×D197C2311yescharacteristic
49D1576C2×C414yescomplemented
50#15691343C2×C418yescomplemented
51#15691341—22no—
52C2×#13096802C2311yescharacteristic
53C2×#13096802C2311yescharacteristic
54C8×D197C2×C414yescomplemented
55C4⋊Dic197C2312yes—
56#15691337D412yes—
57C22⋊Dic197—22no—
58#15691335—22no—
59#15691335—22no—
60#15691335—22no—
61#15691335D412yes—
62C4×Dic197—22no—
63C4×Dic197C2311yescharacteristic
64C2×D4×C197C2312yes—
65C22×C4×C197—22no—
66D8×C197C2×C414yescomplemented
67C2×C8×C197C2311yescharacteristic
68C2×C8×C197C2311yescharacteristic
69C4⋊C4×C197C2312yes—
70C22⋊C4×C197—22no—
71C42×C197C2311yescharacteristic
72C22×D197—22no—
73C22×D197—28no—
74C22×D197—82no—
75C22×D197—82no—
76C22×D197C2×D411yescharacteristic
77D788—22no—
78D788C22×C414yes—
79C22⋊D197—28no—
80C2×Dic197—22no—
81C2×Dic197—82no—
82C2×Dic197C2×D412yes—
83C2×Dic197C2×D411yescharacteristic
84C4×D197—82no—
85C4×D197—82no—
86C4×D197C22×C412yes—
87C4×D197#4112yes—
88#13096802—21no—
89#13096802C22×C412yes—
90C23×C197—22no—
91D4×C197—22no—
92D4×C197C22×C414yes—
93C2×C4×C197—22no—
94C2×C4×C197—82no—
95C2×C4×C197C2×D412yes—
96C2×C4×C197C2411yescharacteristic
97C8×C197—21no—
98C8×C197C22×C412yes—
99C22×C197—24no—
100C22×C197—42no—
101C22×C197—42no—
102C22×C197C22×D411yescharacteristic
103C2×D197—22no—
104C2×D197—24no—
105C2×D197—42no—
106C2×D197—42no—
107C2×D197—88no—
108C2×D197C4×D412yes—
109C4×C197—22no—
110C4×C197—42no—
111C4×C197C23×C411yescharacteristic
112C4×C197C2×#4111yescharacteristic
113Dic197—22no—
114Dic197—42no—
115Dic197C4×D412yes—
116C2×C197—44no—
117C2×C197C2×C4×D411yescharacteristic
118C2×C197#57612yes—
119D197—22no—
120D197—44no—
121C197#294811yescharacteristic, complemented, Sylow
122#2948—1971nomaximal, Sylow
123C22×D8—1971no—
124#518—1972no—
125#442—1971no—
126#442—1971no—
127#442—1971no—
128#442D19711yescharacteristic, complemented
129C4×D8—1972no—
130C4×D8—1972no—
131#427—1971no—
132#418—1972no—
133#405—1971no—
134C22×D4—1972no—
135C2×D8—1972no—
136C2×D8—1978no—
137C2×D8—1971no—
138C2×D8C2×D19711yescharacteristic
139C2×M16—1971no—
140C22×C8—1971no—
141C4×D4—3942no—
142C4×D4—3942no—
143C4×D4—1972no—
144C4×D4—1972no—
145C4×D4—1972no—
146C4×D4C2×D19712yescomplemented
147#117—1972no—
148#117—1971no—
149C2×C22⋊C4—3942no—
150C2×C22⋊C4—3942no—
151#107—1972no—
152#106—1971no—
153#106C2×D19711yescharacteristic, complemented
154#102—1972no—
155#102—1972no—
156#102—1972no—
157#102C2×D19712yescomplemented
158#97—1971no—
159#97C2×D19711yescharacteristic, complemented
160C4×C8—1971no—
161C4×C8—1971no—
162C24—3942no—
163C2×D4—3944no—
164C2×D4—3942no—
165C2×D4—1978no—
166C2×D4—1972no—
167C2×D4C22×D19712yes—
168C22×C4—22no—
169C22×C4—3942no—
170C22×C4—3942no—
171C22×C4—3942no—
172C22×C4—1971no—
173D8—1978no—
174D8—1974no—
175D8C4×D19714yescomplemented
176M16—3942no—
177C2×C8—1971no—
178C2×C8—1971no—
179C2×C8—1974no—
180C2×C8C22×D19711yescharacteristic
181C2×C8C22×D19711yescharacteristic
182C4⋊C4—1972no—
183C4⋊C4—1972no—
184C4⋊C4C22×D19712yes—
185C22⋊C4—22no—
186C22⋊C4—3942no—
187C22⋊C4—3942no—
188C22⋊C4—3942no—
189C22⋊C4—3942no—
190C22⋊C4—1972no—
191C42—3942no—
192C42—1971no—
193C42C22×D19711yescharacteristic
194C23—22no—
195C23—15762no—
196C23—15762no—
197C23—3948no—
198C23—3942no—
199C23—1971no—
200D4—22no—
201D4—3948no—
202D4—3942no—
203D4—1974no—
204D4C2×C4×D19714yes—
205C2×C4—22no—
206C2×C4—82no—
207C2×C4—15762no—
208C2×C4—15762no—
209C2×C4—15762no—
210C2×C4—3942no—
211C2×C4—1972no—
212C2×C4—1971no—
213C2×C4—1972no—
214C2×C4—1972no—
215C2×C4D4×D19712yes—
216C2×C4C23×D19711yescharacteristic
217C8—21no—
218C8—3941no—
219C8—1972no—
220C8C2×C4×D19712yes—
221C22—24no—
222C22—42no—
223C22—42no—
224C22—7882no—
225C22—7882no—
226C22—15768no—
227C22—3944no—
228C22—3942no—
229C22—1972no—
230C22C2×D4×D19711yescharacteristic
231C4—22no—
232C4—42no—
233C4—7882no—
234C4—3942no—
235C4—1972no—
236C4C22×C4×D19711yescharacteristic
237C4#41×D19711yescharacteristic
238C2—44no—
239C2—7884no—
240C2—3942no—
241C2C4×D4×D19711yescharacteristic
242C2#2716188312yes—
subgroupG/H
centerC22C2×D4×D197
commutatorC4×C197C23×C4
FrattiniC2×C4C23×D197
Fitting#442×C197C2
socleC22×C197C22×D4
prime radicalC2×D8×D197C2
cosocleC2×C4×C197C24
hypercenter#442D197
hypocenterC197#2948
LCSG ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #27161551C2semidirect
3. #27161192C2semidirect
4. #27161191C2semidirect
5. #27161190C2semidirect
6. C4×#15691343C2semidirect
7. #27161159C2semidirect
8. #27161114C2semidirect
9. #27161082C2semidirect
10. #27161036C2semidirect
11. #442×C197C2semidirect
15. D8×D197C4semidirect
18. C4×D788C22semidirect
19. D4×Dic197C22semidirect
20. #20263288C22semidirect
27. #20263259C22semidirect
28. #20263258C22semidirect
29. #20263248C22semidirect
30. #20263247C22semidirect
31. #20263246C22semidirect
32. #20263236C22semidirect
33. C4×#13096802C22semidirect
34. C8×Dic197C22semidirect
36. C4×D4×C197C22semidirect
37. #106×C197C22semidirect
38. #102×C197C22semidirect
39. #97×C197C22semidirect
44. D4×D197C2×C4semidirect
49. D1576C2×C4semidirect
50. #15691343C2×C4semidirect
54. C8×D197C2×C4semidirect
66. D8×C197C2×C4semidirect
121. C197#2948semidirect
128. #442D197semidirect
146. C4×D4C2×D197semidirect
153. #106C2×D197semidirect
157. #102C2×D197semidirect
159. #97C2×D197semidirect
175. D8C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1972
#271615512
#271611922
#271611912
#271611902
C4×#156913432
#271611592
#271611142
#271610822
#271610362
#442×C1972
#2948197