Finite Group Databasebeta

← #488×D197 #38753025

Subgroups — #488×D197 #38753025

Order 25216.

412 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#125×D197C211yesmaximal, characteristic, complemented
2#124×D197C211yesmaximal, characteristic
3#27161436C211yesmaximal, characteristic, complemented
4#27161435C212yesmaximal, complemented
5#27161434C211yesmaximal, characteristic, complemented
6#27161433C211yesmaximal, characteristic, complemented
7#27161432C212yesmaximal, complemented
8#103×D197C212yesmaximal, complemented
9#102×D197C212yesmaximal
10C4×C8×D197C211yesmaximal, characteristic, complemented
11#488×C197C211yesmaximal, characteristic, complemented
12C2×Q8×D197C2211yescharacteristic
13C2×D4×D197C2211yescharacteristic
14C2×C8×D197C2212yes—
15#20263318C2211yescharacteristic, complemented
16#20263317C2212yescomplemented
17#20263316—22no—
18#20263315C2211yescharacteristic
19#20263314C2211yescharacteristic
20#20263313C2211yescharacteristic
21#20263312—22no—
22C4⋊C4×D197—22no—
23C4⋊C4×D197C2212yes—
24C22⋊C4×D197—22no—
25C42×D197C2211yescharacteristic
26#20263251C2212yescomplemented
27#20263250C2212yescomplemented
28#20263249C2212yescomplemented
29#20263248C2212yes—
30#20263247C2212yes—
31#20263246C2212yes—
32C4×#13096802C2211yescharacteristic, complemented
33C8×Dic197C2212yescomplemented
34#125×C197C2211yescharacteristic, complemented
35#124×C197C2211yescharacteristic
36#103×C197C2212yescomplemented
37#102×C197C2212yes—
38C4×C8×C197C2211yescharacteristic, complemented
39C23×D197—21no—
40C2×Dic394—21no—
41C2×Dic394C2311yescharacteristic
42Q8×D197—24no—
43C2×D788C2311yescharacteristic
44C2×C22⋊D197—21no—
45D4×D197—24no—
46C2×C4×D197—21no—
47C2×C4×D197—22no—
48C2×C4×D197D412yes—
49C2×C4×D197C2311yescharacteristic
50C2×#13096802C2312yes—
51C8×D197—24no—
52C4⋊Dic197—22no—
53C4⋊Dic197C2312yes—
54#15691337—22no—
55#15691337—22no—
56#15691337—22no—
57C22⋊Dic197—22no—
58#15691335—22no—
59#15691335—22no—
60C4×Dic197D412yes—
61C4×Dic197C2311yescharacteristic
62C2×Q8×C197C2311yescharacteristic
63C2×D4×C197C2311yescharacteristic
64C2×C8×C197C2312yes—
65C4⋊C4×C197—22no—
66C4⋊C4×C197C2312yes—
67C22⋊C4×C197—22no—
68C42×C197C2311yescharacteristic
69C22×D197—21no—
70C22×D197—42no—
71C22×D197—44no—
72C22×D197C2×D411yescharacteristic
73Dic394—22no—
74Dic394—42no—
75D788—22no—
76C22⋊D197—42no—
77C2×Dic197—21no—
78C2×Dic197—22no—
79C2×Dic197C2×D412yes—
80C2×Dic197C2×D411yescharacteristic
81C4×D197—24no—
82C4×D197—42no—
83C4×D197—44no—
84C4×D197#4114yes—
85#13096802—22no—
86C23×C197—21no—
87Q8×C197—22no—
88D4×C197—22no—
89C2×C4×C197—21no—
90C2×C4×C197—22no—
91C2×C4×C197C2×D412yes—
92C2×C4×C197C2411yescharacteristic
93C8×C197—22no—
94C22×C197—41no—
95C22×C197—42no—
96C22×C197C22×D411yescharacteristic
97C2×D197—41no—
98C2×D197—42no—
99C2×D197—84no—
100C2×D197#12412yes—
101C2×D197#13514yes—
102C4×C197—22no—
103C4×C197—41no—
104C4×C197—42no—
105C4×C197C2×#4112yes—
106Dic197—22no—
107Dic197—41no—
108Dic197—42no—
109Dic197#12412yes—
110C2×C197—81no—
111C2×C197C2×#12411yescharacteristic
112C2×C197C2×#13512yes—
113D197—81no—
114D197#48814yesdirect product factor
115C197C2×#48811yescharacteristic, complemented, Sylow
116C2×#488—1971nomaximal, Sylow
117C2×#125—1971no—
118C2×#124—1971no—
119#488—1971no—
120#488—1971no—
121#488—1972no—
122#488—1972no—
123#488—1971no—
124#488D19711yescharacteristic, direct product factor
125C2×#103—1972no—
126C2×#102—1972no—
127C2×C4×C8—1971no—
128C22×Q8—1971no—
129C22×D4—1971no—
130C22×C8—1972no—
131#125—3942no—
132#125—1972no—
133#125—1971no—
134#125C2×D19711yescharacteristic, complemented
135#124—3942no—
136#124—1971no—
137#124—1971no—
138#124—1971no—
139#124C2×D19711yescharacteristic
140C2×C4⋊C4—3942no—
141C2×C4⋊C4—1972no—
142C2×C22⋊C4—3942no—
143C2×C42—1971no—
144#103—1972no—
145#103—1972no—
146#103—1972no—
147#103C2×D19712yescomplemented
148#102—1972no—
149#102—1972no—
150#102—1972no—
151#102C2×D19712yes—
152C4×C8—1972no—
153C4×C8—1971no—
154C4×C8C2×D19711yescharacteristic, complemented
155C24—3941no—
156C2×Q8—3941no—
157C2×Q8—3944no—
158C2×Q8—1971no—
159C2×Q8C22×D19711yescharacteristic
160C2×D4—3944no—
161C2×D4—3941no—
162C2×D4—1971no—
163C2×D4C22×D19711yescharacteristic
164C22×C4—3941no—
165C22×C4—3942no—
166C22×C4—1972no—
167C22×C4—1971no—
168C2×C8—3944no—
169C2×C8—1972no—
170C2×C8C22×D19712yes—
171C4⋊C4—22no—
172C4⋊C4—3942no—
173C4⋊C4—3942no—
174C4⋊C4—3942no—
175C4⋊C4—3942no—
176C4⋊C4—1972no—
177C4⋊C4C22×D19712yes—
178C22⋊C4—22no—
179C22⋊C4—3942no—
180C22⋊C4—3942no—
181C22⋊C4—3942no—
182C42—1972no—
183C42—1971no—
184C42C22×D19711yescharacteristic
185C23—21no—
186C23—7882no—
187C23—7884no—
188C23—3941no—
189C23—1971no—
190Q8—22no—
191Q8—7882no—
192Q8—3942no—
193D4—22no—
194D4—7882no—
195D4—3942no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—7882no—
199C2×C4—7884no—
200C2×C4—3944no—
201C2×C4—3941no—
202C2×C4—3942no—
203C2×C4—1972no—
204C2×C4—1971no—
205C2×C4—1974no—
206C2×C4D4×D19712yes—
207C2×C4C23×D19711yescharacteristic
208C8—22no—
209C8—3942no—
210C22—42no—
211C22—41no—
212C22—7882no—
213C22—7881no—
214C22—15764no—
215C22—1974no—
216C22—1972no—
217C22C2×D4×D19711yescharacteristic
218C4—22no—
219C4—42no—
220C4—41no—
221C4—7882no—
222C4—7881no—
223C4—3942no—
224C4—1972no—
225C4#41×D19712yes—
226C2—81no—
227C2—15761no—
228C2—1974no—
229C2#124×D19711yescharacteristic
230C2#135×D19712yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#488×C197C2
socleC22×C197C22×D4
prime radicalC2×D4×D197C22
hypercenter#488D197
hypocenterC197C2×#488
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #125×D197C2semidirect
3. #27161436C2semidirect
4. #27161435C2semidirect
5. #27161434C2semidirect
6. #27161433C2semidirect
7. #27161432C2semidirect
8. #103×D197C2semidirect
10. C4×C8×D197C2semidirect
11. #488×C197C2semidirect
15. #20263318C22semidirect
16. #20263317C22semidirect
26. #20263251C22semidirect
27. #20263250C22semidirect
28. #20263249C22semidirect
32. C4×#13096802C22semidirect
33. C8×Dic197C22semidirect
34. #125×C197C22semidirect
36. #103×C197C22semidirect
38. C4×C8×C197C22semidirect
114. D197#488direct (N × H)
115. C197C2×#488semidirect
124. #488D197direct (N × H)
134. #125C2×D197semidirect
147. #103C2×D197semidirect
154. C4×C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#125×D1972
#124×D1972
#271614362
#271614352
#271614342
#271614332
#271614322
#103×D1972
#102×D1972
C4×C8×D1972
#488×C1972
C2×#488197