Finite Group Databasebeta

← #38753095

Subgroups — #38753095

Order 25216.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#27161635C211yesmaximal, characteristic, complemented
2#124×D197C211yesmaximal, characteristic
3#27161440C212yesmaximal, complemented
4#27161437C212yesmaximal, complemented
5#27161436C211yesmaximal, characteristic, complemented
6#27161434C211yesmaximal, characteristic, complemented
7#27161433C211yesmaximal, characteristic, complemented
8#27161086C212yesmaximal, complemented
9#27161082C212yesmaximal
10C4×#15691341C211yesmaximal, characteristic, complemented
11#488×C197C211yesmaximal, characteristic, complemented
12C2×Q8×D197C2211yescharacteristic
13C2×D4×D197C2211yescharacteristic
14C2×#15691341C2212yes—
15#20263325C2212yescomplemented
16#20263318C2211yescharacteristic, complemented
17#20263315C2211yescharacteristic
18#20263314C2211yescharacteristic
19#20263313C2211yescharacteristic
20#20263312—22no—
21#20263304—22no—
22#20263291—22no—
23#20263284C2212yes—
24C22⋊C4×D197—22no—
25C42×D197C2211yescharacteristic
26#20263251C2212yescomplemented
27#20263250C2212yescomplemented
28#20263249C2212yescomplemented
29#20263248C2212yes—
30#20263247C2212yes—
31#20263246C2212yes—
32#20263236C2212yescomplemented
33C4×#13096802C2211yescharacteristic, complemented
34#125×C197C2211yescharacteristic, complemented
35#124×C197C2211yescharacteristic
36#103×C197C2212yescomplemented
37#102×C197C2212yes—
38C4×C8×C197C2211yescharacteristic, complemented
39C23×D197—21no—
40C2×Dic394—22no—
41C2×Dic394—21no—
42C2×Dic394C2311yescharacteristic
43Q8×D197—24no—
44C2×D788C2311yescharacteristic
45C2×C22⋊D197—21no—
46D4×D197—24no—
47C2×C4×D197—21no—
48C2×C4×D197D412yes—
49C2×C4×D197C2311yescharacteristic
50#15691341—24no—
51C2×#13096802C2312yes—
52C4⋊Dic197C2312yes—
53#15691337—22no—
54#15691337—22no—
55C22⋊Dic197—22no—
56#15691335—22no—
57#15691335—22no—
58#15691335—22no—
59C4×Dic197—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—44no—
71C22×D197—42no—
72C22×D197C2×D411yescharacteristic
73Dic394—22no—
74Dic394—44no—
75Dic394—42no—
76D788—22no—
77C22⋊D197—42no—
78C2×Dic197—22no—
79C2×Dic197—21no—
80C2×Dic197C2×D412yes—
81C2×Dic197C2×D411yescharacteristic
82C4×D197—42no—
83C4×D197—42no—
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—22no—
98C2×D197—41no—
99C2×D197—42no—
100C2×D197—84no—
101C2×D197#12412yes—
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×C197#57512yes—
113D197—22no—
114D197—81no—
115C197#314011yescharacteristic, complemented, Sylow
116#3140—1971nomaximal, Sylow
117#533—1971no—
118C2×#124—1971no—
119#489—1972no—
120#489—1972no—
121#488—1971no—
122#488—1971no—
123#488—1971no—
124#488D19711yescharacteristic, complemented
125#419—1972no—
126#418—1972no—
127C4×M16—1971no—
128C22×Q8—1971no—
129C22×D4—1971no—
130C2×M16—1972no—
131C4⋊Q8—1972no—
132#125—1971no—
133#125C2×D19711yescharacteristic, complemented
134#124—3942no—
135#124—1971no—
136#124—1971no—
137#124—1971no—
138#124C2×D19711yescharacteristic
139C22⋊Q8—3942no—
140C4×Q8—3942no—
141#117—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—
152#97—1972no—
153C4×C8—1971no—
154C4×C8C2×D19711yescharacteristic, complemented
155C24—3941no—
156C2×Q8—3942no—
157C2×Q8—3941no—
158C2×Q8—3944no—
159C2×Q8—1971no—
160C2×Q8C22×D19711yescharacteristic
161C2×D4—3944no—
162C2×D4—3941no—
163C2×D4—1971no—
164C2×D4C22×D19711yescharacteristic
165C22×C4—3941no—
166C22×C4—1972no—
167C22×C4—1971no—
168M16—3944no—
169C2×C8—1972no—
170C2×C8C22×D19712yes—
171C4⋊C4—22no—
172C4⋊C4—3942no—
173C4⋊C4—3942no—
174C4⋊C4—1972no—
175C4⋊C4C22×D19712yes—
176C22⋊C4—22no—
177C22⋊C4—3942no—
178C22⋊C4—3942no—
179C22⋊C4—3942no—
180C22⋊C4—3942no—
181C42—3942no—
182C42—1972no—
183C42—1971no—
184C42C22×D19711yescharacteristic
185C23—21no—
186C23—7884no—
187C23—7882no—
188C23—3941no—
189C23—1971no—
190Q8—22no—
191Q8—7884no—
192Q8—7882no—
193Q8—3942no—
194D4—22no—
195D4—7882no—
196D4—3942no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—7882no—
200C2×C4—7882no—
201C2×C4—3942no—
202C2×C4—3941no—
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—3942no—
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—3942no—
229C2#124×D19711yescharacteristic
230C2#2716187612yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#488×C197C2
socleC22×C197C22×D4
prime radicalC2×D4×D197C22
hypercenter#488D197
hypocenterC197#3140
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. #27161635C2semidirect
3. #27161440C2semidirect
4. #27161437C2semidirect
5. #27161436C2semidirect
6. #27161434C2semidirect
7. #27161433C2semidirect
8. #27161086C2semidirect
10. C4×#15691341C2semidirect
11. #488×C197C2semidirect
15. #20263325C22semidirect
16. #20263318C22semidirect
26. #20263251C22semidirect
27. #20263250C22semidirect
28. #20263249C22semidirect
32. #20263236C22semidirect
33. C4×#13096802C22semidirect
34. #125×C197C22semidirect
36. #103×C197C22semidirect
38. C4×C8×C197C22semidirect
115. C197#3140semidirect
124. #488D197semidirect
133. #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
#271616352
#124×D1972
#271614402
#271614372
#271614362
#271614342
#271614332
#271610862
#271610822
C4×#156913412
#488×C1972
#3140197