Finite Group Databasebeta

← #38744382

Subgroups — #38744382

Order 25216.

270 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 55 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#27161642C211yesmaximal, characteristic
2#27161051C212yesmaximal, complemented
3#27160872C211yesmaximal, characteristic, complemented
4#27160871C212yesmaximal, complemented
5#353×C197C211yesmaximal, characteristic, complemented
6C2×#15691357C2211yescharacteristic
7C2×D4×D197C411yescharacteristic, complemented
8#20263322C2211yescharacteristic
9#20263314C411yescharacteristic, complemented
10#20263313C411yescharacteristic, complemented
11#20263298—22no—
12#20263294—22no—
13#20263283C411yescharacteristic, complemented
14#20263282—22no—
15#20263241C2212yescomplemented
16#20263240—22no—
17#127×C197C2211yescharacteristic
18#99×C197C2212yescomplemented
19C23×D197—21no—
20#15691357—42no—
21C2×Dic394C2×C411yescharacteristic
22C2×D788C2×C411yescharacteristic
23C2×C22⋊D197—21no—
24C2×C22⋊D197—22no—
25C2×C22⋊D197D412yes—
26D4×D197—42no—
27C22×Dic197D412yes—
28C2×C4×D197C2×C411yescharacteristic
29#15691337—21no—
30#15691337—42no—
31C22⋊Dic197—22no—
32C22⋊Dic197—22no—
33#15691335—21no—
34#15691335—22no—
35C4×Dic197—42no—
36C4×Dic197C2×C411yescharacteristic
37C2×D4×C197—22no—
38C2×D4×C197C2×C411yescharacteristic
39C2×D4×C197C2311yescharacteristic
40C22⋊C4×C197—22no—
41C42×C197C2×C411yescharacteristic
42C22×D197—21no—
43C22×D197—42no—
44C22×D197—82no—
45C22×D197C22⋊C411yescharacteristic
46Dic394—41no—
47D788—41no—
48C22⋊D197—42no—
49C22⋊D197—42no—
50C22⋊D197—82no—
51C2×Dic197—22no—
52C2×Dic197—22no—
53C2×Dic197—21no—
54C2×Dic197—42no—
55C2×Dic197C22⋊C412yes—
56C2×Dic197C22⋊C411yescharacteristic
57C4×D197—22no—
58C23×C197—21no—
59C23×C197C2×D412yes—
60D4×C197—41no—
61D4×C197—41no—
62D4×C197—82no—
63C2×C4×C197—21no—
64C2×C4×C197—42no—
65C2×C4×C197C22×C411yescharacteristic
66C22×C197—22no—
67C22×C197—42no—
68C22×C197—42no—
69C22×C197—41no—
70C22×C197C2×C22⋊C411yescharacteristic
71C2×D197—21no—
72C2×D197—41no—
73C2×D197—42no—
74C2×D197—41no—
75C2×D197—84no—
76C4×C197—21no—
77C4×C197—41no—
78C4×C197—82no—
79Dic197—22no—
80Dic197—41no—
81Dic197—82no—
82Dic197#9912yes—
83C2×C197—21no—
84C2×C197—42no—
85C2×C197—81no—
86C2×C197C2×#9911yescharacteristic
87D197—41no—
88D197—81no—
89C197#212811yescharacteristic, complemented, Sylow
90#2128—1971nomaximal, Sylow
91#535—1971no—
92#410—1972no—
93#353—1971no—
94#353—1972no—
95#353D19711yescharacteristic, complemented
96C2×#41—1971no—
97C22×D4—1971no—
98#127—1971no—
99#127C2×D19711yescharacteristic
100#124—1971no—
101#124—1971no—
102#121—3942no—
103#120—3942no—
104#117—3942no—
105#117—1971no—
106#99—3942no—
107#99—1972no—
108#99C2×D19712yescomplemented
109C24—3941no—
110#41—7882no—
111C2×Q8—1971no—
112C2×D4—22no—
113C2×D4—7882no—
114C2×D4—3942no—
115C2×D4—3941no—
116C2×D4—1972no—
117C2×D4—1971no—
118C2×D4C4×D19711yescharacteristic
119C2×D4C22×D19711yescharacteristic
120C22×C4—1972no—
121C22×C4—1971no—
122C4⋊C4—7882no—
123C4⋊C4—3941no—
124C22⋊C4—22no—
125C22⋊C4—3941no—
126C22⋊C4—3942no—
127C22⋊C4—3942no—
128C22⋊C4—3942no—
129C42—7882no—
130C42—1971no—
131C42C4×D19711yescharacteristic
132C23—21no—
133C23—7882no—
134C23—15762no—
135C23—3941no—
136C23—1971no—
137C23D4×D19712yes—
138Q8—7881no—
139D4—41no—
140D4—41no—
141D4—82no—
142D4—7881no—
143D4—7882no—
144D4—7882no—
145D4—15762no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—7882no—
149C2×C4—3942no—
150C2×C4—3941no—
151C2×C4—3942no—
152C2×C4—3942no—
153C2×C4—1972no—
154C2×C4—1971no—
155C2×C4C2×C4×D19711yescharacteristic
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—7881no—
161C22—7882no—
162C22—7881no—
163C22—15764no—
164C22—3941no—
165C22C22⋊C4×D19711yescharacteristic
166C4—21no—
167C4—41no—
168C4—82no—
169C4—7881no—
170C4—15762no—
171C4—3942no—
172C4—1972no—
173C2—21no—
174C2—42no—
175C2—81no—
176C2—7881no—
177C2—15761no—
178C2#99×D19711yescharacteristic
subgroupG/H
centerC2#99×D197
commutatorC2×C4×C197C22×C4
FrattiniC2×D4C22×D197
Fitting#353×C197C2
socleC2×C197C2×#99
prime radical#27161642C2
cosocleC2×D4×C197C23
hypercenter#353D197
hypocenterC197#2128
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #27161051C2semidirect
3. #27160872C2semidirect
4. #27160871C2semidirect
5. #353×C197C2semidirect
7. C2×D4×D197C4semidirect
9. #20263314C4semidirect
10. #20263313C4semidirect
13. #20263283C4semidirect
15. #20263241C22semidirect
18. #99×C197C22semidirect
89. C197#2128semidirect
95. #353D197semidirect
108. #99C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616422
#271610512
#271608722
#271608712
#353×C1972
#2128197