Finite Group Databasebeta

← #38741683

Subgroups — #38741683

Order 25216.

282 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 53 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#27161660C211yesmaximal, characteristic
2#27161220C212yesmaximal, complemented
3#27160999C211yesmaximal, characteristic, complemented
4#27160814C212yesmaximal, complemented
5#324×C197C211yesmaximal, characteristic, complemented
6C2×#15691358—21no—
7C22×Dic394C2211yescharacteristic
8C2×#15691347—22no—
9#20263306—22no—
10#20263301—22no—
11#20263300—21no—
12#20263300C2212yes—
13C2×#15691337—21no—
14#20263251—22no—
15#20263233—21no—
16#20263233C2211yescharacteristic
17C2×C4⋊C4×C197C2211yescharacteristic
18C22⋊C8×C197C2212yes—
19#15691358—42no—
20#15691358—42no—
21C2×Dic394—21no—
22C2×Dic394—22no—
23C2×Dic394—42no—
24C2×D788—21no—
25C2×D788—21no—
26C2×D788D412yes—
27C2×C22⋊D197—22no—
28C22×Dic197—21no—
29C22×Dic197—21no—
30C2×C4×D197—22no—
31#15691347—44no—
32C4⋊Dic197—21no—
33C4⋊Dic197D412yes—
34#15691337—42no—
35#15691335—22no—
36#15691335—22no—
37C22×C4×C197D411yescharacteristic
38C22×C4×C197D411yescharacteristic
39C22×C4×C197C2311yescharacteristic
40C2×C8×C197—22no—
41C4⋊C4×C197—22no—
42C22×D197—22no—
43Dic394—24no—
44Dic394—42no—
45Dic394—42no—
46D788—42no—
47D788—42no—
48D788—81no—
49C22⋊D197—42no—
50C22⋊D197—42no—
51C2×Dic197—22no—
52C2×Dic197—22no—
53C2×Dic197—22no—
54C2×Dic197—41no—
55C2×Dic197—41no—
56C2×Dic197—41no—
57C2×Dic197—41no—
58C4×D197—44no—
59C23×C197C2×D411yescharacteristic
60C2×C4×C197—21no—
61C2×C4×C197—21no—
62C2×C4×C197—21no—
63C2×C4×C197—41no—
64C2×C4×C197—41no—
65C2×C4×C197SD1612yescomplemented
66C2×C4×C197C2×D412yes—
67C8×C197—42no—
68C22×C197—21no—
69C22×C197—21no—
70C22×C197#12011yescharacteristic
71C22×C197C2×SD1611yescharacteristic
72C22×C197AGL(1,ℤ/8)11yescharacteristic
73C2×D197—42no—
74C2×D197—42no—
75C2×D197—42no—
76C4×C197—22no—
77C4×C197—22no—
78C4×C197—41no—
79Dic197—42no—
80Dic197—42no—
81Dic197—42no—
82C2×C197—21no—
83C2×C197—21no—
84C2×C197#45011yescharacteristic
85C2×C197#45611yescharacteristic
86C2×C197B(4,𝔽2)11yescharacteristic
87D197—82no—
88C197#160411yescharacteristic, complemented, Sylow
89#1604—1971nomaximal, Sylow
90#537—1971no—
91#448—1972no—
92#395—1971no—
93#327—1972no—
94#324D19711yescharacteristic, complemented
95C2×#41—3941no—
96C22×Q8—1971no—
97C2×SD16—3942no—
98C22⋊Q8—3942no—
99#121—3942no—
100#121—3941no—
101#121—1972no—
102C2×C4⋊C4—3941no—
103C2×C4⋊C4C2×D19711yescharacteristic
104#103—3942no—
105C22⋊C8C2×D19712yes—
106#95—3941no—
107#95—1971no—
108#41—7882no—
109#41—7882no—
110C2×Q8—7882no—
111C2×Q8—3941no—
112C2×Q8—3942no—
113C2×D4—3941no—
114C2×D4—3941no—
115C2×D4—3942no—
116C2×D4—1972no—
117C22×C4—3942no—
118C22×C4—3941no—
119C22×C4—3941no—
120C22×C4D78811yescharacteristic
121C22×C4D78811yescharacteristic
122C22×C4C22×D19711yescharacteristic
123SD16—7884no—
124C2×C8—22no—
125C4⋊C4—22no—
126C4⋊C4—7882no—
127C4⋊C4—3941no—
128C4⋊C4—1972no—
129C22⋊C4—3942no—
130C22⋊C4—3942no—
131C23—3942no—
132C23C2×D78811yescharacteristic
133Q8—7882no—
134Q8—7882no—
135Q8—3944no—
136D4—7882no—
137D4—7882no—
138D4—7882no—
139D4—7882no—
140D4—15761no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—7884no—
147C2×C4—7881no—
148C2×C4—7881no—
149C2×C4—7881no—
150C2×C4—7881no—
151C2×C4—3942no—
152C2×C4—3942no—
153C2×C4—3942no—
154C2×C4#1569134712yescomplemented
155C2×C4D4×D19712yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—7882no—
160C22—7882no—
161C22—7882no—
162C22#2026329611yescharacteristic
163C22C2×#1569134711yescharacteristic
164C22#2026335411yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—7882no—
169C4—7882no—
170C4—7882no—
171C2—21no—
172C2—21no—
173C2—15762no—
174C2#2716123211yescharacteristic
175C2#2716126411yescharacteristic
176C2#2716126711yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#324×C197C2
socleC22×C197#120
prime radical#27161660C2
hypercenter#324D197
hypocenterC197#1604
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #27161220C2semidirect
3. #27160999C2semidirect
4. #27160814C2semidirect
5. #324×C197C2semidirect
65. C2×C4×C197SD16semidirect
88. C197#1604semidirect
94. #324D197semidirect
154. C2×C4#15691347semidirect

Maximal subgroups

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

subgroupindex
#271616602
#271612202
#271609992
#271608142
#324×C1972
#1604197