Finite Group Databasebeta

← #38741681

Subgroups — #38741681

Order 25216.

282 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 57 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#27161217C212yesmaximal, complemented
2#27160999C211yesmaximal, characteristic, complemented
3#27160813C212yesmaximal, complemented
4#27160805C211yesmaximal, characteristic, complemented
5#537×C197C211yesmaximal, characteristic
6C22×Dic394C2211yescharacteristic
7C2×#15691346—22no—
8C2×#15691337—21no—
9#20263249—22no—
10C22⋊#13096802C2212yes—
11#20263233—21no—
12#20263233C2211yescharacteristic
13C2×#41×C197—21no—
14C22⋊Q8×C197—22no—
15#121×C197—22no—
16#121×C197—21no—
17#121×C197C2212yes—
18C2×C4⋊C4×C197C2211yescharacteristic
19C2×Dic394—22no—
20C2×Dic394—42no—
21C22×Dic197—21no—
22C22×Dic197—21no—
23#15691346—44no—
24C2×#13096802—22no—
25#15691337—42no—
26#41×C197—42no—
27#41×C197—42no—
28C2×Q8×C197—21no—
29C2×D4×C197—21no—
30C2×D4×C197—22no—
31C2×D4×C197—21no—
32C2×D4×C197D412yes—
33C22×C4×C197—22no—
34C22×C4×C197D411yescharacteristic
35C22×C4×C197D411yescharacteristic
36C22×C4×C197C2311yescharacteristic
37C4⋊C4×C197—21no—
38C4⋊C4×C197—22no—
39C4⋊C4×C197D412yes—
40C22⋊C4×C197—22no—
41C22⋊C4×C197—22no—
42Dic394—24no—
43Dic394—42no—
44C2×Dic197—22no—
45C2×Dic197—22no—
46C2×Dic197—41no—
47C2×Dic197—41no—
48C2×Dic197—41no—
49C2×Dic197—41no—
50#13096802—42no—
51C23×C197—22no—
52C23×C197C2×D411yescharacteristic
53Q8×C197—42no—
54D4×C197—42no—
55D4×C197—42no—
56D4×C197—42no—
57D4×C197—42no—
58D4×C197—81no—
59C2×C4×C197—21no—
60C2×C4×C197—21no—
61C2×C4×C197—21no—
62C2×C4×C197—22no—
63C2×C4×C197—41no—
64C2×C4×C197—44no—
65C2×C4×C197—41no—
66C2×C4×C197SD1612yescomplemented
67C2×C4×C197C2×D412yes—
68C22×C197—21no—
69C22×C197—21no—
70C22×C197—42no—
71C22×C197—42no—
72C22×C197—42no—
73C22×C197#12011yescharacteristic
74C22×C197C2×SD1611yescharacteristic
75C22×C197AGL(1,ℤ/8)11yescharacteristic
76C4×C197—22no—
77C4×C197—22no—
78C4×C197—42no—
79C4×C197—41no—
80Dic197—42no—
81Dic197—42no—
82C2×C197—21no—
83C2×C197—21no—
84C2×C197—82no—
85C2×C197#45011yescharacteristic
86C2×C197#45611yescharacteristic
87C2×C197B(4,𝔽2)11yescharacteristic
88C197#160411yescharacteristic, complemented, Sylow
89#1604—1971nomaximal, Sylow
90#537D19711yescharacteristic
91#448—1972no—
92#395—1971no—
93#327—1972no—
94#324—1971no—
95C2×#41—21no—
96C22×Q8—1971no—
97C2×SD16—3942no—
98C22⋊Q8—22no—
99#121—22no—
100#121—21no—
101#121C2×D19712yes—
102C2×C4⋊C4—3941no—
103C2×C4⋊C4C2×D19711yescharacteristic
104#103—3942no—
105C22⋊C8—1972no—
106#95—3941no—
107#95—1971no—
108#41—42no—
109#41—42no—
110C2×Q8—21no—
111C2×Q8—7882no—
112C2×Q8—3942no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4C22⋊D19712yes—
117C22×C4—22no—
118C22×C4—3941no—
119C22×C4—3941no—
120C22×C4C22⋊D19711yescharacteristic
121C22×C4C22⋊D19711yescharacteristic
122C22×C4C22×D19711yescharacteristic
123SD16—7884no—
124C2×C8—3942no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—7882no—
128C4⋊C4C22⋊D19712yes—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C23—22no—
132C23C2×C22⋊D19711yescharacteristic
133Q8—42no—
134Q8—7882no—
135Q8—3944no—
136D4—42no—
137D4—42no—
138D4—42no—
139D4—42no—
140D4—81no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—44no—
147C2×C4—41no—
148C2×C4—7881no—
149C2×C4—7881no—
150C2×C4—7881no—
151C2×C4—7881no—
152C2×C4—3942no—
153C2×C4—3942no—
154C2×C4#1569134612yescomplemented
155C2×C4C2×C22⋊D19712yes—
156C8—7882no—
157C22—21no—
158C22—21no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22#2026329511yescharacteristic
163C22C2×#1569134611yescharacteristic
164C22#2026335211yescharacteristic
165C4—22no—
166C4—22no—
167C4—42no—
168C4—41no—
169C4—7882no—
170C4—7882no—
171C2—21no—
172C2—21no—
173C2—82no—
174C2#2716123011yescharacteristic
175C2#2716126211yescharacteristic
176C2#2716126611yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting, prime radical#537×C197C2
socleC22×C197#120
hypercenter#537D197
hypocenterC197#1604
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #27161217C2semidirect
2. #27160999C2semidirect
3. #27160813C2semidirect
4. #27160805C2semidirect
66. C2×C4×C197SD16semidirect
88. C197#1604semidirect
154. C2×C4#15691346semidirect

Maximal subgroups

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

subgroupindex
#271612172
#271609992
#271608132
#271608052
#537×C1972
#1604197