Finite Group Databasebeta

← #38741444

Subgroups — #38741444

Order 25216.

254 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 67 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#27161627C211yesmaximal, characteristic
2#27161101C212yesmaximal, complemented
3#27160827C212yesmaximal, complemented
4#27160823C211yesmaximal, characteristic, complemented
5#331×C197C211yesmaximal, characteristic, complemented
6C2×#15691357C412yes—
7C2×#15691341—22no—
8#20263326C2211yescharacteristic
9#20263313C412yes—
10#20263297—22no—
11D4×Dic197—22no—
12C42×D197C2211yescharacteristic
13C4⋊#13096802C2212yes—
14#20263255—22no—
15#127×C197C2211yescharacteristic
16C4⋊C8×C197C2212yes—
17#15691357—44no—
18C2×Dic394C2×C412yes—
19C2×C22⋊D197—22no—
20C22×Dic197—22no—
21C2×C4×D197C2×C411yescharacteristic
22C2×C4×D197D412yes—
23#15691341—44no—
24C2×#13096802—22no—
25C4⋊Dic197—22no—
26C22⋊Dic197—24no—
27C4×Dic197C2×C411yescharacteristic
28C4×Dic197D412yes—
29C2×D4×C197—22no—
30C2×D4×C197C2×C412yes—
31C2×C8×C197—22no—
32C42×C197C2311yescharacteristic
33C22×D197C22⋊C411yescharacteristic
34Dic394—42no—
35C22⋊D197—44no—
36C2×Dic197—22no—
37C2×Dic197—44no—
38C2×Dic197C22⋊C412yes—
39C2×Dic197C22⋊C411yescharacteristic
40C4×D197—22no—
41C4×D197—22no—
42C4×D197—22no—
43#13096802—42no—
44C23×C197—22no—
45D4×C197—42no—
46D4×C197—42no—
47D4×C197—42no—
48C2×C4×C197C22×C411yescharacteristic
49C2×C4×C197C2×D412yes—
50C8×C197—42no—
51C22×C197—42no—
52C22×C197—44no—
53C22×C197C2×C22⋊C411yescharacteristic
54C2×D197—21no—
55C2×D197—22no—
56C4×C197—21no—
57C4×C197AGL(1,ℤ/8)12yes—
58C4×C197#13712yes—
59Dic197—22no—
60Dic197—42no—
61Dic197#10012yes—
62C2×C197—82no—
63C2×C197C2×#10011yescharacteristic
64C2×C197#41712yes—
65D197—41no—
66C197#156011yescharacteristic, complemented, Sylow
67#1560—1971nomaximal, Sylow
68#532—1971no—
69#423—1972no—
70#333—1972no—
71#331—1971no—
72#331D19711yescharacteristic, complemented
73C2×#41—1972no—
74C2×M16—3942no—
75C4⋊Q8—1971no—
76#127C2×D19711yescharacteristic
77#124—1972no—
78#121—3942no—
79C4×D4—3942no—
80C2×C42—1971no—
81C4⋊C8—3942no—
82C4⋊C8—1972no—
83C4⋊C8C2×D19712yes—
84#41—7884no—
85C2×Q8—1972no—
86C2×D4—22no—
87C2×D4—3942no—
88C2×D4C4×D19712yes—
89C22×C4—3942no—
90C22×C4—1971no—
91C22×C4—1972no—
92M16—7884no—
93C2×C8—22no—
94C2×C8—3942no—
95C4⋊C4—3942no—
96C22⋊C4—3944no—
97C42—1971no—
98C42—1972no—
99C42C22×D19711yescharacteristic
100C23—22no—
101C23—1971no—
102Q8—7882no—
103D4—42no—
104D4—42no—
105D4—42no—
106D4—7884no—
107C2×C4—7884no—
108C2×C4—3942no—
109C2×C4—3942no—
110C2×C4—3942no—
111C2×C4—3942no—
112C2×C4—1972no—
113C2×C4—1971no—
114C2×C4C2×C4×D19711yescharacteristic
115C2×C4D4×D19712yes—
116C8—42no—
117C8—7882no—
118C22—42no—
119C22—44no—
120C22—3941no—
121C22—3942no—
122C22C22⋊C4×D19711yescharacteristic
123C4—21no—
124C4—7882no—
125C4—3942no—
126C4—1972no—
127C4#2026335012yes—
128C4#2026335512yes—
129C2—82no—
130C2—7881no—
131C2#100×D19711yescharacteristic
132C2#2716107712yes—
subgroupG/H
centerC22C22⋊C4×D197
commutatorC2×C4×C197C22×C4
FrattiniC42C22×D197
Fitting#331×C197C2
socleC22×C197C2×C22⋊C4
prime radical#27161627C2
cosocleC42×C197C23
hypercenter#331D197
hypocenterC197#1560
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27161101C2semidirect
3. #27160827C2semidirect
4. #27160823C2semidirect
5. #331×C197C2semidirect
66. C197#1560semidirect
72. #331D197semidirect

Maximal subgroups

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

subgroupindex
#271616272
#271611012
#271608272
#271608232
#331×C1972
#1560197