Finite Group Databasebeta

← #38741305

Subgroups — #38741305

Order 25216.

226 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 63 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
2#27161031C212yesmaximal
3#27160821C211yesmaximal, characteristic, complemented
4#27160820C212yesmaximal, complemented
5#330×C197C211yesmaximal, characteristic, complemented
6C2×#15691341—22no—
7#20263325C412yes—
8#20263318C2211yescharacteristic
9#20263304—22no—
10#20263291—22no—
11#20263284C412yes—
12C42×D197C2211yescharacteristic
13#20263237C2212yes—
14C8×Dic197—22no—
15#125×C197C2211yescharacteristic
16#97×C197C2212yes—
17C2×Dic394—22no—
18C2×C4×D197C2×C411yescharacteristic
19C2×C4×D197D412yes—
20#15691341—28no—
21C2×#13096802—22no—
22C4⋊Dic197C2×C412yes—
23#15691337—22no—
24#15691337—22no—
25#15691335—22no—
26C4×Dic197—22no—
27C4×Dic197C2×C411yescharacteristic
28C4×Dic197D412yes—
29C2×C8×C197—22no—
30C4⋊C4×C197—22no—
31C4⋊C4×C197C2×C412yes—
32C42×C197C2311yescharacteristic
33C22×D197C22⋊C411yescharacteristic
34Dic394—44no—
35C2×Dic197—22no—
36C2×Dic197C22⋊C411yescharacteristic
37C2×Dic197C22⋊C412yes—
38C4×D197—22no—
39C4×D197—24no—
40#13096802—24no—
41C2×C4×C197—22no—
42C2×C4×C197C22×C411yescharacteristic
43C2×C4×C197C2×D412yes—
44C8×C197—24no—
45C22×C197C2×C22⋊C411yescharacteristic
46C2×D197—21no—
47C2×D197—22no—
48C4×C197—21no—
49C4×C197—22no—
50C4×C197—42no—
51Dic197—42no—
52Dic197#10112yes—
53Dic197#10414yes—
54C2×C197C2×#10111yescharacteristic
55C2×C197C2×#10412yes—
56D197—41no—
57C197#152611yescharacteristic, complemented, Sylow
58#1526—1971nomaximal, Sylow
59#533—1971no—
60C4×M16—1972no—
61#330—1971no—
62#330—1972no—
63#330D19711yescharacteristic, complemented
64C2×M16—3942no—
65C4⋊Q8—1972no—
66#125—1971no—
67#125C2×D19711yescharacteristic
68C22⋊Q8—3942no—
69C4×Q8—3942no—
70#117—1972no—
71C2×C42—1971no—
72#97—1972no—
73#97C2×D19712yes—
74C4×C8—3942no—
75C2×Q8—3942no—
76C22×C4—1971no—
77C22×C4—1972no—
78M16—3948no—
79C2×C8—22no—
80C2×C8—3942no—
81C4⋊C4—22no—
82C4⋊C4—3942no—
83C4⋊C4—3942no—
84C4⋊C4—1972no—
85C4⋊C4C4×D19712yes—
86C22⋊C4—3942no—
87C42—3942no—
88C42—1971no—
89C42—1972no—
90C42C22×D19711yescharacteristic
91C23—1971no—
92Q8—7884no—
93C2×C4—22no—
94C2×C4—3944no—
95C2×C4—3942no—
96C2×C4—3942no—
97C2×C4—1971no—
98C2×C4—1972no—
99C2×C4C2×C4×D19711yescharacteristic
100C2×C4D4×D19712yes—
101C8—24no—
102C8—3944no—
103C22—3942no—
104C22—3941no—
105C22C22⋊C4×D19711yescharacteristic
106C4—22no—
107C4—21no—
108C4—42no—
109C4—7882no—
110C4—1974no—
111C4—1972no—
112C2—7881no—
113C2#101×D19711yescharacteristic
114C2#104×D19712yes—
subgroupG/H
centerC22C22⋊C4×D197
commutatorC2×C4×C197C22×C4
FrattiniC42C22×D197
Fitting#330×C197C2
socleC22×C197C2×C22⋊C4
prime radicalC22×D197C22⋊C4
cosocleC42×C197C23
hypercenter#330D197
hypocenterC197#1526
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
3. #27160821C2semidirect
4. #27160820C2semidirect
5. #330×C197C2semidirect
57. C197#1526semidirect
63. #330D197semidirect

Maximal subgroups

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

subgroupindex
#271616352
#271610312
#271608212
#271608202
#330×C1972
#1526197