Finite Group Databasebeta

← #38742036

Subgroups — #38742036

Order 25216.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#27161827C211yesmaximal, characteristic
2#27161449C211yesmaximal, characteristic, complemented
3#27161439C212yesmaximal, complemented
4#27160818C212yesmaximal, complemented
5#330×C197C211yesmaximal, characteristic, complemented
6#20263326C2211yescharacteristic
7#20263314C2212yes—
8#20263311—22no—
9#20263304—22no—
10#20263301—22no—
11#20263251—22no—
12#20263249—22no—
13#20263248—22no—
14#20263237C2211yescharacteristic
15#125×C197C2211yescharacteristic
16#97×C197C2212yes—
17C2×Dic394—21no—
18C2×Dic394D412yes—
19C2×D788—22no—
20C2×D788D412yes—
21C2×C4×D197—22no—
22C2×#13096802—21no—
23C4⋊Dic197—22no—
24#15691337—22no—
25#15691335—22no—
26#15691335—22no—
27#15691335—22no—
28C2×C8×C197—22no—
29C4⋊C4×C197—22no—
30C4⋊C4×C197D412yes—
31C42×C197C2311yescharacteristic
32C22×D197—22no—
33Dic394—42no—
34Dic394—42no—
35D788—42no—
36D788—82no—
37C2×Dic197—22no—
38C2×Dic197—41no—
39C4×D197—82no—
40#13096802—41no—
41C2×C4×C197—22no—
42C2×C4×C197C2×D412yes—
43C2×C4×C197C2×D411yescharacteristic
44C8×C197—42no—
45C22×C197#12011yescharacteristic
46C2×D197—42no—
47C2×D197—42no—
48C2×D197—42no—
49C4×C197—21no—
50C4×C197—22no—
51C4×C197—42no—
52Dic197—42no—
53Dic197—81no—
54C2×C197#45411yescharacteristic
55C2×C197#45512yes—
56D197—82no—
57C197#166211yescharacteristic, complemented, Sylow
58#1662—1971nomaximal, Sylow
59#562—1971no—
60#491—1971no—
61#489—1972no—
62#330D19711yescharacteristic, complemented
63#329—1972no—
64C4⋊Q8—1971no—
65#125C2×D19711yescharacteristic
66#124—1972no—
67#123—3942no—
68C22⋊Q8—3942no—
69#121—3942no—
70#103—3942no—
71#103—3942no—
72#102—3942no—
73#97—1971no—
74#97C2×D19712yes—
75C2×Q8—3941no—
76C2×Q8—1972no—
77C2×D4—3942no—
78C2×D4—1972no—
79C22×C4—3942no—
80C2×C8—22no—
81C2×C8—3941no—
82C4⋊C4—22no—
83C4⋊C4—3942no—
84C4⋊C4—3942no—
85C4⋊C4D78812yes—
86C22⋊C4—3942no—
87C22⋊C4—3942no—
88C22⋊C4—3942no—
89C42C22×D19711yescharacteristic
90C23—3942no—
91Q8—7882no—
92Q8—7882no—
93D4—7882no—
94D4—15762no—
95C2×C4—22no—
96C2×C4—7881no—
97C2×C4—15762no—
98C2×C4—3942no—
99C2×C4D4×D19712yes—
100C2×C4C2×D78811yescharacteristic
101C8—42no—
102C8—7881no—
103C22—7882no—
104C22—7882no—
105C22—7882no—
106C22#2026329611yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—7882no—
111C4—15761no—
112C2—15762no—
113C2#2716125411yescharacteristic
114C2#2716125812yes—
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#330×C197C2
socleC22×C197#120
prime radical#27161827C2
hypercenter#330D197
hypocenterC197#1662
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #27161449C2semidirect
3. #27161439C2semidirect
4. #27160818C2semidirect
5. #330×C197C2semidirect
57. C197#1662semidirect
62. #330D197semidirect

Maximal subgroups

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

subgroupindex
#271618272
#271614492
#271614392
#271608182
#330×C1972
#1662197