Finite Group Databasebeta

← #38741829

Subgroups — #38741829

Order 25216.

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

normal subgroups only

This group has 314 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#27161714C211yesmaximal, characteristic
2#27161451C211yesmaximal, characteristic, complemented
3#27161370C212yesmaximal, complemented
4#27160806C212yesmaximal, complemented
5#331×C197C211yesmaximal, characteristic, complemented
6C2×D4×D197—22no—
7C2×#15691346—22no—
8#20263326C2211yescharacteristic, complemented
9#20263297—22no—
10#20263294—22no—
11C4×D788C2212yes—
12#20263258—22no—
13#20263248—22no—
14C4×#13096802C2211yescharacteristic, complemented
15#127×C197C2211yescharacteristic
16C4⋊C8×C197C2212yescomplemented
17C23×D197—22no—
18C2×Dic394—21no—
19C2×D788D412yes—
20C2×C22⋊D197—24no—
21D4×D197—44no—
22D4×D197—44no—
23C2×C4×D197—22no—
24#15691346—44no—
25C2×#13096802—21no—
26C4⋊Dic197—22no—
27C4⋊Dic197D412yes—
28C22⋊Dic197—22no—
29#15691335—22no—
30C2×D4×C197—22no—
31C2×D4×C197D412yes—
32C2×C8×C197—22no—
33C42×C197C2311yescharacteristic
34C22×D197—22no—
35C22×D197—48no—
36C22×D197—82no—
37Dic394—42no—
38D788—22no—
39D788SD1614yescomplemented
40C22⋊D197—48no—
41C2×Dic197—22no—
42C2×Dic197—41no—
43C4×D197—42no—
44C4×D197—42no—
45#13096802—22no—
46C23×C197—22no—
47D4×C197—42no—
48D4×C197—42no—
49D4×C197—42no—
50C2×C4×C197C2×D412yes—
51C2×C4×C197C2×D411yescharacteristic
52C8×C197—42no—
53C22×C197—44no—
54C22×C197—42no—
55C22×C197#12011yescharacteristic
56C2×D197—24no—
57C2×D197—42no—
58C2×D197—42no—
59C2×D197—88no—
60C4×C197—21no—
61C4×C197C2×SD1612yes—
62C4×C197AGL(1,ℤ/8)12yes—
63Dic197—42no—
64Dic197—81no—
65C2×C197—82no—
66C2×C197#45012yes—
67C2×C197#45311yescharacteristic
68D197—44no—
69C197#162911yescharacteristic, complemented, Sylow
70#1629—1971nomaximal, Sylow
71D42—1971no—
72#492—1971no—
73#476—1972no—
74#331D19711yescharacteristic, complemented
75D4⋊C8—1972no—
76C22×D4—3942no—
77C2×SD16—3942no—
78C4⋊Q8—1971no—
79#127C2×D19711yescharacteristic
80#121—3942no—
81#120—3942no—
82C4×D4—1972no—
83#106—3942no—
84C4⋊C8C2×D19712yescomplemented
85#102—3942no—
86C4×C8—1971no—
87C24—3942no—
88C2×Q8—3941no—
89C2×D4—22no—
90C2×D4—7884no—
91C2×D4—7884no—
92C2×D4—3944no—
93C2×D4—1972no—
94C2×D4D78812yes—
95C22×C4—3942no—
96SD16—7884no—
97C2×C8—22no—
98C2×C8—3941no—
99C4⋊C4—3942no—
100C4⋊C4—1972no—
101C22⋊C4—3942no—
102C22⋊C4—3942no—
103C42C22×D19711yescharacteristic
104C23—22no—
105C23—7888no—
106C23—15762no—
107C23—3942no—
108Q8—7882no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—7888no—
113D4—3942no—
114D4—1974no—
115C2×C4—7882no—
116C2×C4—7882no—
117C2×C4—7881no—
118C2×C4—3942no—
119C2×C4D4×D19712yes—
120C2×C4C2×D78811yescharacteristic
121C8—42no—
122C8—3942no—
123C22—44no—
124C22—42no—
125C22—7882no—
126C22—7882no—
127C22—15768no—
128C22—3944no—
129C22#2026329611yescharacteristic
130C4—21no—
131C4—7882no—
132C4—15761no—
133C4SD16×D19712yes—
134C4#2026335012yes—
135C2—82no—
136C2—7884no—
137C2#2716123112yes—
138C2#2716124911yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#331×C197C2
socleC22×C197#120
prime radical#27161714C2
hypercenter#331D197
hypocenterC197#1629
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27161451C2semidirect
3. #27161370C2semidirect
4. #27160806C2semidirect
5. #331×C197C2semidirect
8. #20263326C22semidirect
14. C4×#13096802C22semidirect
16. C4⋊C8×C197C22semidirect
39. D788SD16semidirect
69. C197#1629semidirect
74. #331D197semidirect
84. C4⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617142
#271614512
#271613702
#271608062
#331×C1972
#1629197