Finite Group Databasebeta

← #27161270

Subgroups — #27161270

Order 12608.

160 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 39 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#20263371C211yesmaximal, characteristic
2#20263310C212yesmaximal, complemented
3#20263307C211yesmaximal, characteristic, complemented
4#20263240C212yesmaximal, complemented
5#99×C197C211yesmaximal, characteristic, complemented
6#15691358—21no—
7#15691357—22no—
8C2×C22⋊D197—21no—
9C2×C22⋊D197C2212yes—
10D4×D197—22no—
11C22×Dic197C2211yescharacteristic, complemented
12C4⋊Dic197—22no—
13#15691337—21no—
14C22⋊Dic197—21no—
15C22⋊Dic197C2211yescharacteristic, complemented
16#15691335—22no—
17C2×D4×C197C2211yescharacteristic
18C22⋊C4×C197C2212yescomplemented
19C22×D197—21no—
20C22×D197D412yes—
21Dic394—21no—
22D788—21no—
23C22⋊D197—22no—
24C22⋊D197—22no—
25C22⋊D197—22no—
26C2×Dic197—21no—
27C2×Dic197—22no—
28C2×Dic197—21no—
29C2×Dic197—21no—
30C2×Dic197D412yes—
31C4×D197—22no—
32C23×C197D411yescharacteristic
33C23×C197C2311yescharacteristic
34D4×C197—22no—
35C2×C4×C197—22no—
36C2×C4×C197D411yescharacteristic
37C22×C197—21no—
38C22×C197—41no—
39C22×C197—41no—
40C22×C197C2×D412yes—
41C22×C197C2×D411yescharacteristic
42C2×D197—22no—
43C2×D197—42no—
44C2×D197—81no—
45C4×C197—21no—
46C4×C197—42no—
47Dic197—22no—
48Dic197—41no—
49Dic197—41no—
50C2×C197—21no—
51C2×C197—22no—
52C2×C197—41no—
53C2×C197#12011yescharacteristic
54D197—42no—
55C197#45811yescharacteristic, complemented, Sylow
56#458—1971nomaximal, Sylow
57H(5,𝔽2)—1971no—
58#123—1971no—
59#123—1972no—
60#99—1972no—
61#99D19711yescharacteristic, complemented
62#41—3942no—
63#41—3941no—
64C2×D4—3942no—
65C2×D4—3941no—
66C2×D4—1972no—
67C2×D4C2×D19711yescharacteristic
68C22×C4—1971no—
69C4⋊C4—3942no—
70C4⋊C4—3941no—
71C22⋊C4—3942no—
72C22⋊C4—3941no—
73C22⋊C4—1971no—
74C22⋊C4C2×D19712yescomplemented
75C23—3941no—
76C23—1972no—
77C23D78811yescharacteristic
78C23C22×D19711yescharacteristic
79Q8—3941no—
80D4—22no—
81D4—3941no—
82D4—3942no—
83D4—3942no—
84D4—3942no—
85C2×C4—22no—
86C2×C4—3942no—
87C2×C4—3942no—
88C2×C4—3941no—
89C2×C4—3941no—
90C2×C4—3941no—
91C2×C4—1972no—
92C2×C4D78811yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—7882no—
97C22—15761no—
98C22—3942no—
99C22D4×D19712yes—
100C22C2×D78811yescharacteristic
101C4—21no—
102C4—42no—
103C4—7881no—
104C4—7881no—
105C4—3942no—
106C2—22no—
107C2—21no—
108C2—41no—
109C2—7882no—
110C2#2026329611yescharacteristic
subgroupG/H
centerC2#20263296
commutator, cosocleC23×C197C23
FrattiniC23C22×D197
Fitting#99×C197C2
socleC2×C197#120
prime radical#20263371C2
hypercenter#99D197
hypocenterC197#458
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
2. #20263310C2semidirect
3. #20263307C2semidirect
4. #20263240C2semidirect
5. #99×C197C2semidirect
11. C22×Dic197C22semidirect
15. C22⋊Dic197C22semidirect
18. C22⋊C4×C197C22semidirect
55. C197#458semidirect
61. #99D197semidirect
74. C22⋊C4C2×D197semidirect

Maximal subgroups

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

subgroupindex
#202633712
#202633102
#202633072
#202632402
#99×C1972
#458197