Finite Group Databasebeta

← #38742018

Subgroups — #38742018

Order 25216.

228 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 49 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#27161474C211yesmaximal, characteristic, complemented
2#27161438C212yesmaximal, complemented
3#27160819C212yesmaximal, complemented
4#27160819C211yesmaximal, characteristic, complemented
5H(3,𝔽4)×C197C211yesmaximal, characteristic
6C2×#15691348—21no—
7C2×#15691346—21no—
8#20263326C2211yescharacteristic, complemented
9#20263250—22no—
10#20263247—22no—
11#20263237C2212yescomplemented
12#20263237C2211yescharacteristic, complemented
13#124×C197—22no—
14#124×C197—21no—
15#124×C197C2212yes—
16#124×C197C2211yescharacteristic
17#120×C197—22no—
18#120×C197—21no—
19C2×Dic394—21no—
20#15691348—42no—
21#15691346—42no—
22C2×#13096802—21no—
23C2×#13096802—22no—
24C4⋊Dic197—22no—
25C24×C197—21no—
26C2×Q8×C197D412yes—
27C2×Q8×C197D411yescharacteristic
28C2×D4×C197—21no—
29C2×D4×C197—22no—
30C2×D4×C197D412yes—
31C2×D4×C197D411yescharacteristic
32C22⋊C4×C197—22no—
33C22⋊C4×C197—21no—
34C22⋊C4×C197—22no—
35C22⋊C4×C197—22no—
36C22⋊C4×C197—22no—
37C42×C197—21no—
38C42×C197C2311yescharacteristic
39Dic394—41no—
40Dic394—41no—
41C2×Dic197—41no—
42#13096802—22no—
43#13096802—42no—
44C23×C197—21no—
45C23×C197—22no—
46C23×C197—81no—
47C23×C197—82no—
48Q8×C197—42no—
49Q8×C197—41no—
50D4×C197—42no—
51D4×C197—41no—
52D4×C197—81no—
53D4×C197—82no—
54C2×C4×C197—21no—
55C2×C4×C197—22no—
56C2×C4×C197C2×D412yes—
57C2×C4×C197C2×D411yescharacteristic
58C22×C197—41no—
59C22×C197—42no—
60C22×C197—42no—
61C22×C197—42no—
62C22×C197—42no—
63C22×C197—84no—
64C22×C197#12011yescharacteristic
65C4×C197—21no—
66C4×C197—22no—
67C4×C197—41no—
68C4×C197—42no—
69Dic197—81no—
70C2×C197—81no—
71C2×C197—82no—
72C2×C197#45311yescharacteristic
73C2×C197#45512yes—
74C197#165911yescharacteristic, complemented, Sylow
75#1659—1971nomaximal, Sylow
76H(3,𝔽4)D19711yescharacteristic
77#497—1971no—
78#489—1972no—
79#329—1971no—
80#329—1972no—
81C2×Q16—3941no—
82C2×SD16—3941no—
83C4⋊Q8—1971no—
84#124—22no—
85#124—21no—
86#124C2×D19712yes—
87#124C2×D19711yescharacteristic
88#120—22no—
89#120—21no—
90#103—3942no—
91#102—3942no—
92#97—1972no—
93#97—1971no—
94C24—21no—
95C2×Q8—3941no—
96C2×Q8C22⋊D19712yes—
97C2×Q8C22⋊D19711yescharacteristic
98C2×D4—21no—
99C2×D4—22no—
100C2×D4C22⋊D19712yes—
101C2×D4C22⋊D19711yescharacteristic
102Q16—7882no—
103SD16—7882no—
104C2×C8—3942no—
105C2×C8—3941no—
106C4⋊C4—3942no—
107C22⋊C4—22no—
108C22⋊C4—21no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C42—21no—
113C42C22×D19711yescharacteristic
114C23—21no—
115C23—22no—
116C23—81no—
117C23—82no—
118Q8—41no—
119Q8—42no—
120Q8—7881no—
121Q8—7881no—
122D4—41no—
123D4—42no—
124D4—81no—
125D4—82no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—7881no—
129C2×C4C2×C22⋊D19712yes—
130C2×C4C2×C22⋊D19711yescharacteristic
131C8—7882no—
132C8—3942no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—42no—
138C22—84no—
139C22#2026329511yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—42no—
144C4—15761no—
145C2—81no—
146C2—82no—
147C2#2716124711yescharacteristic
148C2#2716125712yes—
subgroupG/H
centerC22#20263295
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting, prime radicalH(3,𝔽4)×C197C2
socleC22×C197#120
hypercenterH(3,𝔽4)D197
hypocenterC197#1659
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #27161474C2semidirect
2. #27161438C2semidirect
3. #27160819C2semidirect
4. #27160819C2semidirect
8. #20263326C22semidirect
11. #20263237C22semidirect
12. #20263237C22semidirect
74. C197#1659semidirect

Maximal subgroups

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

subgroupindex
#271614742
#271614382
#271608192
H(3,𝔽4)×C1972
#1659197