Finite Group Databasebeta

← #38741885

Subgroups — #38741885

Order 25216.

194 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 53 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#27161803C211yesmaximal, characteristic
2#27161431C211yesmaximal, characteristic, complemented
3#27161375C211yesmaximal, characteristic, complemented
4#27161370C211yesmaximal, characteristic
5#27160809C211yesmaximal, characteristic, complemented
6#27160806C211yesmaximal, characteristic
7#332×C197C211yesmaximal, characteristic, complemented
8#20263326C2211yescharacteristic
9#20263321—22no—
10#20263315—21no—
11#20263306—21no—
12Q8×Dic197—21no—
13C4×Dic394C2211yescharacteristic
14C4×D788C2211yescharacteristic
15#20263284—21no—
16#20263260—21no—
17#20263258—21no—
18#20263251—21no—
19#20263249—22no—
20#20263248—21no—
21C4×#13096802C2211yescharacteristic
22C4⋊Q8×C197C2211yescharacteristic
23C4⋊C8×C197C2211yescharacteristic
24C4⋊C8×C197C2211yescharacteristic
25C2×Dic394—21no—
26C2×Dic394D411yescharacteristic
27C2×D788D411yescharacteristic
28C2×C4×D197—21no—
29C2×#13096802—21no—
30C4⋊Dic197—22no—
31C4⋊Dic197—21no—
32C4⋊Dic197—21no—
33C4⋊Dic197D411yescharacteristic
34C4⋊Dic197D411yescharacteristic
35#15691337—21no—
36#15691335—22no—
37#15691335—22no—
38#15691335—21no—
39C4×Dic197—22no—
40C4×Dic197—21no—
41C2×Q8×C197—21no—
42C2×C8×C197—21no—
43C2×C8×C197—21no—
44C4⋊C4×C197—21no—
45C4⋊C4×C197D412yes—
46C42×C197C2311yescharacteristic
47C22×D197—21no—
48Dic394—21no—
49Dic394—21no—
50Dic394—42no—
51D788—21no—
52D788—21no—
53C2×Dic197—21no—
54C2×Dic197—21no—
55C2×Dic197—21no—
56C2×Dic197—41no—
57C4×D197—41no—
58C4×D197—41no—
59#13096802—41no—
60Q8×C197—41no—
61Q8×C197—41no—
62C2×C4×C197—22no—
63C2×C4×C197C2×D411yescharacteristic
64C2×C4×C197C2×D411yescharacteristic
65C2×C4×C197C2×D411yescharacteristic
66C8×C197—41no—
67C8×C197—41no—
68C22×C197#12011yescharacteristic
69C2×D197—41no—
70C2×D197—41no—
71C2×D197—41no—
72C4×C197—21no—
73C4×C197—42no—
74C4×C197#13511yescharacteristic
75C4×C197#13511yescharacteristic
76C4×C197#13711yescharacteristic
77C4×C197#13711yescharacteristic
78Dic197—41no—
79Dic197—41no—
80Dic197—41no—
81Dic197—81no—
82C2×C197#45211yescharacteristic
83C2×C197#45211yescharacteristic
84C2×C197#45411yescharacteristic
85D197—81no—
86C197#163811yescharacteristic, complemented, Sylow
87#1638—1971nomaximal, Sylow
88#556—1971no—
89#487—1971no—
90#477—1971no—
91#476—1971no—
92#332D19711yescharacteristic, complemented
93Q8⋊C8—1971no—
94D4⋊C8—1971no—
95C4⋊Q8—1971no—
96C4⋊Q8C2×D19711yescharacteristic
97#126—3942no—
98#124—3941no—
99C22⋊Q8—3941no—
100C4×Q8—3941no—
101C4×Q8—1971no—
102C4×D4—1971no—
103#117—3941no—
104#107—3941no—
105#106—3941no—
106C4⋊C8C2×D19711yescharacteristic
107C4⋊C8C2×D19711yescharacteristic
108#103—3941no—
109#103—3942no—
110#102—3941no—
111C4×C8—1971no—
112C2×Q8—21no—
113C2×Q8—3941no—
114C2×Q8—1971no—
115C2×D4—1971no—
116C22×C4—3941no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—3941no—
120C4⋊C4—21no—
121C4⋊C4—3942no—
122C4⋊C4—3941no—
123C4⋊C4—3941no—
124C4⋊C4—3941no—
125C4⋊C4—1971no—
126C4⋊C4—1971no—
127C4⋊C4D78812yes—
128C22⋊C4—3942no—
129C22⋊C4—3942no—
130C22⋊C4—3941no—
131C42—3942no—
132C42—3941no—
133C42C22×D19711yescharacteristic
134C23—3941no—
135Q8—41no—
136Q8—41no—
137Q8—7882no—
138Q8—3941no—
139Q8—3941no—
140D4—3941no—
141D4—3941no—
142C2×C4—22no—
143C2×C4—7881no—
144C2×C4—7881no—
145C2×C4—7881no—
146C2×C4—3941no—
147C2×C4—3941no—
148C2×C4—3941no—
149C2×C4D4×D19711yescharacteristic
150C2×C4D4×D19711yescharacteristic
151C2×C4C2×D78811yescharacteristic
152C8—41no—
153C8—41no—
154C8—7881no—
155C22—7881no—
156C22—7881no—
157C22—7881no—
158C22#2026329611yescharacteristic
159C4—21no—
160C4—42no—
161C4—7881no—
162C4—7881no—
163C4—7881no—
164C4—15761no—
165C4#2026334511yescharacteristic
166C4#2026334711yescharacteristic
167C4#2026335511yescharacteristic
168C4#2026335811yescharacteristic
169C2—15761no—
170C2#2716124211yescharacteristic
171C2#2716124411yescharacteristic
172C2#2716125411yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#332×C197C2
socleC22×C197#120
prime radicalC2×D788D4
hypercenter#332D197
hypocenterC197#1638
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #27161431C2semidirect
3. #27161375C2semidirect
5. #27160809C2semidirect
7. #332×C197C2semidirect
86. C197#1638semidirect
92. #332D197semidirect

Maximal subgroups

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

subgroupindex
#271618032
#271614312
#271613752
#271613702
#271608092
#271608062
#332×C1972
#1638197