Finite Group Databasebeta

← #38742044

Subgroups — #38742044

Order 25216.

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

normal subgroups only

This group has 184 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#27161833C211yesmaximal, characteristic
2#27161447C211yesmaximal, characteristic, complemented
3#27161441C211yesmaximal, characteristic
4#27160821C211yesmaximal, characteristic, complemented
5#27160820C211yesmaximal, characteristic, complemented
6#27160818C211yesmaximal, characteristic
7#501×C197C211yesmaximal, characteristic, complemented
8#20263325—21no—
9#20263321—21no—
10#20263321—21no—
11#20263318C2211yescharacteristic
12#20263314C2211yescharacteristic
13#20263311—21no—
14#20263306—21no—
15#20263304—21no—
16#20263250—21no—
17#20263250—21no—
18#20263249—21no—
19#20263246—21no—
20#20263237C2211yescharacteristic
21#20263237C2211yescharacteristic
22C4⋊Q8×C197C2211yescharacteristic
23#125×C197C2211yescharacteristic
24#107×C197—21no—
25#106×C197—21no—
26#97×C197C2211yescharacteristic
27C2×Dic394—21no—
28C2×Dic394D411yescharacteristic
29C2×D788D411yescharacteristic
30C2×C4×D197—21no—
31C2×#13096802—21no—
32C2×#13096802—21no—
33C4⋊Dic197—21no—
34C4⋊Dic197D411yescharacteristic
35C4⋊Dic197D411yescharacteristic
36#15691337—21no—
37#15691337—21no—
38#15691337—21no—
39#15691335—21no—
40#15691335—21no—
41#15691335—21no—
42#15691335—21no—
43#15691335—21no—
44C4×Dic197—21no—
45C2×Q8×C197—21no—
46C2×C8×C197—21no—
47C4⋊C4×C197—21no—
48C4⋊C4×C197—21no—
49C4⋊C4×C197—21no—
50C4⋊C4×C197—21no—
51C4⋊C4×C197D411yescharacteristic
52C4⋊C4×C197D411yescharacteristic
53C42×C197C2311yescharacteristic
54C22×D197—21no—
55Dic394—41no—
56Dic394—81no—
57D788—41no—
58C2×Dic197—21no—
59C2×Dic197—21no—
60C2×Dic197—21no—
61C4×D197—81no—
62#13096802—41no—
63#13096802—41no—
64Q8×C197—42no—
65C2×C4×C197—21no—
66C2×C4×C197—21no—
67C2×C4×C197—41no—
68C2×C4×C197C2×D411yescharacteristic
69C2×C4×C197C2×D411yescharacteristic
70C2×C4×C197C2×D411yescharacteristic
71C8×C197—22no—
72C22×C197#12011yescharacteristic
73C2×D197—41no—
74C2×D197—41no—
75C2×D197—41no—
76C4×C197—21no—
77C4×C197—21no—
78C4×C197—21no—
79C4×C197—41no—
80C4×C197—41no—
81C4×C197—81no—
82Dic197—41no—
83Dic197—41no—
84Dic197—41no—
85C2×C197#45411yescharacteristic
86C2×C197#45511yescharacteristic
87C2×C197#45611yescharacteristic
88D197—81no—
89C197#166411yescharacteristic, complemented, Sylow
90#1664—1971nomaximal, Sylow
91#563—1971no—
92#501D19711yescharacteristic, complemented
93#491—1971no—
94#489—1971no—
95#330—1971no—
96#330—1971no—
97#329—1971no—
98C4⋊Q8—3941no—
99C4⋊Q8C2×D19711yescharacteristic
100#126—3941no—
101#126—3941no—
102#125—1971no—
103#125C2×D19711yescharacteristic
104#124—1971no—
105#123—3941no—
106C22⋊Q8—3941no—
107C22⋊Q8—3941no—
108#107—21no—
109#106—21no—
110#103—3941no—
111#103—3941no—
112#103—3941no—
113#102—3941no—
114#97—1971no—
115#97—1971no—
116#97C2×D19711yescharacteristic
117C2×Q8—21no—
118C2×Q8—3941no—
119C2×Q8—1971no—
120C2×D4—1971no—
121C22×C4—3941no—
122C2×C8—21no—
123C2×C8—3941no—
124C2×C8—3941no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—3941no—
130C4⋊C4—3941no—
131C4⋊C4—3941no—
132C4⋊C4—3941no—
133C4⋊C4—1971no—
134C4⋊C4—1971no—
135C4⋊C4C22⋊D19711yescharacteristic
136C4⋊C4C22⋊D19711yescharacteristic
137C22⋊C4—3941no—
138C22⋊C4—3941no—
139C22⋊C4—3941no—
140C22⋊C4—3941no—
141C22⋊C4—3941no—
142C42—3941no—
143C42C22×D19711yescharacteristic
144C23—3941no—
145Q8—42no—
146Q8—7881no—
147Q8—15761no—
148D4—7881no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—15761no—
153C2×C4—3941no—
154C2×C4—3941no—
155C2×C4—3941no—
156C2×C4D4×D19711yescharacteristic
157C2×C4D4×D19711yescharacteristic
158C2×C4C2×C22⋊D19711yescharacteristic
159C8—22no—
160C8—7881no—
161C8—7881no—
162C22—7881no—
163C22—7881no—
164C22—7881no—
165C22#2026329411yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—81no—
172C4—7881no—
173C4—7881no—
174C4—7881no—
175C2—15761no—
176C2#2716125011yescharacteristic
177C2#2716125511yescharacteristic
178C2#2716126111yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#501×C197C2
socleC22×C197#120
prime radicalC2×D788D4
hypercenter#501D197
hypocenterC197#1664
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #27161447C2semidirect
4. #27160821C2semidirect
5. #27160820C2semidirect
7. #501×C197C2semidirect
89. C197#1664semidirect
92. #501D197semidirect

Maximal subgroups

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

subgroupindex
#271618332
#271614472
#271614412
#271608212
#271608202
#271608182
#501×C1972
#1664197