Finite Group Databasebeta

← #38742025

Subgroups — #38742025

Order 25216.

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

normal subgroups only

This group has 196 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#27161489C211yesmaximal, characteristic, complemented
3#27161488C211yesmaximal, characteristic, complemented
4#27161473C211yesmaximal, characteristic
5#27160820C211yesmaximal, characteristic, complemented
6#27160818C211yesmaximal, characteristic
7#330×C197C211yesmaximal, characteristic, complemented
8C2×Dic788—21no—
9C2×#15691347—21no—
10#20263326C2211yescharacteristic
11#20263325—21no—
12#20263321—21no—
13#20263321—21no—
14#20263318C2211yescharacteristic
15#20263314C2211yescharacteristic
16#20263311—21no—
17#20263306—21no—
18#20263304—21no—
19#20263260—21no—
20#20263259—21no—
21#20263258—21no—
22#20263257—21no—
23#20263237C2211yescharacteristic
24#125×C197C2211yescharacteristic
25#97×C197C2211yescharacteristic
26#97×C197C2211yescharacteristic
27C2×Dic394—21no—
28C2×Dic394—21no—
29C2×Dic394D411yescharacteristic
30C2×D788D411yescharacteristic
31C2×C4×D197—21no—
32Dic788—42no—
33#15691347—42no—
34C2×#13096802—21no—
35C4⋊Dic197—21no—
36C4⋊Dic197—21no—
37C4⋊Dic197—21no—
38C4⋊Dic197D411yescharacteristic
39C4⋊Dic197D411yescharacteristic
40#15691337—21no—
41#15691337—21no—
42#15691337—21no—
43#15691335—21no—
44#15691335—21no—
45#15691335—21no—
46#15691335—21no—
47#15691335—21no—
48C4×Dic197—21no—
49C2×C8×C197—21no—
50C2×C8×C197—21no—
51C4⋊C4×C197—21no—
52C4⋊C4×C197—21no—
53C4⋊C4×C197D411yescharacteristic
54C4⋊C4×C197D411yescharacteristic
55C42×C197C2311yescharacteristic
56C22×D197—21no—
57Dic394—41no—
58Dic394—41no—
59Dic394—41no—
60Dic394—81no—
61D788—41no—
62C2×Dic197—21no—
63C2×Dic197—21no—
64C2×Dic197—21no—
65C2×Dic197—41no—
66C4×D197—81no—
67#13096802—22no—
68C2×C4×C197—21no—
69C2×C4×C197—21no—
70C2×C4×C197C2×D411yescharacteristic
71C2×C4×C197C2×D411yescharacteristic
72C2×C4×C197C2×D411yescharacteristic
73C8×C197—22no—
74C8×C197—22no—
75C22×C197#12011yescharacteristic
76C2×D197—41no—
77C2×D197—41no—
78C2×D197—41no—
79C4×C197—21no—
80C4×C197—21no—
81C4×C197—21no—
82C4×C197—41no—
83C4×C197—41no—
84Dic197—41no—
85Dic197—41no—
86Dic197—41no—
87Dic197—81no—
88C2×C197#45311yescharacteristic
89C2×C197#45611yescharacteristic
90C2×C197#45611yescharacteristic
91D197—81no—
92C197#166011yescharacteristic, complemented, Sylow
93#1660—1971nomaximal, Sylow
94#563—1971no—
95#501—1971no—
96#501—1971no—
97#497—1971no—
98#330—1971no—
99#330D19711yescharacteristic, complemented
100#329—1971no—
101C2×Q16—3941no—
102C2×SD16—3941no—
103C4⋊Q8—3941no—
104C4⋊Q8—1971no—
105#126—3941no—
106#126—3941no—
107#125—1971no—
108#125C2×D19711yescharacteristic
109#124—1971no—
110#123—3941no—
111C22⋊Q8—3941no—
112C22⋊Q8—3941no—
113#107—3941no—
114#107—3941no—
115#106—3941no—
116#106—3941no—
117#97—1971no—
118#97C2×D19711yescharacteristic
119#97C2×D19711yescharacteristic
120C2×Q8—3941no—
121C2×Q8—3941no—
122C2×Q8—1971no—
123C2×D4—1971no—
124C22×C4—3941no—
125Q16—7882no—
126SD16—7882no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—3941no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—3941no—
133C4⋊C4—3941no—
134C4⋊C4—3941no—
135C4⋊C4—3941no—
136C4⋊C4—3941no—
137C4⋊C4—3941no—
138C4⋊C4—1971no—
139C4⋊C4—1971no—
140C4⋊C4D78811yescharacteristic
141C4⋊C4D78811yescharacteristic
142C22⋊C4—3941no—
143C22⋊C4—3941no—
144C22⋊C4—3941no—
145C22⋊C4—3941no—
146C22⋊C4—3941no—
147C42—3941no—
148C42C22×D19711yescharacteristic
149C23—3941no—
150Q8—7881no—
151Q8—7881no—
152Q8—7881no—
153Q8—15761no—
154D4—7881no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—7881no—
158C2×C4—15761no—
159C2×C4—3941no—
160C2×C4—3941no—
161C2×C4—3941no—
162C2×C4D4×D19711yescharacteristic
163C2×C4D4×D19711yescharacteristic
164C2×C4C2×D78811yescharacteristic
165C8—22no—
166C8—22no—
167C8—3942no—
168C22—7881no—
169C22—7881no—
170C22—7881no—
171C22#2026329611yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—7881no—
178C4—7881no—
179C4—7881no—
180C4—15761no—
181C2—15761no—
182C2#2716124811yescharacteristic
183C2#2716126311yescharacteristic
184C2#2716126411yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#330×C197C2
socleC22×C197#120
prime radicalC2×D788D4
hypercenter#330D197
hypocenterC197#1660
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #27161489C2semidirect
3. #27161488C2semidirect
5. #27160820C2semidirect
7. #330×C197C2semidirect
92. C197#1660semidirect
99. #330D197semidirect

Maximal subgroups

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

subgroupindex
#271618332
#271614892
#271614882
#271614732
#271608202
#271608182
#330×C1972
#1660197