Finite Group Databasebeta

← #38741289

Subgroups — #38741289

Order 25216.

218 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 77 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#27161132C211yesmaximal, characteristic, complemented
2#27160947C211yesmaximal, characteristic, complemented
3#27160816C212yesmaximal, complemented
4#27160813C212yesmaximal
5#529×C197C211yesmaximal, characteristic
6C4⋊#13096802—21no—
7C22⋊#13096802C2212yes—
8C4×#13096802—21no—
9#20263233—21no—
10#20263233—21no—
11#20263233C2212yes—
12#126×C197—21no—
13#125×C197C411yescharacteristic, complemented
14#124×C197C411yescharacteristic, complemented
15#123×C197—21no—
16C22⋊Q8×C197C2211yescharacteristic
17#121×C197C2211yescharacteristic
18C4×Q8×C197C411yescharacteristic, complemented
19C4×D4×C197—21no—
20C4×D4×C197C411yescharacteristic, complemented
21#117×C197—21no—
22C2×C42×C197C2211yescharacteristic
23C22×Dic197—22no—
24C2×#13096802—22no—
25C2×Q8×C197C2×C411yescharacteristic
26C2×D4×C197—21no—
27C2×D4×C197C2×C411yescharacteristic
28C22×C4×C197—21no—
29C22×C4×C197D411yescharacteristic
30C22×C4×C197D411yescharacteristic
31C22×C4×C197C2311yescharacteristic
32C4⋊C4×C197—21no—
33C4⋊C4×C197—21no—
34C4⋊C4×C197—21no—
35C4⋊C4×C197—21no—
36C4⋊C4×C197C2×C411yescharacteristic
37C4⋊C4×C197C2×C411yescharacteristic
38C22⋊C4×C197—21no—
39C22⋊C4×C197—21no—
40C22⋊C4×C197—21no—
41C22⋊C4×C197—21no—
42C22⋊C4×C197—21no—
43C42×C197—21no—
44C42×C197C2×C411yescharacteristic
45C42×C197C2×C411yescharacteristic
46C42×C197D411yescharacteristic
47C42×C197D411yescharacteristic
48C2×Dic197—42no—
49C2×Dic197—42no—
50C2×Dic197—42no—
51#13096802—42no—
52C23×C197—21no—
53C23×C197C2×D411yescharacteristic
54Q8×C197—41no—
55D4×C197—41no—
56D4×C197—42no—
57C2×C4×C197—22no—
58C2×C4×C197—21no—
59C2×C4×C197—21no—
60C2×C4×C197—21no—
61C2×C4×C197—41no—
62C2×C4×C197—41no—
63C2×C4×C197—42no—
64C2×C4×C197C22⋊C411yescharacteristic
65C2×C4×C197C22⋊C411yescharacteristic
66C2×C4×C197C22⋊C411yescharacteristic
67C2×C4×C197C22⋊C411yescharacteristic
68C2×C4×C197C22×C411yescharacteristic
69C2×C4×C197C2×D411yescharacteristic
70C22×C197—21no—
71C22×C197—21no—
72C22×C197—41no—
73C22×C197—41no—
74C22×C197—41no—
75C22×C197C2×C22⋊C411yescharacteristic
76C22×C197#13512yes—
77C4×C197—21no—
78C4×C197—21no—
79C4×C197—21no—
80C4×C197—21no—
81C4×C197—21no—
82C4×C197—21no—
83C4×C197—41no—
84C4×C197—41no—
85C4×C197—41no—
86Dic197—82no—
87C2×C197—22no—
88C2×C197—81no—
89C2×C197#41011yescharacteristic
90C2×C197#41611yescharacteristic
91C2×C197#42111yescharacteristic
92C197#152111yescharacteristic, complemented, Sylow
93#1521—1971nomaximal, Sylow
94#529D19711yescharacteristic
95#431—1971no—
96#383—1971no—
97#328—1972no—
98#327—1972no—
99#126—21no—
100#125Dic19711yescharacteristic, complemented
101#124Dic19711yescharacteristic, complemented
102#123—21no—
103C22⋊Q8C2×D19711yescharacteristic
104#121C2×D19711yescharacteristic
105C4×Q8Dic19711yescharacteristic, complemented
106C4×D4—21no—
107C4×D4Dic19711yescharacteristic, complemented
108#117—21no—
109C2×C42C2×D19711yescharacteristic
110C4⋊C8—3941no—
111C22⋊C8—1972no—
112C4×C8—3941no—
113#95—3941no—
114#95—3941no—
115#95—1972no—
116C2×Q8C2×Dic19711yescharacteristic
117C2×D4—21no—
118C2×D4C2×Dic19711yescharacteristic
119C22×C4—21no—
120C22×C4—3942no—
121C22×C4C22⋊D19711yescharacteristic
122C22×C4C22⋊D19711yescharacteristic
123C22×C4C22×D19711yescharacteristic
124C2×C8—3942no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4C2×Dic19711yescharacteristic
130C4⋊C4C2×Dic19711yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C42—21no—
137C42C2×Dic19711yescharacteristic
138C42C2×Dic19711yescharacteristic
139C42C22⋊D19711yescharacteristic
140C42C22⋊D19711yescharacteristic
141C23—21no—
142C23C2×C22⋊D19711yescharacteristic
143Q8—41no—
144D4—41no—
145D4—42no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—7882no—
154C2×C4—7882no—
155C2×C4—7882no—
156C2×C4C22⋊Dic19711yescharacteristic
157C2×C4C22⋊Dic19711yescharacteristic
158C2×C4C22⋊Dic19711yescharacteristic
159C2×C4C22⋊Dic19711yescharacteristic
160C2×C4C22×Dic19711yescharacteristic
161C2×C4C2×C22⋊D19711yescharacteristic
162C8—7882no—
163C22—21no—
164C22—21no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22C2×C22⋊Dic19711yescharacteristic
169C22#2026334612yes—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—41no—
177C4—41no—
178C4—41no—
179C4—15762no—
180C2—22no—
181C2—81no—
182C2#2716105311yescharacteristic
183C2#2716107511yescharacteristic
184C2#2716109611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic197
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#529×C197C2
socleC22×C197C2×C22⋊C4
prime radical#121×C197C22
cosocleC22×C4×C197C23
hypercenter#529D197
hypocenterC197#1521
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #27161132C2semidirect
2. #27160947C2semidirect
3. #27160816C2semidirect
13. #125×C197C4semidirect
14. #124×C197C4semidirect
18. C4×Q8×C197C4semidirect
20. C4×D4×C197C4semidirect
92. C197#1521semidirect
100. #125Dic197semidirect
101. #124Dic197semidirect
105. C4×Q8Dic197semidirect
107. C4×D4Dic197semidirect

Maximal subgroups

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

subgroupindex
#271611322
#271609472
#271608162
#271608132
#529×C1972
#1521197