Finite Group Databasebeta

← #38741334

Subgroups — #38741334

Order 25216.

248 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 59 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#124×D197C211yesmaximal, characteristic
2#27161134C212yesmaximal, complemented
3#27160819C211yesmaximal, characteristic, complemented
4#27160818C212yesmaximal, complemented
5#329×C197C211yesmaximal, characteristic, complemented
6C2×Q8×D197C411yescharacteristic
7C2×D4×D197C411yescharacteristic
8#20263315C411yescharacteristic
9#20263314C2211yescharacteristic
10#20263313C411yescharacteristic
11#20263312—22no—
12C22⋊C4×D197—22no—
13C42×D197C2211yescharacteristic
14#20263255—22no—
15#20263238—22no—
16#20263237C2212yescomplemented
17#124×C197C2211yescharacteristic
18#97×C197C2212yescomplemented
19C23×D197—21no—
20C2×Dic394—21no—
21C2×Dic394C2×C411yescharacteristic
22Q8×D197—42no—
23C2×D788C2×C411yescharacteristic
24C2×C22⋊D197—21no—
25D4×D197—42no—
26C2×C4×D197—21no—
27C2×C4×D197C2×C411yescharacteristic
28C2×C4×D197D412yes—
29C2×#13096802—22no—
30C22⋊Dic197—22no—
31#15691335—22no—
32#15691335—22no—
33C4×Dic197C2×C411yescharacteristic
34C4×Dic197D412yes—
35C2×Q8×C197C2×C411yescharacteristic
36C2×D4×C197C2×C411yescharacteristic
37C2×C8×C197—22no—
38C22⋊C4×C197—22no—
39C42×C197C2311yescharacteristic
40C22×D197—21no—
41C22×D197—44no—
42C22×D197—42no—
43C22×D197C22⋊C411yescharacteristic
44Dic394—42no—
45Dic394—41no—
46D788—41no—
47C22⋊D197—42no—
48C2×Dic197—21no—
49C2×Dic197C22⋊C412yes—
50C2×Dic197C22⋊C411yescharacteristic
51C4×D197—22no—
52C4×D197—42no—
53C4×D197—42no—
54#13096802—42no—
55C23×C197—21no—
56Q8×C197—41no—
57D4×C197—41no—
58C2×C4×C197—21no—
59C2×C4×C197C22×C411yescharacteristic
60C2×C4×C197C2×D412yes—
61C8×C197—42no—
62C22×C197—41no—
63C22×C197—42no—
64C22×C197C2×C22⋊C411yescharacteristic
65C2×D197—22no—
66C2×D197—41no—
67C2×D197—42no—
68C2×D197—84no—
69C2×D197#10012yes—
70C4×C197—21no—
71C4×C197—22no—
72C4×C197—41no—
73Dic197—21no—
74Dic197—22no—
75Dic197—41no—
76C2×C197—81no—
77C2×C197C2×#10011yescharacteristic
78C2×C197#42112yes—
79D197—22no—
80D197—81no—
81C197#153311yescharacteristic, complemented, Sylow
82#1533—1971nomaximal, Sylow
83C2×#124—1971no—
84#432—1972no—
85#329—1971no—
86#329—1972no—
87#329D19711yescharacteristic, complemented
88C22×Q8—1971no—
89C22×D4—1971no—
90#124—3942no—
91#124—1971no—
92#124—1971no—
93#124—1971no—
94#124C2×D19711yescharacteristic
95C2×C22⋊C4—3942no—
96C2×C42—1971no—
97C4⋊C8—3942no—
98C22⋊C8—3942no—
99#97—1972no—
100#97C2×D19712yescomplemented
101C24—3941no—
102C2×Q8—7882no—
103C2×Q8—3941no—
104C2×Q8—1971no—
105C2×Q8C4×D19711yescharacteristic
106C2×D4—7882no—
107C2×D4—3941no—
108C2×D4—1971no—
109C2×D4C4×D19711yescharacteristic
110C22×C4—3941no—
111C22×C4—1972no—
112C22×C4—1971no—
113C2×C8—22no—
114C2×C8—3942no—
115C22⋊C4—22no—
116C22⋊C4—3942no—
117C22⋊C4—3942no—
118C22⋊C4—3942no—
119C42—1971no—
120C42—1972no—
121C42C22×D19711yescharacteristic
122C23—21no—
123C23—7884no—
124C23—7882no—
125C23—3941no—
126C23—1971no—
127Q8—41no—
128Q8—7881no—
129Q8—7882no—
130D4—41no—
131D4—7881no—
132D4—7882no—
133C2×C4—21no—
134C2×C4—7882no—
135C2×C4—7882no—
136C2×C4—3942no—
137C2×C4—3941no—
138C2×C4—1972no—
139C2×C4—1971no—
140C2×C4C2×C4×D19711yescharacteristic
141C2×C4D4×D19712yes—
142C8—42no—
143C8—7882no—
144C22—42no—
145C22—41no—
146C22—7882no—
147C22—7881no—
148C22—15764no—
149C22—3942no—
150C22—1972no—
151C22C22⋊C4×D19711yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—7881no—
156C4—3942no—
157C4—3941no—
158C2—81no—
159C2—15761no—
160C2—3942no—
161C2#100×D19711yescharacteristic
162C2#2716109412yes—
subgroupG/H
centerC22C22⋊C4×D197
commutatorC2×C4×C197C22×C4
FrattiniC42C22×D197
Fitting#329×C197C2
socleC22×C197C2×C22⋊C4
prime radicalC2×D4×D197C4
cosocleC42×C197C23
hypercenter#329D197
hypocenterC197#1533
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #27161134C2semidirect
3. #27160819C2semidirect
4. #27160818C2semidirect
5. #329×C197C2semidirect
16. #20263237C22semidirect
18. #97×C197C22semidirect
81. C197#1533semidirect
87. #329D197semidirect
100. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
#124×D1972
#271611342
#271608192
#271608182
#329×C1972
#1533197