Finite Group Databasebeta

← #38741302

Subgroups — #38741302

Order 25216.

270 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 63 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#27161628C211yesmaximal, characteristic
2#27161031C212yesmaximal, complemented
3#27160819C211yesmaximal, characteristic, complemented
4#27160818C212yesmaximal, complemented
5#329×C197C211yesmaximal, characteristic, complemented
6C2×#15691359C411yescharacteristic
7C2×#15691357C411yescharacteristic
8C2×#15691341—22no—
9#20263324C411yescharacteristic
10#20263322C411yescharacteristic
11#20263314C2211yescharacteristic
12#20263299—22no—
13#20263286—22no—
14C42×D197C2211yescharacteristic
15#20263237C2212yes—
16C8×Dic197—22no—
17#124×C197C2211yescharacteristic
18#97×C197C2212yes—
19#15691359—42no—
20#15691357—42no—
21C2×Dic394C2×C411yescharacteristic
22C2×D788—21no—
23C2×D788C2×C411yescharacteristic
24C2×C22⋊D197—21no—
25C2×C22⋊D197—22no—
26C22×Dic197—21no—
27C2×C4×D197—21no—
28C2×C4×D197C2×C411yescharacteristic
29C2×C4×D197D412yes—
30#15691341—28no—
31C2×#13096802—22no—
32#15691337—22no—
33#15691335—22no—
34C4×Dic197C2×C411yescharacteristic
35C4×Dic197D412yes—
36C2×Q8×C197C2×C411yescharacteristic
37C2×D4×C197C2×C411yescharacteristic
38C2×C8×C197—22no—
39C22⋊C4×C197—22no—
40C42×C197C2311yescharacteristic
41C22×D197—21no—
42C22×D197C22⋊C411yescharacteristic
43Dic394—41no—
44D788—41no—
45D788—42no—
46C22⋊D197—42no—
47C22⋊D197—44no—
48C2×Dic197—21no—
49C2×Dic197—42no—
50C2×Dic197C22⋊C411yescharacteristic
51C2×Dic197C22⋊C412yes—
52C4×D197—22no—
53C4×D197—24no—
54C4×D197—42no—
55#13096802—24no—
56C23×C197—21no—
57Q8×C197—41no—
58D4×C197—41no—
59C2×C4×C197—21no—
60C2×C4×C197C22×C411yescharacteristic
61C2×C4×C197C2×D412yes—
62C8×C197—24no—
63C22×C197—41no—
64C22×C197—42no—
65C22×C197C2×C22⋊C411yescharacteristic
66C2×D197—21no—
67C2×D197—22no—
68C2×D197—41no—
69C2×D197—42no—
70C4×C197—21no—
71C4×C197—22no—
72C4×C197—41no—
73Dic197—41no—
74Dic197#10012yes—
75Dic197#10414yes—
76C2×C197—81no—
77C2×C197C2×#10011yescharacteristic
78C2×C197C2×#10412yes—
79D197—41no—
80D197—81no—
81C197#152511yescharacteristic, complemented, Sylow
82#1525—1971nomaximal, Sylow
83#532—1971no—
84C4×M16—1972no—
85#329—1971no—
86#329—1972no—
87#329D19711yescharacteristic, complemented
88C2×#41—1971no—
89C2×#41—1971no—
90C2×M16—3942no—
91C4⋊Q8—1971no—
92#127—1971no—
93#124—1971no—
94#124C2×D19711yescharacteristic
95#121—3942no—
96C4×D4—3942no—
97C2×C42—1971no—
98#97—1972no—
99#97C2×D19712yes—
100C4×C8—3942no—
101#41—7882no—
102#41—7882no—
103C2×Q8—1971no—
104C2×Q8C4×D19711yescharacteristic
105C2×D4—3941no—
106C2×D4—3941no—
107C2×D4—3942no—
108C2×D4—1971no—
109C2×D4C4×D19711yescharacteristic
110C22×C4—3941no—
111C22×C4—3941no—
112C22×C4—1971no—
113C22×C4—1972no—
114M16—3948no—
115C2×C8—22no—
116C2×C8—3942no—
117C4⋊C4—3942no—
118C22⋊C4—22no—
119C22⋊C4—3942no—
120C42—1971no—
121C42—1972no—
122C42C22×D19711yescharacteristic
123C23—21no—
124C23—3941no—
125C23—1971no—
126Q8—41no—
127Q8—7881no—
128D4—41no—
129D4—7881no—
130D4—7882no—
131D4—7882no—
132D4—7884no—
133C2×C4—21no—
134C2×C4—7882no—
135C2×C4—7882no—
136C2×C4—3944no—
137C2×C4—3942no—
138C2×C4—3941no—
139C2×C4—1972no—
140C2×C4—1971no—
141C2×C4C2×C4×D19711yescharacteristic
142C2×C4D4×D19712yes—
143C8—24no—
144C8—3944no—
145C22—42no—
146C22—41no—
147C22—7882no—
148C22—7881no—
149C22—3942no—
150C22—3941no—
151C22C22⋊C4×D19711yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—7881no—
156C4—1974no—
157C4—1972no—
158C2—81no—
159C2—7881no—
160C2—15761no—
161C2#100×D19711yescharacteristic
162C2#104×D19712yes—
subgroupG/H
centerC22C22⋊C4×D197
commutatorC2×C4×C197C22×C4
FrattiniC42C22×D197
Fitting#329×C197C2
socleC22×C197C2×C22⋊C4
prime radical#27161628C2
cosocleC42×C197C23
hypercenter#329D197
hypocenterC197#1525
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. #27161031C2semidirect
3. #27160819C2semidirect
4. #27160818C2semidirect
5. #329×C197C2semidirect
81. C197#1525semidirect
87. #329D197semidirect

Maximal subgroups

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

subgroupindex
#271616282
#271610312
#271608192
#271608182
#329×C1972
#1525197