Finite Group Databasebeta

← #38741260

Subgroups — #38741260

Order 25216.

252 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 69 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#27161629C211yesmaximal, characteristic
2#27160947C211yesmaximal, characteristic, complemented
3#27160814C214yesmaximal, complemented
4#432×C197C211yesmaximal, characteristic, complemented
5C2×#15691358—21no—
6#20263326C411yescharacteristic, complemented
7#20263323C411yescharacteristic, complemented
8#20263314—21no—
9#20263300—21no—
10#20263300C2212yes—
11C4×D788—21no—
12C4×D788C412yescomplemented
13#20263233—21no—
14#20263233—21no—
15#20263233C2212yes—
16C2×C42×C197C2211yescharacteristic
17C4⋊C8×C197—21no—
18C22⋊C8×C197C2212yes—
19#97×C197—21no—
20#15691358—44no—
21C2×Dic394—21no—
22C2×D788—21no—
23C2×D788—21no—
24C2×D788C2×C412yes—
25C2×C22⋊D197—22no—
26C22×Dic197—22no—
27C2×C4×D197—22no—
28C4⋊Dic197—21no—
29C4⋊Dic197C2×C412yes—
30#15691335—22no—
31#15691335—22no—
32C22×C4×C197D411yescharacteristic
33C22×C4×C197D411yescharacteristic
34C22×C4×C197C2311yescharacteristic
35C2×C8×C197—22no—
36C42×C197C2×C411yescharacteristic
37C42×C197C2×C411yescharacteristic
38C42×C197D411yescharacteristic
39C42×C197D411yescharacteristic
40C22×D197—22no—
41Dic394—42no—
42D788—42no—
43D788—42no—
44D788—42no—
45C22⋊D197—44no—
46C2×Dic197—22no—
47C2×Dic197—42no—
48C2×Dic197—42no—
49C2×Dic197—42no—
50C4×D197—44no—
51C23×C197C2×D411yescharacteristic
52C2×C4×C197—22no—
53C2×C4×C197—41no—
54C2×C4×C197—41no—
55C2×C4×C197C22⋊C411yescharacteristic
56C2×C4×C197C22⋊C411yescharacteristic
57C2×C4×C197C22⋊C411yescharacteristic
58C2×C4×C197C22⋊C411yescharacteristic
59C2×C4×C197C22×C411yescharacteristic
60C2×C4×C197C2×D411yescharacteristic
61C8×C197—42no—
62C22×C197—21no—
63C22×C197—21no—
64C22×C197C2×C22⋊C411yescharacteristic
65C22×C197AGL(1,ℤ/8)12yes—
66C2×D197—42no—
67C2×D197—42no—
68C2×D197—42no—
69C4×C197—21no—
70C4×C197—21no—
71C4×C197—21no—
72C4×C197—21no—
73C4×C197—21no—
74C4×C197—21no—
75Dic197—42no—
76Dic197—82no—
77C2×C197—22no—
78C2×C197#41011yescharacteristic
79C2×C197#41811yescharacteristic
80C2×C197#42111yescharacteristic
81D197—82no—
82C197#151411yescharacteristic, complemented, Sylow
83#1514—1971nomaximal, Sylow
84#532—1971no—
85#432D19711yescharacteristic, complemented
86#383—1971no—
87#327—1974no—
88C2×#41—3941no—
89C4⋊Q8—1971no—
90#127—1971no—
91#124—3941no—
92#121—3941no—
93#121—1972no—
94C4×D4—3941no—
95C4×D4—1972no—
96C2×C42C2×D19711yescharacteristic
97C4⋊C8—21no—
98C22⋊C8C2×D19712yes—
99#97—21no—
100#95—3941no—
101#95—3941no—
102#95—1972no—
103#41—7884no—
104C2×Q8—3941no—
105C2×D4—3942no—
106C2×D4—3941no—
107C2×D4—3941no—
108C2×D4—1972no—
109C22×C4—3942no—
110C22×C4—3942no—
111C22×C4D78811yescharacteristic
112C22×C4D78811yescharacteristic
113C22×C4C22×D19711yescharacteristic
114C2×C8—22no—
115C4⋊C4—3941no—
116C4⋊C4—1972no—
117C22⋊C4—3942no—
118C22⋊C4—3942no—
119C42C4×D19711yescharacteristic
120C42C4×D19711yescharacteristic
121C42C22⋊D19711yescharacteristic
122C42C22⋊D19711yescharacteristic
123C23—3942no—
124C23C2×D78811yescharacteristic
125Q8—7882no—
126D4—7882no—
127D4—7882no—
128D4—7882no—
129D4—7884no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—7884no—
134C2×C4—7882no—
135C2×C4—7882no—
136C2×C4—7882no—
137C2×C4—3942no—
138C2×C4#1569133511yescharacteristic
139C2×C4#1569133511yescharacteristic
140C2×C4#1569133511yescharacteristic
141C2×C4#1569133511yescharacteristic
142C2×C4C2×C4×D19711yescharacteristic
143C2×C4C2×C22⋊D19711yescharacteristic
144C8—42no—
145C22—21no—
146C22—21no—
147C22—7882no—
148C22—7882no—
149C22—7882no—
150C22C2×#1569133511yescharacteristic
151C22#2026335412yes—
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—7882no—
159C4—15762no—
160C2—22no—
161C2—15762no—
162C2#2716105211yescharacteristic
163C2#2716108511yescharacteristic
164C2#2716109511yescharacteristic
subgroupG/H
centerC22C2×#15691335
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#432×C197C2
socleC22×C197C2×C22⋊C4
prime radical#27161629C2
cosocleC22×C4×C197C23
hypercenter#432D197
hypocenterC197#1514
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #27160947C2semidirect
3. #27160814C2semidirect
4. #432×C197C2semidirect
6. #20263326C4semidirect
7. #20263323C4semidirect
12. C4×D788C4semidirect
82. C197#1514semidirect
85. #432D197semidirect

Maximal subgroups

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

subgroupindex
#271616292
#271609472
#271608142
#432×C1972
#1514197