Finite Group Databasebeta

← #38741235

Subgroups — #38741235

Order 25216.

242 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 73 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#27161613C211yesmaximal, characteristic
2#27161136C211yesmaximal, characteristic, complemented
3#27160815C212yesmaximal, complemented
4#27160812C212yesmaximal
5C4×C4⋊C4×C197C211yesmaximal, characteristic, complemented
6#20263320—21no—
7#20263318C411yescharacteristic, complemented
8#20263314C411yescharacteristic, complemented
9#20263309—21no—
10#20263305C2211yescharacteristic
11#20263300C2211yescharacteristic
12C4×Dic394C411yescharacteristic, complemented
13C4×D788C411yescharacteristic, complemented
14C4×C22⋊D197—21no—
15#20263283—21no—
16C4⋊#13096802—21no—
17C22⋊#13096802C2212yes—
18#20263237—21no—
19C2×C4⋊C4×C197—21no—
20C2×C42×C197—21no—
21C2×C42×C197C2211yescharacteristic
22#95×C197C2212yes—
23C2×Dic394C2×C411yescharacteristic
24C2×D788C2×C411yescharacteristic
25C2×C22⋊D197—21no—
26C2×C4×D197—21no—
27C2×#13096802—22no—
28C4⋊Dic197C2×C411yescharacteristic
29C4⋊Dic197C2×C411yescharacteristic
30#15691337—21no—
31#15691337—21no—
32#15691337—21no—
33#15691337—21no—
34C22⋊Dic197—21no—
35#15691335—21no—
36#15691335—21no—
37#15691335—21no—
38#15691335—21no—
39C4×Dic197—21no—
40C22×C4×C197—22no—
41C22×C4×C197D411yescharacteristic
42C22×C4×C197D411yescharacteristic
43C22×C4×C197C2311yescharacteristic
44C4⋊C4×C197—24no—
45C42×C197—42no—
46C42×C197C2×C411yescharacteristic
47C42×C197C2×C411yescharacteristic
48C42×C197D411yescharacteristic
49C42×C197D411yescharacteristic
50C22×D197—21no—
51Dic394—41no—
52D788—41no—
53C22⋊D197—41no—
54C22⋊D197—41no—
55C2×Dic197—21no—
56C2×Dic197—21no—
57C2×Dic197—21no—
58C4×D197—42no—
59#13096802—42no—
60C23×C197C2×D411yescharacteristic
61C2×C4×C197—21no—
62C2×C4×C197—21no—
63C2×C4×C197—22no—
64C2×C4×C197—24no—
65C2×C4×C197—41no—
66C2×C4×C197—42no—
67C2×C4×C197—42no—
68C2×C4×C197C22⋊C411yescharacteristic
69C2×C4×C197C22⋊C411yescharacteristic
70C2×C4×C197C22⋊C411yescharacteristic
71C2×C4×C197C22⋊C411yescharacteristic
72C2×C4×C197C22×C411yescharacteristic
73C2×C4×C197C2×D411yescharacteristic
74C22×C197—21no—
75C22×C197—21no—
76C22×C197C2×C22⋊C411yescharacteristic
77C22×C197AGL(1,ℤ/8)11yescharacteristic
78C22×C197#13711yescharacteristic
79C2×D197—41no—
80C2×D197—41no—
81C2×D197—41no—
82C4×C197—21no—
83C4×C197—21no—
84C4×C197—21no—
85C4×C197—21no—
86C4×C197—21no—
87C4×C197—44no—
88C4×C197#10412yes—
89Dic197—41no—
90Dic197—41no—
91Dic197—41no—
92C2×C197—21no—
93C2×C197—21no—
94C2×C197#41011yescharacteristic
95C2×C197#41711yescharacteristic
96C2×C197C2×#10411yescharacteristic
97D197—81no—
98C197#150911yescharacteristic, complemented, Sylow
99#1509—1971nomaximal, Sylow
100#529—1971no—
101#432—1971no—
102C4×C4⋊C4D19711yescharacteristic, complemented
103#328—1972no—
104#327—1972no—
105#126—3941no—
106#125—1971no—
107#124—1971no—
108#123—3941no—
109C22⋊Q8—1971no—
110#121—1971no—
111C4×Q8—1971no—
112C4×D4—3941no—
113C4×D4—1971no—
114#117—3941no—
115C2×C4⋊C4—21no—
116C2×C42—21no—
117C2×C42C2×D19711yescharacteristic
118C4⋊C8—3941no—
119C22⋊C8—1972no—
120#97—3941no—
121#95C2×D19712yes—
122C2×Q8—1971no—
123C2×D4—3941no—
124C2×D4—1971no—
125C22×C4—22no—
126C22×C4—3941no—
127C22×C4C22⋊D19711yescharacteristic
128C22×C4C22⋊D19711yescharacteristic
129C22×C4C22×D19711yescharacteristic
130C2×C8—3942no—
131C4⋊C4—24no—
132C4⋊C4—3941no—
133C4⋊C4—3941no—
134C4⋊C4—3941no—
135C4⋊C4—3941no—
136C4⋊C4—1971no—
137C4⋊C4—1971no—
138C22⋊C4—3941no—
139C22⋊C4—3941no—
140C22⋊C4—3941no—
141C22⋊C4—3941no—
142C22⋊C4—3941no—
143C42—42no—
144C42—3941no—
145C42C4×D19711yescharacteristic
146C42C4×D19711yescharacteristic
147C42D78811yescharacteristic
148C42D78811yescharacteristic
149C23—3941no—
150C23C2×C22⋊D19711yescharacteristic
151Q8—7881no—
152D4—7881no—
153D4—7881no—
154D4—7881no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—7882no—
163C2×C4—3941no—
164C2×C4—3941no—
165C2×C4—3941no—
166C2×C4#1569133511yescharacteristic
167C2×C4#1569133511yescharacteristic
168C2×C4#1569133511yescharacteristic
169C2×C4#1569133511yescharacteristic
170C2×C4C2×C4×D19711yescharacteristic
171C2×C4C2×D78811yescharacteristic
172C8—7882no—
173C22—21no—
174C22—21no—
175C22—7881no—
176C22—7881no—
177C22—7881no—
178C22C2×#1569133511yescharacteristic
179C22#2026335211yescharacteristic
180C22#2026335711yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—21no—
186C4—44no—
187C4—7881no—
188C4—7881no—
189C4—7881no—
190C4#2026325412yes—
191C2—21no—
192C2—21no—
193C2—15761no—
194C2#2716105211yescharacteristic
195C2#2716107811yescharacteristic
196C2C2×#2026325411yescharacteristic
subgroupG/H
centerC22C2×#15691335
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
FittingC4×C4⋊C4×C197C2
socleC22×C197C2×C22⋊C4
prime radical#20263300C22
cosocleC22×C4×C197C23
hypercenterC4×C4⋊C4D197
hypocenterC197#1509
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #27161136C2semidirect
3. #27160815C2semidirect
5. C4×C4⋊C4×C197C2semidirect
7. #20263318C4semidirect
8. #20263314C4semidirect
12. C4×Dic394C4semidirect
13. C4×D788C4semidirect
98. C197#1509semidirect
102. C4×C4⋊C4D197semidirect

Maximal subgroups

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

subgroupindex
#271616132
#271611362
#271608152
#271608122
C4×C4⋊C4×C1972
#1509197