Finite Group Databasebeta

← #38741209

Subgroups — #38741209

Order 25216.

240 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 71 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#27161631C211yesmaximal, characteristic
2#27161129C211yesmaximal, characteristic
3#27160930C211yesmaximal, characteristic
4#27160817C211yesmaximal, characteristic
5#27160816C211yesmaximal, characteristic
6#27160815C211yesmaximal, characteristic
7#328×C197C211yesmaximal, characteristic
8#20263325C411yescharacteristic, complemented
9#20263324C411yescharacteristic, complemented
10#20263316—21no—
11#20263305C2211yescharacteristic
12#20263303—21no—
13Q8×Dic197C411yescharacteristic, complemented
14#20263291—21no—
15#20263291C411yescharacteristic, complemented
16#20263282—21no—
17C2×#15691337—21no—
18C2×C4×Dic197—21no—
19C2×C4×Dic197C2211yescharacteristic
20#20263255—21no—
21C22⋊#13096802C2211yescharacteristic
22C8×Dic197—21no—
23#20263233C2211yescharacteristic
24C22⋊Q8×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic
26#95×C197C2211yescharacteristic
27C2×Dic394—21no—
28C2×Dic394C2×C411yescharacteristic
29C22×Dic197—21no—
30C22×Dic197D411yescharacteristic
31C22×Dic197D411yescharacteristic
32C2×#13096802—21no—
33C4⋊Dic197—21no—
34C4⋊Dic197C2×C411yescharacteristic
35#15691337—24no—
36#15691337—21no—
37#15691337—21no—
38#15691337—21no—
39#15691337—21no—
40#15691337—21no—
41C22⋊Dic197—21no—
42C4×Dic197—21no—
43C4×Dic197—21no—
44C4×Dic197—42no—
45C4×Dic197C2×C411yescharacteristic
46C4×Dic197C2×C411yescharacteristic
47C4×Dic197D411yescharacteristic
48C4×Dic197D411yescharacteristic
49C2×Q8×C197C2×C411yescharacteristic
50C22×C4×C197—21no—
51C22×C4×C197C2311yescharacteristic
52C2×C8×C197—21no—
53C4⋊C4×C197—21no—
54C4⋊C4×C197C2×C411yescharacteristic
55C22⋊C4×C197—21no—
56Dic394—42no—
57Dic394—41no—
58C2×Dic197—22no—
59C2×Dic197—22no—
60C2×Dic197—21no—
61C2×Dic197—21no—
62C2×Dic197—41no—
63C2×Dic197—41no—
64C2×Dic197C22⋊C411yescharacteristic
65C2×Dic197C22⋊C411yescharacteristic
66C2×Dic197C22⋊C411yescharacteristic
67C2×Dic197C22⋊C411yescharacteristic
68C2×Dic197SD1612yes—
69C2×Dic197Q1612yes—
70#13096802—41no—
71C23×C197C2×D411yescharacteristic
72Q8×C197—41no—
73C2×C4×C197—22no—
74C2×C4×C197—21no—
75C2×C4×C197—21no—
76C2×C4×C197—41no—
77C2×C4×C197—41no—
78C2×C4×C197—41no—
79C2×C4×C197C22×C411yescharacteristic
80C2×C4×C197C2×D411yescharacteristic
81C8×C197—41no—
82C22×C197—21no—
83C22×C197—21no—
84C22×C197C2×C22⋊C411yescharacteristic
85C22×C197C2×SD1611yescharacteristic
86C22×C197C2×Q1611yescharacteristic
87C4×C197—21no—
88C4×C197—21no—
89C4×C197—42no—
90C4×C197—41no—
91C4×C197—41no—
92Dic197—21no—
93Dic197—42no—
94Dic197—41no—
95Dic197—41no—
96Dic197#10312yes—
97Dic197#10312yes—
98Dic197#10412yes—
99C2×C197—21no—
100C2×C197—21no—
101C2×C197#41011yescharacteristic
102C2×C197C2×#10311yescharacteristic
103C2×C197C2×#10411yescharacteristic
104C197#150311yescharacteristic, complemented, Sylow
105#1503—1971nomaximal, Sylow
106#533—1971no—
107#431—1971no—
108C4×C4⋊C4—1971no—
109#328—1971no—
110#328—1971no—
111#328—1971no—
112#328D19711yescharacteristic
113C4⋊Q8—1971no—
114C4⋊Q8—1971no—
115#125—3941no—
116C22⋊Q8—3941no—
117C22⋊Q8—1971no—
118C22⋊Q8C2×D19711yescharacteristic
119C4×Q8—3941no—
120C4×Q8—1971no—
121C4×Q8—1971no—
122#117—3941no—
123C2×C4⋊C4—3941no—
124C2×C42—3941no—
125C2×C42—1971no—
126C4⋊C8—3941no—
127C22⋊C8—1971no—
128C22⋊C8C2×D19711yescharacteristic
129C4×C8—3941no—
130#95—1971no—
131#95C2×D19711yescharacteristic
132C2×Q8—3941no—
133C2×Q8—1971no—
134C2×Q8C4×D19711yescharacteristic
135C22×C4—21no—
136C22×C4—3941no—
137C22×C4—1971no—
138C22×C4—1971no—
139C22×C4C22×D19711yescharacteristic
140C2×C8—21no—
141C2×C8—3941no—
142C4⋊C4—21no—
143C4⋊C4—3944no—
144C4⋊C4—3941no—
145C4⋊C4—3941no—
146C4⋊C4—3941no—
147C4⋊C4—3941no—
148C4⋊C4—3941no—
149C4⋊C4—3941no—
150C4⋊C4—1971no—
151C4⋊C4C4×D19711yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—3941no—
154C42—7882no—
155C42—3941no—
156C42—3941no—
157C42—1971no—
158C42—1971no—
159C42—1971no—
160C42—1971no—
161C23D4×D19711yescharacteristic
162Q8—41no—
163Q8—7881no—
164Q8—7882no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—7881no—
172C2×C4—7881no—
173C2×C4—3942no—
174C2×C4—3941no—
175C2×C4—3941no—
176C2×C4—3942no—
177C2×C4—1971no—
178C2×C4—1971no—
179C2×C4—1971no—
180C2×C4—1971no—
181C2×C4—1972no—
182C2×C4—1972no—
183C2×C4C2×C4×D19711yescharacteristic
184C2×C4D4×D19711yescharacteristic
185C8—41no—
186C8—7881no—
187C22—21no—
188C22—21no—
189C22C22⋊C4×D19711yescharacteristic
190C22SD16×D19711yescharacteristic
191C22Q16×D19711yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—41no—
196C4—41no—
197C4—7882no—
198C4—7881no—
199C4—7881no—
200C4—3941no—
201C4—1972no—
202C4—1972no—
203C4—1972no—
204C2—21no—
205C2—21no—
206C2#2716105111yescharacteristic
207C2#103×D19711yescharacteristic
208C2#104×D19711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D197
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#328×C197C2
socleC22×C197C2×C22⋊C4
prime radicalC23×C197C2×D4
cosocleC22×C4×C197C23
hypercenter#328D197
hypocenterC197#1503
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
8. #20263325C4semidirect
9. #20263324C4semidirect
13. Q8×Dic197C4semidirect
15. #20263291C4semidirect
104. C197#1503semidirect

Maximal subgroups

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

subgroupindex
#271616312
#271611292
#271609302
#271608172
#271608162
#271608152
#328×C1972
#1503197