Finite Group Databasebeta

← #38741222

Subgroups — #38741222

Order 25216.

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

normal subgroups only

This group has 228 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#27161631C211yesmaximal, characteristic
2#27161133C211yesmaximal, characteristic
3#27160924C211yesmaximal, 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×C4×Dic197—21no—
18C2×C4×Dic197C2211yescharacteristic
19#20263255—21no—
20C22⋊#13096802C2211yescharacteristic
21#20263236—21no—
22#20263233—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—21no—
36#15691337—21no—
37#15691337—21no—
38#15691337—21no—
39#15691337—21no—
40C22⋊Dic197—21no—
41C4×Dic197—24no—
42C4×Dic197—21no—
43C4×Dic197—21no—
44C4×Dic197C2×C411yescharacteristic
45C4×Dic197C2×C411yescharacteristic
46C4×Dic197D411yescharacteristic
47C4×Dic197D411yescharacteristic
48C2×Q8×C197C2×C411yescharacteristic
49C22×C4×C197—21no—
50C22×C4×C197C2311yescharacteristic
51C2×C8×C197—21no—
52C4⋊C4×C197—21no—
53C4⋊C4×C197C2×C411yescharacteristic
54C22⋊C4×C197—21no—
55Dic394—42no—
56Dic394—41no—
57C2×Dic197—21no—
58C2×Dic197—21no—
59C2×Dic197—22no—
60C2×Dic197—22no—
61C2×Dic197—21no—
62C2×Dic197—21no—
63C2×Dic197—41no—
64C2×Dic197—41no—
65C2×Dic197C22⋊C411yescharacteristic
66C2×Dic197C22⋊C411yescharacteristic
67C2×Dic197C22⋊C411yescharacteristic
68C2×Dic197C22⋊C411yescharacteristic
69#13096802—41no—
70C23×C197C2×D411yescharacteristic
71Q8×C197—41no—
72C2×C4×C197—22no—
73C2×C4×C197—21no—
74C2×C4×C197—21no—
75C2×C4×C197—41no—
76C2×C4×C197—41no—
77C2×C4×C197—41no—
78C2×C4×C197C22×C411yescharacteristic
79C2×C4×C197C2×D411yescharacteristic
80C8×C197—41no—
81C22×C197—21no—
82C22×C197—21no—
83C22×C197C2×C22⋊C411yescharacteristic
84C22×C197#13711yescharacteristic
85C22×C197#13711yescharacteristic
86C4×C197—21no—
87C4×C197—21no—
88C4×C197—42no—
89C4×C197—41no—
90C4×C197—41no—
91Dic197—21no—
92Dic197—21no—
93Dic197—21no—
94Dic197—42no—
95Dic197—41no—
96Dic197—41no—
97Dic197#10412yes—
98C2×C197—21no—
99C2×C197—21no—
100C2×C197#41011yescharacteristic
101C2×C197#41911yescharacteristic
102C2×C197C2×#10411yescharacteristic
103C197#150611yescharacteristic, complemented, Sylow
104#1506—1971nomaximal, Sylow
105#533—1971no—
106#432—1971no—
107#376—1971no—
108#328—1971no—
109#328—1971no—
110#328—1971no—
111#328D19711yescharacteristic
112C4⋊Q8—1971no—
113C4⋊Q8—1971no—
114#125—3941no—
115C22⋊Q8—3941no—
116C22⋊Q8—1971no—
117C22⋊Q8C2×D19711yescharacteristic
118C4×Q8—3941no—
119C4×Q8—1971no—
120C4×Q8—1971no—
121#117—3941no—
122C2×C42—3941no—
123C2×C42—1971no—
124C4⋊C8—3941no—
125C22⋊C8—1971no—
126C22⋊C8C2×D19711yescharacteristic
127#97—3941no—
128#95—3941no—
129#95—1971no—
130#95C2×D19711yescharacteristic
131C2×Q8—3941no—
132C2×Q8—1971no—
133C2×Q8C4×D19711yescharacteristic
134C22×C4—21no—
135C22×C4—3941no—
136C22×C4—1971no—
137C22×C4—1971no—
138C22×C4C22×D19711yescharacteristic
139C2×C8—21no—
140C2×C8—3941no—
141C4⋊C4—21no—
142C4⋊C4—3941no—
143C4⋊C4—3941no—
144C4⋊C4—3941no—
145C4⋊C4—3941no—
146C4⋊C4—3941no—
147C4⋊C4—3941no—
148C4⋊C4—1971no—
149C4⋊C4C4×D19711yescharacteristic
150C22⋊C4—21no—
151C22⋊C4—3941no—
152C42—3944no—
153C42—3941no—
154C42—3941no—
155C42—1971no—
156C42—1971no—
157C42—1971no—
158C42—1971no—
159C23D4×D19711yescharacteristic
160Q8—41no—
161Q8—7881no—
162Q8—7882no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—7881no—
170C2×C4—7881no—
171C2×C4—3942no—
172C2×C4—3941no—
173C2×C4—3941no—
174C2×C4—3941no—
175C2×C4—3941no—
176C2×C4—3942no—
177C2×C4—1971no—
178C2×C4—1971no—
179C2×C4—1971no—
180C2×C4—1971no—
181C2×C4C2×C4×D19711yescharacteristic
182C2×C4D4×D19711yescharacteristic
183C8—41no—
184C8—7881no—
185C22—21no—
186C22—21no—
187C22C22⋊C4×D19711yescharacteristic
188C22#2026335511yescharacteristic
189C22#2026335811yescharacteristic
190C4—21no—
191C4—21no—
192C4—42no—
193C4—41no—
194C4—41no—
195C4—7882no—
196C4—7881no—
197C4—7881no—
198C4—3941no—
199C4—3941no—
200C4—3941no—
201C4—1972no—
202C2—21no—
203C2—21no—
204C2#2716105111yescharacteristic
205C2#2716108611yescharacteristic
206C2#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#1506
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
103. C197#1506semidirect

Maximal subgroups

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

subgroupindex
#271616312
#271611332
#271609242
#271608172
#271608162
#271608152
#328×C1972
#1506197