Finite Group Databasebeta

← #38753102

Subgroups — #38753102

Order 25216.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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#27161627C211yesmaximal, characteristic
2C4⋊Q8×D197C211yesmaximal, characteristic, complemented
3#27161437C214yesmaximal, complemented
4#27161431C211yesmaximal, characteristic, complemented
5#27161430C211yesmaximal, characteristic, complemented
6#27161425C211yesmaximal, characteristic, complemented
7#27161077C214yesmaximal, complemented
8C4×#15691341C211yesmaximal, characteristic, complemented
9#486×C197C211yesmaximal, characteristic, complemented
10C2×#15691357C2212yes—
11C2×Q8×D197—21no—
12C2×#15691341C2212yes—
13#20263326C2211yescharacteristic
14#20263326C2211yescharacteristic
15#20263325—21no—
16#20263325C2212yes—
17#20263324—21no—
18#20263313C2212yes—
19#20263297—21no—
20#20263297—21no—
21D4×Dic197—21no—
22D4×Dic197—21no—
23C4⋊C4×D197—21no—
24C4⋊C4×D197C2212yes—
25C42×D197C2211yescharacteristic
26#20263251C2214yes—
27#20263249C2214yes—
28#20263247C2214yes—
29#20263236C2212yes—
30C4×#13096802C2211yescharacteristic
31C4⋊Q8×C197C2211yescharacteristic
32#127×C197C2211yescharacteristic
33#102×C197C2214yes—
34C4×C8×C197C2211yescharacteristic
35#15691357—28no—
36C2×Dic394—22no—
37C2×Dic394—21no—
38C2×Dic394C2312yes—
39Q8×D197—44no—
40C2×C22⋊D197—22no—
41C22×Dic197—22no—
42C2×C4×D197—22no—
43C2×C4×D197D411yescharacteristic
44C2×C4×D197D411yescharacteristic
45C2×C4×D197C2311yescharacteristic
46#15691341—24no—
47C2×#13096802C2312yes—
48C4⋊Dic197—21no—
49C4⋊Dic197—21no—
50C4⋊Dic197—21no—
51C4⋊Dic197C2312yes—
52#15691337—22no—
53#15691337—22no—
54C22⋊Dic197—22no—
55C22⋊Dic197—22no—
56C4×Dic197D411yescharacteristic
57C4×Dic197D411yescharacteristic
58C4×Dic197C2311yescharacteristic
59C2×Q8×C197—21no—
60C2×D4×C197—21no—
61C2×D4×C197—21no—
62C2×D4×C197C2312yes—
63C2×C8×C197C2312yes—
64C4⋊C4×C197—21no—
65C4⋊C4×C197C2312yes—
66C42×C197C2311yescharacteristic
67C22×D197C2×D411yescharacteristic
68Dic394—24no—
69Dic394—44no—
70Dic394—42no—
71C22⋊D197—44no—
72C2×Dic197—22no—
73C2×Dic197—22no—
74C2×Dic197—44no—
75C2×Dic197C2×D411yescharacteristic
76C2×Dic197C2×D411yescharacteristic
77C2×Dic197C2×D411yescharacteristic
78C4×D197—22no—
79C4×D197—22no—
80C4×D197—44no—
81C4×D197#4114yes—
82#13096802—22no—
83C23×C197—22no—
84Q8×C197—42no—
85D4×C197—24no—
86D4×C197—42no—
87D4×C197—42no—
88C2×C4×C197—22no—
89C2×C4×C197C2×D411yescharacteristic
90C2×C4×C197C2×D411yescharacteristic
91C2×C4×C197C2411yescharacteristic
92C8×C197—22no—
93C22×C197—42no—
94C22×C197—42no—
95C22×C197—42no—
96C22×C197C22×D411yescharacteristic
97C2×D197—21no—
98C2×D197—21no—
99C2×D197#12412yes—
100C4×C197—42no—
101C4×C197AGL(1,ℤ/8)12yes—
102C4×C197#13712yes—
103C4×C197C2×#4112yes—
104Dic197—21no—
105Dic197—21no—
106Dic197—42no—
107Dic197—42no—
108Dic197#12412yes—
109C2×C197—82no—
110C2×C197C2×#12411yescharacteristic
111C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
112C2×C197C2×#13711yescharacteristic
113D197—22no—
114C197#314111yescharacteristic, complemented, Sylow
115#3141—1971nomaximal, Sylow
116#532—1971no—
117C2×C4⋊Q8—1971no—
118#489—1974no—
119#487—1971no—
120#487—1971no—
121#486—1971no—
122#486D19711yescharacteristic, complemented
123#417—1974no—
124C4×M16—1971no—
125C2×#41—1972no—
126C22×Q8—3941no—
127C2×M16—1972no—
128C4⋊Q8—3941no—
129C4⋊Q8—3941no—
130C4⋊Q8—1972no—
131C4⋊Q8—1971no—
132C4⋊Q8—1971no—
133C4⋊Q8C2×D19711yescharacteristic
134#127C2×D19711yescharacteristic
135#124—1972no—
136#121—3941no—
137#121—3941no—
138C4×D4—3941no—
139C4×D4—3941no—
140C2×C4⋊C4—3941no—
141C2×C4⋊C4—1972no—
142C2×C42—1971no—
143#103—1974no—
144#103—1974no—
145#102—1974no—
146#102C2×D19714yes—
147#97—1972no—
148C4×C8—1971no—
149C4×C8C2×D19711yescharacteristic
150#41—3948no—
151C2×Q8—21no—
152C2×Q8—7884no—
153C2×Q8—3942no—
154C2×Q8—3941no—
155C2×Q8—1972no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—3942no—
159C2×D4C22×D19712yes—
160C22×C4—3942no—
161C22×C4—3942no—
162C22×C4—1971no—
163C22×C4—1971no—
164C22×C4—1971no—
165M16—3944no—
166C2×C8—1972no—
167C2×C8C22×D19712yes—
168C4⋊C4—21no—
169C4⋊C4—3942no—
170C4⋊C4—3942no—
171C4⋊C4—3941no—
172C4⋊C4—3941no—
173C4⋊C4—3941no—
174C4⋊C4—1972no—
175C4⋊C4C22×D19712yes—
176C22⋊C4—3942no—
177C22⋊C4—3942no—
178C42—1971no—
179C42—1971no—
180C42—1971no—
181C42C22×D19711yescharacteristic
182C23—22no—
183C23—1971no—
184Q8—42no—
185Q8—7884no—
186Q8—7882no—
187Q8—3944no—
188D4—24no—
189D4—42no—
190D4—42no—
191D4—7884no—
192C2×C4—22no—
193C2×C4—7884no—
194C2×C4—7884no—
195C2×C4—3942no—
196C2×C4—3942no—
197C2×C4—3942no—
198C2×C4—3942no—
199C2×C4—1971no—
200C2×C4—1971no—
201C2×C4—1971no—
202C2×C4—1974no—
203C2×C4D4×D19711yescharacteristic
204C2×C4D4×D19711yescharacteristic
205C2×C4C23×D19711yescharacteristic
206C8—22no—
207C8—3942no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—3941no—
212C22—3941no—
213C22—1972no—
214C22C2×D4×D19711yescharacteristic
215C4—42no—
216C4—7882no—
217C4—7882no—
218C4—3941no—
219C4—3941no—
220C4—1972no—
221C4#2026335012yes—
222C4#2026335512yes—
223C4#41×D19712yes—
224C2—82no—
225C2—3942no—
226C2#124×D19711yescharacteristic
227C2C2×#2026335011yescharacteristic
228C2C2×#2026335511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#486×C197C2
socleC22×C197C22×D4
prime radical#27161627C2
hypercenter#486D197
hypocenterC197#3141
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D197C2semidirect
3. #27161437C2semidirect
4. #27161431C2semidirect
5. #27161430C2semidirect
6. #27161425C2semidirect
7. #27161077C2semidirect
8. C4×#15691341C2semidirect
9. #486×C197C2semidirect
114. C197#3141semidirect
122. #486D197semidirect

Maximal subgroups

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

subgroupindex
#271616272
C4⋊Q8×D1972
#271614372
#271614312
#271614302
#271614252
#271610772
C4×#156913412
#486×C1972
#3141197