Finite Group Databasebeta

← #38746004

Subgroups — #38746004

Order 25216.

504 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 163 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
1C2×#20263291C211yesmaximal, characteristic
2C2×#20263291C211yesmaximal, characteristic
3#27161013C212yesmaximal
4#27161010C211yesmaximal, characteristic
5#27161000C211yesmaximal, characteristic
6#27160999C212yesmaximal
7#27160999C211yesmaximal, characteristic
8C4⋊C4×Dic197C212yesmaximal
9#27160930C212yesmaximal
10#27160924C211yesmaximal, characteristic
11#398×C197C211yesmaximal, characteristic
12C22×Dic394C2211yescharacteristic
13C22×Dic394C2211yescharacteristic
14#20263291—24no—
15#20263291—24no—
16C2×C4⋊Dic197C2212yes—
17C2×#15691337C2212yes—
18C2×#15691337C2212yes—
19C2×#15691337C2212yes—
20C2×#15691337C2211yescharacteristic
21C2×#15691337C2211yescharacteristic
22C2×C4×Dic197C2211yescharacteristic
23C2×C4×Dic197C2212yes—
24C2×C4×Dic197C2211yescharacteristic
25C2×C4×Dic197C2211yescharacteristic
26C2×C4×Dic197C2212yes—
27#20263233C2211yescharacteristic
28#20263233C2212yes—
29#20263233C2212yes—
30#20263233C2212yes—
31#20263233C2212yes—
32C2×C4⋊C4×C197C2212yes—
33C2×C4⋊C4×C197C2211yescharacteristic
34C2×C4⋊C4×C197C2211yescharacteristic
35#95×C197C2211yescharacteristic
36#95×C197C2212yes—
37C2×Dic394—22no—
38C2×Dic394—22no—
39C2×Dic394—24no—
40C2×Dic394—22no—
41C2×Dic394—22no—
42C22×Dic197C2311yescharacteristic
43C22×Dic197C2311yescharacteristic
44C22×Dic197C2312yes—
45C22×Dic197C2311yescharacteristic
46C22×Dic197C2312yes—
47C22×Dic197C2311yescharacteristic
48C4⋊Dic197—24no—
49#15691337—24no—
50#15691337—22no—
51#15691337—24no—
52#15691337—24no—
53#15691337—22no—
54C4×Dic197—22no—
55C4×Dic197—22no—
56C4×Dic197—24no—
57C4×Dic197D414yes—
58C4×Dic197Q818yescomplemented
59C22×C4×C197C2311yescharacteristic
60C22×C4×C197C2312yes—
61C22×C4×C197C2311yescharacteristic
62C22×C4×C197C2312yes—
63C22×C4×C197C2311yescharacteristic
64C4⋊C4×C197—22no—
65C4⋊C4×C197—22no—
66C4⋊C4×C197—24no—
67Dic394—28no—
68Dic394—28no—
69C2×Dic197—22no—
70C2×Dic197—21no—
71C2×Dic197—22no—
72C2×Dic197—22no—
73C2×Dic197—21no—
74C2×Dic197—22no—
75C2×Dic197—21no—
76C2×Dic197—21no—
77C2×Dic197—21no—
78C2×Dic197—22no—
79C2×Dic197C2×D412yes—
80C2×Dic197C2×D412yes—
81C2×Dic197C2×Q814yes—
82C2×Dic197C2×Q814yes—
83C2×Dic197#4112yes—
84C2×Dic197#4114yes—
85C23×C197C2411yescharacteristic
86C2×C4×C197—21no—
87C2×C4×C197—22no—
88C2×C4×C197—22no—
89C2×C4×C197—22no—
90C2×C4×C197—21no—
91C2×C4×C197—21no—
92C2×C4×C197—22no—
93C2×C4×C197—22no—
94C2×C4×C197—22no—
95C2×C4×C197—22no—
96C2×C4×C197C2×D412yes—
97C2×C4×C197C2×Q814yes—
98C2×C4×C197#4112yes—
99C22×C197C22×D411yescharacteristic
100C22×C197C22×Q812yes—
101C22×C197C2×#4112yes—
102C22×C197C2×#4111yescharacteristic
103C22×C197C2×#4111yescharacteristic
104C4×C197—22no—
105C4×C197—22no—
106C4×C197—24no—
107C4×C197—41no—
108C4×C197—42no—
109Dic197—22no—
110Dic197—24no—
111Dic197—22no—
112Dic197—24no—
113Dic197—41no—
114Dic197C4⋊Q814yes—
115C2×C197#52511yescharacteristic
116C2×C197#52511yescharacteristic
117C2×C197C2×C4⋊Q811yescharacteristic
118C2×C197#53312yes—
119C2×C197#53312yes—
120C197#239911yescharacteristic, complemented, Sylow
121#2399—1971nomaximal, Sylow
122C2×C4×Q8—1971no—
123C2×C4×Q8—1971no—
124#398—1971no—
125#398—1972no—
126#398D19711yescharacteristic
127#395—1972no—
128#395—1971no—
129#395—1971no—
130C4×C4⋊C4—1972no—
131C4×C4⋊C4—1972no—
132#376—1971no—
133C22×Q8—1971no—
134C22×Q8—1971no—
135C4×Q8—3944no—
136C4×Q8—3944no—
137C2×C4⋊C4—1972no—
138C2×C4⋊C4—1971no—
139C2×C4⋊C4—1971no—
140C2×C4⋊C4—1972no—
141C2×C4⋊C4—1972no—
142C2×C4⋊C4—1972no—
143C2×C4⋊C4C2×D19711yescharacteristic
144C2×C4⋊C4C2×D19711yescharacteristic
145C2×C4⋊C4C2×D19712yes—
146C2×C42—1971no—
147C2×C42—1972no—
148C2×C42—1971no—
149C2×C42—1971no—
150C2×C42—1972no—
151#95—1971no—
152#95—1972no—
153#95—1972no—
154#95—1972no—
155#95—1972no—
156#95C2×D19711yescharacteristic
157#95C2×D19712yes—
158C2×Q8—3942no—
159C2×Q8—3944no—
160C2×Q8—3942no—
161C2×Q8—3942no—
162C2×Q8—3942no—
163C22×C4—1971no—
164C22×C4—1972no—
165C22×C4—1972no—
166C22×C4—1971no—
167C22×C4—1971no—
168C22×C4—1971no—
169C22×C4C22×D19712yes—
170C22×C4C22×D19711yescharacteristic
171C22×C4C22×D19711yescharacteristic
172C22×C4C22×D19712yes—
173C22×C4C22×D19711yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—24no—
177C4⋊C4—3944no—
178C4⋊C4—3944no—
179C4⋊C4—3942no—
180C4⋊C4—3944no—
181C4⋊C4—3944no—
182C4⋊C4—3942no—
183C42—3942no—
184C42—3942no—
185C42—3944no—
186C42—1974no—
187C42—1978no—
188C23C23×D19711yescharacteristic
189Q8—3948no—
190Q8—3948no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—3941no—
202C2×C4—3942no—
203C2×C4—3942no—
204C2×C4—3941no—
205C2×C4—3942no—
206C2×C4—3942no—
207C2×C4—3942no—
208C2×C4—3941no—
209C2×C4—3941no—
210C2×C4—3941no—
211C2×C4—1972no—
212C2×C4—1974no—
213C2×C4—1974no—
214C2×C4—1972no—
215C2×C4—1972no—
216C2×C4—1974no—
217C2×C4D4×D19712yes—
218C2×C4Q8×D19714yes—
219C2×C4#1569135712yes—
220C22C2×D4×D19711yescharacteristic
221C22C2×Q8×D19712yes—
222C22#41×D19712yes—
223C22#41×D19711yescharacteristic
224C22C2×#1569135711yescharacteristic
225C4—24no—
226C4—22no—
227C4—22no—
228C4—41no—
229C4—42no—
230C4—7881no—
231C4—3944no—
232C4—3944no—
233C4—3942no—
234C4—3942no—
235C4—1974no—
236C2#2716158711yescharacteristic
237C2#2716158911yescharacteristic
238C2C4⋊Q8×D19711yescharacteristic
239C2#2716163112yes—
240C2#2716163512yes—
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, prime radical, cosocleC23×C197C24
Fitting#398×C197C2
hypercenter#398D197
hypocenterC197#2399
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
58. C4×Dic197Q8semidirect
120. C197#2399semidirect

Maximal subgroups

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

subgroupindex
C2×#202632912
#271610132
#271610102
#271610002
#271609992
C4⋊C4×Dic1972
#271609302
#271609242
#398×C1972
#2399197