Finite Group Databasebeta

← #38741037

Subgroups — #38741037

Order 25216.

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

normal subgroups only

This group has 440 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
1C2×C22⋊C4×D197C211yesmaximal, characteristic
2#27161041C212yesmaximal, complemented
3#27160803C211yesmaximal, characteristic, complemented
4#27160802C212yesmaximal, complemented
5#323×C197C211yesmaximal, characteristic, complemented
6C24×D197C411yescharacteristic
7C22×C4×D197C411yescharacteristic
8C22×C4×D197C2211yescharacteristic
9C2×#15691341—22no—
10C2×C22⋊Dic197C411yescharacteristic
11C2×#15691335C411yescharacteristic
12C2×#15691335C2211yescharacteristic
13C22⋊C4×D197—22no—
14C22⋊C4×D197—22no—
15C22⋊#13096802C2212yescomplemented
16#20263238—22no—
17C2×C22⋊C4×C197C2211yescharacteristic
18C22⋊C8×C197C2212yescomplemented
19C23×D197—21no—
20C23×D197—21no—
21C23×D197—22no—
22C23×D197—22no—
23C23×D197—42no—
24C23×D197—42no—
25C23×D197C2×C411yescharacteristic
26C23×D197C2×C411yescharacteristic
27C22×Dic197C2×C411yescharacteristic
28C22×Dic197C2×C411yescharacteristic
29C2×C4×D197—21no—
30C2×C4×D197—21no—
31C2×C4×D197—42no—
32C2×C4×D197—42no—
33C2×C4×D197D412yes—
34C2×C4×D197D412yes—
35#15691341—44no—
36C2×#13096802—22no—
37C22⋊Dic197—22no—
38#15691335—22no—
39#15691335—22no—
40C24×C197C2×C411yescharacteristic
41C22×C4×C197C2×C411yescharacteristic
42C22×C4×C197C2311yescharacteristic
43C2×C8×C197—22no—
44C22⋊C4×C197—22no—
45C22×D197—21no—
46C22×D197—21no—
47C22×D197—22no—
48C22×D197—22no—
49C22×D197—24no—
50C22×D197—24no—
51C22×D197—21no—
52C22×D197—21no—
53C22×D197—48no—
54C22×D197—48no—
55C22×D197—42no—
56C22×D197—42no—
57C22×D197—42no—
58C22×D197—42no—
59C22×D197—41no—
60C22×D197—41no—
61C22×D197C22⋊C411yescharacteristic
62C22×D197C22⋊C411yescharacteristic
63C22×D197M1612yescomplemented
64C2×Dic197—21no—
65C2×Dic197—41no—
66C2×Dic197—41no—
67C2×Dic197C22⋊C412yes—
68C4×D197—24no—
69C4×D197—42no—
70C4×D197—42no—
71C4×D197—42no—
72#13096802—42no—
73C23×C197—21no—
74C23×C197—21no—
75C23×C197—41no—
76C23×C197—41no—
77C23×C197M1612yescomplemented
78C23×C197C22×C411yescharacteristic
79C2×C4×C197—21no—
80C2×C4×C197—41no—
81C2×C4×C197—41no—
82C2×C4×C197C2×D412yes—
83C8×C197—42no—
84C22×C197—21no—
85C22×C197—21no—
86C22×C197—22no—
87C22×C197—42no—
88C22×C197—42no—
89C22×C197—41no—
90C22×C197—41no—
91C22×C197C2×C22⋊C411yescharacteristic
92C22×C197C2×M1611yescharacteristic
93C22×C197C2×M1611yescharacteristic
94C2×D197—24no—
95C2×D197—24no—
96C2×D197—21no—
97C2×D197—21no—
98C2×D197—21no—
99C2×D197—21no—
100C2×D197—22no—
101C2×D197—48no—
102C2×D197—48no—
103C2×D197—42no—
104C2×D197—41no—
105C2×D197—41no—
106C2×D197—42no—
107C2×D197#9912yes—
108C2×D197#10012yes—
109C4×C197—22no—
110C4×C197—41no—
111Dic197—22no—
112Dic197—41no—
113C2×C197—21no—
114C2×C197—21no—
115C2×C197—42no—
116C2×C197#40711yescharacteristic
117C2×C197C2×#9911yescharacteristic
118C2×C197C2×#10011yescharacteristic
119D197—22no—
120D197—22no—
121D197—42no—
122C197#146211yescharacteristic, complemented, Sylow
123#1462—1971nomaximal, Sylow
124C22×C22⋊C4—1971no—
125#407—1972no—
126#323—1971no—
127#323—1972no—
128#323D19711yescharacteristic, complemented
129C25—1971no—
130C23×C4—1971no—
131C23×C4—1971no—
132C2×M16—3942no—
133C2×C22⋊C4—3942no—
134C2×C22⋊C4—3942no—
135C2×C22⋊C4—1971no—
136C2×C22⋊C4—1971no—
137C2×C22⋊C4—1971no—
138C2×C22⋊C4C2×D19711yescharacteristic
139C22⋊C8—3942no—
140C22⋊C8—1972no—
141C22⋊C8C2×D19712yescomplemented
142C24—7882no—
143C24—7882no—
144C24—3941no—
145C24—3941no—
146C24—3942no—
147C24—3942no—
148C24—1971no—
149C24—1971no—
150C24C4×D19711yescharacteristic
151C22×C4—7882no—
152C22×C4—7882no—
153C22×C4—3941no—
154C22×C4—3941no—
155C22×C4—1971no—
156C22×C4—1971no—
157C22×C4—1972no—
158C22×C4—1972no—
159C22×C4C4×D19711yescharacteristic
160C22×C4C22×D19711yescharacteristic
161M16—7884no—
162C2×C8—22no—
163C2×C8—3942no—
164C22⋊C4—22no—
165C22⋊C4—3942no—
166C22⋊C4—3942no—
167C22⋊C4—3942no—
168C23—21no—
169C23—21no—
170C23—41no—
171C23—41no—
172C23—7881no—
173C23—7881no—
174C23—7888no—
175C23—7888no—
176C23—7882no—
177C23—7882no—
178C23—7882no—
179C23—7882no—
180C23—3944no—
181C23—3944no—
182C23—3942no—
183C23—3942no—
184C23—3941no—
185C23—3941no—
186C23—3941no—
187C23—3941no—
188C23—1972no—
189C23—1971no—
190C23—1971no—
191C23#1569134112yescomplemented
192C23C2×C4×D19711yescharacteristic
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—7882no—
197C2×C4—7882no—
198C2×C4—7882no—
199C2×C4—7881no—
200C2×C4—7881no—
201C2×C4—3944no—
202C2×C4—3941no—
203C2×C4—1972no—
204C2×C4D4×D19712yes—
205C8—42no—
206C8—7882no—
207C22—21no—
208C22—21no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—7882no—
215C22—7882no—
216C22—7888no—
217C22—7888no—
218C22—7881no—
219C22—7881no—
220C22—3944no—
221C22—3944no—
222C22—3942no—
223C22—3941no—
224C22—3941no—
225C22—3941no—
226C22—3941no—
227C22—1972no—
228C22—1972no—
229C22C22⋊C4×D19711yescharacteristic
230C22M16×D19711yescharacteristic
231C22C2×#1569134111yescharacteristic
232C4—22no—
233C4—41no—
234C4—7881no—
235C4—3942no—
236C2—21no—
237C2—21no—
238C2—42no—
239C2—7882no—
240C2—3942no—
241C2—3942no—
242C2#2716104111yescharacteristic
243C2#99×D19711yescharacteristic
244C2#100×D19711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D197
commutatorC23×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#323×C197C2
socleC22×C197C2×C22⋊C4
prime radicalC24×D197C4
cosocleC22×C4×C197C23
hypercenter#323D197
hypocenterC197#1462
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #27161041C2semidirect
3. #27160803C2semidirect
4. #27160802C2semidirect
5. #323×C197C2semidirect
15. C22⋊#13096802C22semidirect
18. C22⋊C8×C197C22semidirect
63. C22×D197M16semidirect
77. C23×C197M16semidirect
122. C197#1462semidirect
128. #323D197semidirect
141. C22⋊C8C2×D197semidirect
191. C23#15691341semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D1972
#271610412
#271608032
#271608022
#323×C1972
#1462197