Finite Group Databasebeta

← #38751155

Subgroups — #38751155

Order 25216.

764 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 213 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
1C22×#15691358C211yesmaximal, characteristic
2C2×M16×D197C212yesmaximal, complemented
3#27161044C218yesmaximal, complemented
4#27161043C211yesmaximal, characteristic, complemented
5#27161042C212yesmaximal, complemented
6#407×C197C211yesmaximal, characteristic, complemented
7C2×#15691358—28no—
8C2×#15691358C414yes—
9C2×#15691358C2214yes—
10C22×Dic394C411yescharacteristic
11C22×D788C411yescharacteristic
12C22×C22⋊D197C411yescharacteristic
13C22×C22⋊D197C411yescharacteristic
14C22×C4×D197C2211yescharacteristic
15C22×C4×D197C2211yescharacteristic
16C2×#15691342C2212yes—
17C2×#15691341C2214yescomplemented
18M16×D197—28no—
19C2×C8×D197C2214yescomplemented
20C22⋊#13096802C2214yescomplemented
21#20263238C2218yescomplemented
22C23×C4×C197C2211yescharacteristic
23C2×M16×C197C2212yes—
24C22⋊C8×C197C2214yescomplemented
25C23×D197C2×C411yescharacteristic
26C23×D197C2×C411yescharacteristic
27#15691358—216no—
28#15691358—28no—
29#15691358—28no—
30C2×Dic394—24no—
31C2×Dic394C2×C414yes—
32C2×D788—24no—
33C2×D788C2×C414yes—
34C2×C22⋊D197—24no—
35C2×C22⋊D197—24no—
36C2×C22⋊D197C2×C414yes—
37C2×C22⋊D197C2×C414yes—
38C22×Dic197C2×C411yescharacteristic
39C22×Dic197C2×C411yescharacteristic
40C2×C4×D197—24no—
41C2×C4×D197C2×C412yes—
42C2×C4×D197C2×C412yes—
43C2×C4×D197D418yes—
44C2×C4×D197C2312yes—
45C2×C4×D197C2312yes—
46#15691342—24no—
47#15691341—28no—
48C2×#13096802C2314yes—
49C8×D197—28no—
50C24×C197C2×C411yescharacteristic
51C22×C4×C197—24no—
52C22×C4×C197C2×C411yescharacteristic
53C22×C4×C197C2×C412yes—
54C22×C4×C197C2312yes—
55C22×C4×C197C2311yescharacteristic
56M16×C197—24no—
57C2×C8×C197C2314yes—
58C22×D197—22no—
59C22×D197—42no—
60C22×D197—42no—
61C22×D197C22⋊C414yes—
62C22×D197C22×C412yes—
63C22×D197C22×C412yes—
64Dic394—28no—
65D788—28no—
66C22⋊D197—28no—
67C22⋊D197—28no—
68C2×Dic197—22no—
69C2×Dic197C22⋊C414yes—
70C2×Dic197C22×C412yes—
71C2×Dic197C22×C412yes—
72C4×D197—24no—
73C4×D197—24no—
74C4×D197C22⋊C418yes—
75C4×D197C2×D414yes—
76C4×D197C2×D414yes—
77#13096802—24no—
78C23×C197—22no—
79C23×C197—42no—
80C23×C197C22×C412yes—
81C23×C197C22×C411yescharacteristic
82C2×C4×C197—22no—
83C2×C4×C197—24no—
84C2×C4×C197—24no—
85C2×C4×C197C22×C412yes—
86C2×C4×C197C22×C411yescharacteristic
87C2×C4×C197C2×D414yes—
88C2×C4×C197C2411yescharacteristic
89C8×C197—24no—
90C22×C197—22no—
91C22×C197—22no—
92C22×C197—24no—
93C22×C197—44no—
94C22×C197C2×C22⋊C412yes—
95C22×C197C23×C411yescharacteristic
96C2×D197—24no—
97C2×D197—22no—
98C2×D197—24no—
99C2×D197—44no—
100C2×D197—44no—
101C2×D197C2×C22⋊C414yes—
102C4×C197—22no—
103C4×C197C2×C22⋊C412yes—
104C4×C197C22×D412yes—
105Dic197—22no—
106Dic197C2×C22⋊C414yes—
107C2×C197—22no—
108C2×C197—42no—
109C2×C197C22×C22⋊C411yescharacteristic
110C2×C197#56812yes—
111D197—24no—
112D197—42no—
113C197#287911yescharacteristic, complemented, Sylow
114#2879—1971nomaximal, Sylow
115C22×#41—1971no—
116C22×M16—1972no—
117#408—1978no—
118#407—1971no—
119#407—1972no—
120#407D19711yescharacteristic, complemented
121C2×#41—3948no—
122C2×#41—1974no—
123C2×#41—1974no—
124C22×Q8—1971no—
125C22×D4—1971no—
126C22×D4—1971no—
127C22×D4—1971no—
128C23×C4—1971no—
129C23×C4—1971no—
130C23×C4C2×D19711yescharacteristic
131C2×M16—3948no—
132C2×M16—1972no—
133C2×M16—1974no—
134C2×M16C2×D19712yes—
135C22×C8—1974no—
136C22⋊C8—1974no—
137C22⋊C8—1978no—
138C22⋊C8C2×D19714yescomplemented
139C24—1971no—
140C24—1971no—
141C24C4×D19711yescharacteristic
142#41—39416no—
143#41—3948no—
144#41—3948no—
145C2×Q8—3944no—
146C2×Q8—1974no—
147C2×D4—3944no—
148C2×D4—3944no—
149C2×D4—3944no—
150C2×D4—1974no—
151C2×D4—1974no—
152C2×D4—1974no—
153C22×C4—24no—
154C22×C4—3944no—
155C22×C4—1971no—
156C22×C4—1972no—
157C22×C4—1972no—
158C22×C4—1971no—
159C22×C4—1972no—
160C22×C4—1978no—
161C22×C4—1972no—
162C22×C4C4×D19711yescharacteristic
163C22×C4C4×D19712yes—
164C22×C4C22×D19712yes—
165C22×C4C22×D19711yescharacteristic
166M16—24no—
167M16—3948no—
168M16—3944no—
169C2×C8—3948no—
170C2×C8—1974no—
171C2×C8C22×D19714yes—
172C23—22no—
173C23—42no—
174C23—7882no—
175C23—7882no—
176C23—3942no—
177C23—1974no—
178C23—1972no—
179C23—1972no—
180C23C2×C4×D19712yes—
181C23C2×C4×D19711yescharacteristic
182Q8—3948no—
183D4—3948no—
184D4—3948no—
185D4—3948no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—3942no—
190C2×C4—3944no—
191C2×C4—3944no—
192C2×C4—1972no—
193C2×C4—1974no—
194C2×C4—1972no—
195C2×C4—1978no—
196C2×C4—1974no—
197C2×C4—1974no—
198C2×C4C2×C4×D19712yes—
199C2×C4C2×C4×D19711yescharacteristic
200C2×C4D4×D19714yes—
201C2×C4C23×D19711yescharacteristic
202C8—24no—
203C8—3944no—
204C22—22no—
205C22—22no—
206C22—24no—
207C22—44no—
208C22—7884no—
209C22—7884no—
210C22—3942no—
211C22—3944no—
212C22—3944no—
213C22—1974no—
214C22C22⋊C4×D19712yes—
215C22C22×C4×D19711yescharacteristic
216C4—22no—
217C4—3942no—
218C4—1974no—
219C4C22⋊C4×D19712yes—
220C4C2×D4×D19712yes—
221C2—22no—
222C2—42no—
223C2—7882no—
224C2—3944no—
225C2C2×C22⋊C4×D19711yescharacteristic
226C2#2716184512yes—
subgroupG/H
center, FrattiniC2×C4C23×D197
commutator, socleC22×C197C23×C4
Fitting#407×C197C2
prime radicalC22×#15691358C2
cosocleC2×C4×C197C24
hypercenter#407D197
hypocenterC197#2879
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×M16×D197C2semidirect
3. #27161044C2semidirect
4. #27161043C2semidirect
5. #27161042C2semidirect
6. #407×C197C2semidirect
17. C2×#15691341C22semidirect
19. C2×C8×D197C22semidirect
20. C22⋊#13096802C22semidirect
21. #20263238C22semidirect
24. C22⋊C8×C197C22semidirect
113. C197#2879semidirect
120. #407D197semidirect
138. C22⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
C22×#156913582
C2×M16×D1972
#271610442
#271610432
#271610422
#407×C1972
#2879197