Finite Group Databasebeta

← #38745288

Subgroups — #38745288

Order 25216.

1160 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C23×C22⋊D197C211yesmaximal, characteristic
2C2×C22⋊C4×D197C212yesmaximal, complemented
3#27160955C218yesmaximal, complemented
4#27160940C212yesmaximal, complemented
5#27160935C211yesmaximal, characteristic, complemented
6#379×C197C211yesmaximal, characteristic, complemented
7C24×D197C2211yescharacteristic
8C22×C22⋊D197—28no—
9C22×C22⋊D197C418yescomplemented
10C22×C22⋊D197C2214yes—
11C23×Dic197C2211yescharacteristic
12C22×C4×D197C2214yescomplemented
13C2×C22⋊Dic197C2212yes—
14C2×C22⋊Dic197C2214yescomplemented
15C2×#15691335C2214yescomplemented
16C22⋊C4×D197—28no—
17#20263233C2218yescomplemented
18C25×C197C2211yescharacteristic
19C2×C22⋊C4×C197C2212yes—
20C2×C22⋊C4×C197C2214yescomplemented
21C23×D197—44no—
22C23×D197C2×C414yes—
23C23×D197D418yes—
24C23×D197C2312yes—
25C2×C22⋊D197—232no—
26C2×C22⋊D197—28no—
27C2×C22⋊D197—28no—
28C2×C22⋊D197C2×C4116yescomplemented
29C22×Dic197—24no—
30C22×Dic197C2×C414yescomplemented
31C22×Dic197C2314yes—
32C22×Dic197C2312yes—
33C2×C4×D197—28no—
34C2×C4×D197—48no—
35C22⋊Dic197—24no—
36C22⋊Dic197—44no—
37#15691335—28no—
38C24×C197—24no—
39C24×C197—44no—
40C24×C197C2×C414yescomplemented
41C24×C197C2312yes—
42C24×C197C2311yescharacteristic
43C22×C4×C197C2314yes—
44C22⋊C4×C197—24no—
45C22⋊C4×C197—44no—
46C22×D197—28no—
47C22×D197—48no—
48C22×D197—416no—
49C22×D197C22⋊C4116yescomplemented
50C22×D197C22×C414yes—
51C22×D197C2×D414yes—
52C22×D197C2×D414yes—
53C22⋊D197—232no—
54C2×Dic197—28no—
55C2×Dic197—24no—
56C2×Dic197—24no—
57C2×Dic197—28no—
58C2×Dic197C22×C414yes—
59C4×D197—416no—
60C23×C197—24no—
61C23×C197—28no—
62C23×C197—24no—
63C23×C197—24no—
64C23×C197—24no—
65C23×C197—48no—
66C23×C197—416no—
67C23×C197C22×C414yes—
68C23×C197C22×C412yes—
69C23×C197C2×D414yes—
70C23×C197C2411yescharacteristic
71C2×C4×C197—24no—
72C2×C4×C197—28no—
73C22×C197—24no—
74C22×C197—28no—
75C22×C197—24no—
76C22×C197—24no—
77C22×C197—24no—
78C22×C197—416no—
79C22×C197—48no—
80C22×C197C2×C22⋊C414yes—
81C22×C197C23×C411yescharacteristic
82C22×C197C22×D412yes—
83C22×C197H(5,𝔽2)14yes—
84C2×D197—28no—
85C2×D197—28no—
86C2×D197—28no—
87C2×D197—416no—
88C2×D197C2×C22⋊C418yes—
89C4×C197—44no—
90Dic197—24no—
91Dic197—44no—
92C2×C197—24no—
93C2×C197—44no—
94C2×C197C22×C22⋊C411yescharacteristic
95C2×C197#51812yes—
96C2×C197#53414yes—
97D197—28no—
98C197#230811yescharacteristic, complemented, Sylow
99#2308—1971nomaximal, Sylow
100C23×D4—1971no—
101C22×C22⋊C4—1972no—
102#386—1978no—
103#381—1972no—
104#379—1971no—
105#379D19711yescharacteristic, complemented
106C25—1971no—
107C25C2×D19711yescharacteristic
108C22×D4—3948no—
109C22×D4—1978no—
110C22×D4—1974no—
111C23×C4—1974no—
112C23×C4—1971no—
113C2×C22⋊C4—3948no—
114C2×C22⋊C4—1974no—
115C2×C22⋊C4—1972no—
116C2×C22⋊C4—1974no—
117C2×C22⋊C4C2×D19712yes—
118C2×C22⋊C4C2×D19714yescomplemented
119#95—1978no—
120C24—24no—
121C24—44no—
122C24—7884no—
123C24—1974no—
124C24—1978no—
125C24—1972no—
126C24C4×D19714yescomplemented
127C24C22×D19712yes—
128C24C22×D19711yescharacteristic
129C2×D4—39432no—
130C2×D4—3948no—
131C2×D4—3948no—
132C2×D4—19716no—
133C22×C4—7888no—
134C22×C4—3948no—
135C22×C4—3944no—
136C22×C4—1974no—
137C22×C4—1974no—
138C22×C4—1972no—
139C22×C4C22×D19714yes—
140C22⋊C4—24no—
141C22⋊C4—44no—
142C22⋊C4—7884no—
143C22⋊C4—3948no—
144C22⋊C4—3944no—
145C23—24no—
146C23—28no—
147C23—24no—
148C23—24no—
149C23—24no—
150C23—48no—
151C23—416no—
152C23—78816no—
153C23—7888no—
154C23—3948no—
155C23—19716no—
156C23—1974no—
157C23—1974no—
158C23—1974no—
159C23C2×C4×D19714yes—
160C23C2×C4×D19712yes—
161C23D4×D19714yes—
162C23C23×D19711yescharacteristic
163D4—39432no—
164C2×C4—24no—
165C2×C4—28no—
166C2×C4—78816no—
167C2×C4—3944no—
168C2×C4—3948no—
169C2×C4—3948no—
170C2×C4—3944no—
171C2×C4—1974no—
172C22—28no—
173C22—24no—
174C22—24no—
175C22—24no—
176C22—24no—
177C22—416no—
178C22—48no—
179C22—78816no—
180C22—3948no—
181C22—3948no—
182C22—3948no—
183C22—1978no—
184C22C22⋊C4×D19714yes—
185C22C22×C4×D19711yescharacteristic
186C22C2×D4×D19712yes—
187C22#2026337114yes—
188C4—44no—
189C4—7884no—
190C4—3944no—
191C2—24no—
192C2—44no—
193C2—3948no—
194C2C2×C22⋊C4×D19711yescharacteristic
195C2#2716155212yes—
196C2#2716163914yes—
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#379×C197C2
socle, cosocleC23×C197C24
prime radicalC23×C22⋊D197C2
hypercenter#379D197
hypocenterC197#2308
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×D197C2semidirect
3. #27160955C2semidirect
4. #27160940C2semidirect
5. #27160935C2semidirect
6. #379×C197C2semidirect
9. C22×C22⋊D197C4semidirect
12. C22×C4×D197C22semidirect
14. C2×C22⋊Dic197C22semidirect
15. C2×#15691335C22semidirect
17. #20263233C22semidirect
20. C2×C22⋊C4×C197C22semidirect
28. C2×C22⋊D197C2×C4semidirect
30. C22×Dic197C2×C4semidirect
40. C24×C197C2×C4semidirect
49. C22×D197C22⋊C4semidirect
98. C197#2308semidirect
105. #379D197semidirect
118. C2×C22⋊C4C2×D197semidirect
126. C24C4×D197semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1972
C2×C22⋊C4×D1972
#271609552
#271609402
#271609352
#379×C1972
#2308197