Finite Group Databasebeta

← C2×#469×C197 #38739986

Subgroups — C2×#469×C197 #38739986

Order 25216.

596 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 216 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×D8×C197C211yesmaximal, characteristic, complemented
2C22×M16×C197C211yesmaximal, characteristic, complemented
3C2×#121×C197C212yesmaximal, complemented
4#469×C197C218yesmaximal, direct product factor
5C2×#106×C197C211yesmaximal, characteristic, complemented
6C2×#102×C197C212yesmaximal, complemented
7C22×D4×C197—22no—
8C22×D4×C197C2212yes—
9C23×C4×C197C2211yescharacteristic
10C2×D8×C197—24no—
11C2×D8×C197C2214yescomplemented
12C2×M16×C197—24no—
13C2×M16×C197C2214yescomplemented
14C22×C8×C197C2211yescharacteristic
15C22×C8×C197C2211yescharacteristic
16#121×C197—24no—
17#121×C197C2218yescomplemented
18C2×C4⋊C4×C197C2212yes—
19C2×C22⋊C4×C197—22no—
20#106×C197C2214yescomplemented
21#102×C197C2218yescomplemented
22C24×C197—22no—
23C24×C197D411yescharacteristic
24C2×D4×C197—28no—
25C2×D4×C197—24no—
26C2×D4×C197—22no—
27C2×D4×C197—24no—
28C2×D4×C197—48no—
29C2×D4×C197C2314yes—
30C22×C4×C197—22no—
31C22×C4×C197—22no—
32C22×C4×C197—22no—
33C22×C4×C197D412yes—
34C22×C4×C197D411yescharacteristic
35C22×C4×C197C2312yes—
36C22×C4×C197C2311yescharacteristic
37D8×C197—28no—
38M16×C197—28no—
39C2×C8×C197—22no—
40C2×C8×C197D414yescomplemented
41C2×C8×C197C2312yes—
42C2×C8×C197C2312yes—
43C4⋊C4×C197—22no—
44C4⋊C4×C197C2314yes—
45C22⋊C4×C197—28no—
46C23×C197—22no—
47C23×C197—24no—
48C23×C197—24no—
49C23×C197—44no—
50C23×C197—44no—
51C23×C197—44no—
52C23×C197C2×D412yes—
53C23×C197C2×D411yescharacteristic
54D4×C197—24no—
55D4×C197—24no—
56D4×C197—416no—
57D4×C197—44no—
58C2×C4×C197—22no—
59C2×C4×C197—24no—
60C2×C4×C197—22no—
61C2×C4×C197—22no—
62C2×C4×C197—24no—
63C2×C4×C197—44no—
64C2×C4×C197C2×D412yes—
65C2×C4×C197C2×D411yescharacteristic
66C2×C4×C197C2×D412yes—
67C2×C4×C197#4112yes—
68C2×C4×C197C2411yescharacteristic
69C8×C197—22no—
70C8×C197C2×D414yescomplemented
71C22×C197—22no—
72C22×C197—24no—
73C22×C197—44no—
74C22×C197—44no—
75C22×C197—44no—
76C22×C197—44no—
77C22×C197—88no—
78C22×C197#12112yes—
79C22×C197AGL(1,ℤ/8)14yes—
80C22×C197C22×D411yescharacteristic
81C4×C197—22no—
82C4×C197—44no—
83C4×C197#12112yes—
84C4×C197C22×D411yescharacteristic
85C4×C197C2×#4111yescharacteristic
86C2×C197—42no—
87C2×C197—84no—
88C2×C197#46914yesdirect product factor
89C2×C197C2×#12111yescharacteristic
90C2×C197C2×AGL(1,ℤ/8)12yes—
91C197C2×#46911yescharacteristic, direct product factor, Sylow
92C2×#469C19711yesmaximal, characteristic, direct product factor, Sylow
93C22×D8C2×C19711yescharacteristic, complemented
94C22×M16C2×C19711yescharacteristic, complemented
95C2×#121C2×C19712yescomplemented
96#469C2×C19718yesdirect product factor
97C2×#106C2×C19711yescharacteristic, complemented
98C2×#102C2×C19712yescomplemented
99C22×D4—22no—
100C22×D4C22×C19712yes—
101C23×C4C22×C19711yescharacteristic
102C2×D8—24no—
103C2×D8C22×C19714yescomplemented
104C2×M16—24no—
105C2×M16C22×C19714yescomplemented
106C22×C8C22×C19711yescharacteristic
107C22×C8C22×C19711yescharacteristic
108#121—24no—
109#121C22×C19718yescomplemented
110C2×C4⋊C4C22×C19712yes—
111C2×C22⋊C4—22no—
112#106C22×C19714yescomplemented
113#102C22×C19718yescomplemented
114C24—22no—
115C24D4×C19711yescharacteristic
116C2×D4—28no—
117C2×D4—24no—
118C2×D4—22no—
119C2×D4—24no—
120C2×D4—48no—
121C2×D4C23×C19714yes—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4D4×C19711yescharacteristic
126C22×C4D4×C19712yes—
127C22×C4C23×C19712yes—
128C22×C4C23×C19711yescharacteristic
129D8—28no—
130M16—28no—
131C2×C8—22no—
132C2×C8D4×C19714yescomplemented
133C2×C8C23×C19712yes—
134C2×C8C23×C19712yes—
135C4⋊C4—22no—
136C4⋊C4C23×C19714yes—
137C22⋊C4—28no—
138C23—22no—
139C23—24no—
140C23—24no—
141C23—44no—
142C23—44no—
143C23—44no—
144C23C2×D4×C19712yes—
145C23C2×D4×C19711yescharacteristic
146D4—24no—
147D4—24no—
148D4—416no—
149D4—44no—
150C2×C4—22no—
151C2×C4—24no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—24no—
155C2×C4—44no—
156C2×C4C2×D4×C19712yes—
157C2×C4C2×D4×C19711yescharacteristic
158C2×C4C2×D4×C19712yes—
159C2×C4#41×C19712yes—
160C2×C4C24×C19711yescharacteristic
161C8—22no—
162C8C2×D4×C19714yescomplemented
163C22—22no—
164C22—24no—
165C22—44no—
166C22—44no—
167C22—44no—
168C22—44no—
169C22—88no—
170C22#121×C19712yes—
171C22AGL(1,ℤ/8)×C19714yes—
172C22C22×D4×C19711yescharacteristic
173C4—22no—
174C4—44no—
175C4#121×C19712yes—
176C4C22×D4×C19711yescharacteristic
177C4C2×#41×C19711yescharacteristic
178C2—42no—
179C2—84no—
180C2#469×C19714yesdirect product factor
181C2C2×#121×C19711yescharacteristic
182C2C2×AGL(1,ℤ/8)×C19712yes—
subgroupG/H
center, socleC23×C197C2×D4
commutator, Frattini, cosocleC2×C4C24×C197
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C197 ◁ C23×C4×C197 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×D8×C197C2semidirect
2. C22×M16×C197C2semidirect
3. C2×#121×C197C2semidirect
4. #469×C197C2direct (N × H)
5. C2×#106×C197C2semidirect
6. C2×#102×C197C2semidirect
11. C2×D8×C197C22semidirect
13. C2×M16×C197C22semidirect
17. #121×C197C22semidirect
20. #106×C197C22semidirect
21. #102×C197C22semidirect
40. C2×C8×C197D4semidirect
70. C8×C197C2×D4semidirect
88. C2×C197#469direct (N × H)
91. C197C2×#469direct (N × H)
92. C2×#469C197direct (N × H)
93. C22×D8C2×C197semidirect
94. C22×M16C2×C197semidirect
95. C2×#121C2×C197semidirect
96. #469C2×C197direct (N × H)
97. C2×#106C2×C197semidirect
98. C2×#102C2×C197semidirect
103. C2×D8C22×C197semidirect
105. C2×M16C22×C197semidirect
109. #121C22×C197semidirect
112. #106C22×C197semidirect
113. #102C22×C197semidirect
132. C2×C8D4×C197semidirect
162. C8C2×D4×C197semidirect
180. C2#469×C197direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D8×C1972
C22×M16×C1972
C2×#121×C1972
#469×C1972
C2×#106×C1972
C2×#102×C1972
C2×#469197