Finite Group Databasebeta

← #2613×C197 #38739547

Subgroups — #2613×C197 #38739547

Order 25216.

1134 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 232 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×D4×C197C211yesmaximal, characteristic, complemented
2C2×#123×C197C211yesmaximal, characteristic, complemented
3C2×#120×C197C212yesmaximal, complemented
4#394×C197C214yesmaximal, complemented
5#392×C197C214yesmaximal, complemented
6#381×C197C211yesmaximal, characteristic, complemented
7#379×C197C212yesmaximal, complemented
8C25×C197C2212yes—
9C22×D4×C197—28no—
10C22×D4×C197—24no—
11C22×D4×C197C2214yes—
12C22×D4×C197C2211yescharacteristic
13C22×D4×C197C2214yescomplemented
14C23×C4×C197C2211yescharacteristic
15#123×C197—24no—
16#120×C197—28no—
17C2×C4⋊C4×C197C2212yescomplemented
18C2×C22⋊C4×C197C2211yescharacteristic
19C2×C22⋊C4×C197C2212yes—
20C2×C22⋊C4×C197C2212yescomplemented
21C2×C22⋊C4×C197C2214yescomplemented
22C2×C22⋊C4×C197C2218yescomplemented
23#95×C197C2214yescomplemented
24C24×C197—24no—
25C24×C197—24no—
26C24×C197—48no—
27C24×C197D418yescomplemented
28C24×C197C2311yescharacteristic
29C24×C197C2314yes—
30C24×C197C2311yescharacteristic
31C2×D4×C197—232no—
32C2×D4×C197—216no—
33C2×D4×C197—24no—
34C2×D4×C197—28no—
35C2×D4×C197—22no—
36C2×D4×C197—28no—
37C2×D4×C197—48no—
38C22×C4×C197—24no—
39C22×C4×C197D414yescomplemented
40C22×C4×C197C2314yes—
41C22×C4×C197C2312yes—
42C22×C4×C197C2311yescharacteristic
43C22×C4×C197C2312yes—
44C4⋊C4×C197—24no—
45C22⋊C4×C197—24no—
46C22⋊C4×C197—22no—
47C22⋊C4×C197—28no—
48C22⋊C4×C197—24no—
49C22⋊C4×C197—48no—
50C23×C197—24no—
51C23×C197—216no—
52C23×C197—24no—
53C23×C197—24no—
54C23×C197—28no—
55C23×C197—28no—
56C23×C197—21no—
57C23×C197—22no—
58C23×C197—24no—
59C23×C197—416no—
60C23×C197—416no—
61C23×C197—416no—
62C23×C197C2×D418yes—
63C23×C197C2×D414yes—
64C23×C197C2×D412yes—
65C23×C197C2411yescharacteristic
66D4×C197—232no—
67D4×C197—416no—
68D4×C197—44no—
69C2×C4×C197—28no—
70C2×C4×C197—24no—
71C2×C4×C197—22no—
72C2×C4×C197—22no—
73C2×C4×C197—24no—
74C2×C4×C197—21no—
75C2×C4×C197—24no—
76C2×C4×C197—28no—
77C2×C4×C197C2×D414yes—
78C22×C197—28no—
79C22×C197—24no—
80C22×C197—28no—
81C22×C197—28no—
82C22×C197—28no—
83C22×C197—432no—
84C22×C197—48no—
85C22×C197—48no—
86C22×C197—41no—
87C22×C197—44no—
88C22×C197—42no—
89C22×C197#12014yes—
90C22×C197C22×D412yes—
91C22×C197C22×D411yescharacteristic
92C22×C197H(5,𝔽2)14yes—
93C4×C197—24no—
94C4×C197—41no—
95C4×C197—44no—
96C4×C197—42no—
97C2×C197—24no—
98C2×C197—48no—
99C2×C197—81no—
100C2×C197C2×#12011yescharacteristic
101C2×C197#53414yes—
102C2×C197#53512yes—
103C197#261311yescharacteristic, direct product factor, Sylow
104#2613C19711yesmaximal, characteristic, direct product factor, Sylow
105C23×D4C2×C19711yescharacteristic, complemented
106C2×#123C2×C19711yescharacteristic, complemented
107C2×#120C2×C19712yescomplemented
108#394C2×C19714yescomplemented
109#392C2×C19714yescomplemented
110#381C2×C19711yescharacteristic, complemented
111#379C2×C19712yescomplemented
112C25C22×C19712yes—
113C22×D4—28no—
114C22×D4—24no—
115C22×D4C22×C19711yescharacteristic
116C22×D4C22×C19714yes—
117C22×D4C22×C19714yescomplemented
118C23×C4C22×C19711yescharacteristic
119#123—24no—
120#120—28no—
121C2×C4⋊C4C22×C19712yescomplemented
122C2×C22⋊C4C22×C19712yes—
123C2×C22⋊C4C22×C19711yescharacteristic
124C2×C22⋊C4C22×C19714yescomplemented
125C2×C22⋊C4C22×C19712yescomplemented
126C2×C22⋊C4C22×C19718yescomplemented
127#95C22×C19714yescomplemented
128C24—24no—
129C24—24no—
130C24—48no—
131C24D4×C19718yescomplemented
132C24C23×C19711yescharacteristic
133C24C23×C19714yes—
134C24C23×C19711yescharacteristic
135C2×D4—232no—
136C2×D4—216no—
137C2×D4—24no—
138C2×D4—28no—
139C2×D4—22no—
140C2×D4—28no—
141C2×D4—48no—
142C22×C4—24no—
143C22×C4D4×C19714yescomplemented
144C22×C4C23×C19714yes—
145C22×C4C23×C19712yes—
146C22×C4C23×C19711yescharacteristic
147C22×C4C23×C19712yes—
148C4⋊C4—24no—
149C22⋊C4—24no—
150C22⋊C4—22no—
151C22⋊C4—28no—
152C22⋊C4—24no—
153C22⋊C4—48no—
154C23—24no—
155C23—216no—
156C23—24no—
157C23—24no—
158C23—28no—
159C23—28no—
160C23—21no—
161C23—22no—
162C23—24no—
163C23—416no—
164C23—416no—
165C23—416no—
166C23C2×D4×C19718yes—
167C23C2×D4×C19714yes—
168C23C2×D4×C19712yes—
169C23C24×C19711yescharacteristic
170D4—232no—
171D4—416no—
172D4—44no—
173C2×C4—28no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—21no—
179C2×C4—24no—
180C2×C4—28no—
181C2×C4C2×D4×C19714yes—
182C22—28no—
183C22—24no—
184C22—28no—
185C22—28no—
186C22—28no—
187C22—432no—
188C22—48no—
189C22—48no—
190C22—41no—
191C22—44no—
192C22—42no—
193C22#120×C19714yes—
194C22C22×D4×C19712yes—
195C22C22×D4×C19711yescharacteristic
196C22H(5,𝔽2)×C19714yes—
197C4—24no—
198C4—41no—
199C4—44no—
200C4—42no—
201C2—24no—
202C2—48no—
203C2—81no—
204C2C2×#120×C19711yescharacteristic
205C2#534×C19714yes—
206C2#535×C19712yes—
subgroupG/H
center, socleC23×C197C24
commutator, Frattini, cosocleC23C24×C197
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C197 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×D4×C197C2semidirect
2. C2×#123×C197C2semidirect
3. C2×#120×C197C2semidirect
4. #394×C197C2semidirect
5. #392×C197C2semidirect
6. #381×C197C2semidirect
7. #379×C197C2semidirect
13. C22×D4×C197C22semidirect
17. C2×C4⋊C4×C197C22semidirect
20. C2×C22⋊C4×C197C22semidirect
21. C2×C22⋊C4×C197C22semidirect
22. C2×C22⋊C4×C197C22semidirect
23. #95×C197C22semidirect
27. C24×C197D4semidirect
39. C22×C4×C197D4semidirect
103. C197#2613direct (N × H)
104. #2613C197direct (N × H)
105. C23×D4C2×C197semidirect
106. C2×#123C2×C197semidirect
107. C2×#120C2×C197semidirect
108. #394C2×C197semidirect
109. #392C2×C197semidirect
110. #381C2×C197semidirect
111. #379C2×C197semidirect
117. C22×D4C22×C197semidirect
121. C2×C4⋊C4C22×C197semidirect
124. C2×C22⋊C4C22×C197semidirect
125. C2×C22⋊C4C22×C197semidirect
126. C2×C22⋊C4C22×C197semidirect
127. #95C22×C197semidirect
131. C24D4×C197semidirect
143. C22×C4D4×C197semidirect

Maximal subgroups

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

subgroupindex
C23×D4×C1972
C2×#123×C1972
C2×#120×C1972
#394×C1972
#392×C1972
#381×C1972
#379×C1972
#2613197