Finite Group Databasebeta

← C2×#453×C197 #38739948

Subgroups — C2×#453×C197 #38739948

Order 25216.

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

normal subgroups only

This group has 844 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
1C2×H(5,𝔽2)×C197C211yesmaximal, characteristic, complemented
2C2×AGL(1,ℤ/8)×C197C212yesmaximal, complemented
3C2×#127×C197C211yesmaximal, characteristic, complemented
4#453×C197C218yesmaximal, direct product factor
5C2×#104×C197C212yesmaximal, complemented
6C2×#100×C197C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C197—24no—
8H(5,𝔽2)×C197—22no—
9H(5,𝔽2)×C197C2214yescomplemented
10C2×#41×C197—22no—
11C2×#41×C197C2212yes—
12C22×D4×C197—22no—
13C22×D4×C197—22no—
14C22×D4×C197—22no—
15C22×D4×C197C2211yescharacteristic
16C22×D4×C197C2211yescharacteristic, complemented
17AGL(1,ℤ/8)×C197—24no—
18AGL(1,ℤ/8)×C197C2218yescomplemented
19C2×SD16×C197—22no—
20C2×D8×C197—22no—
21C2×M16×C197C2212yescomplemented
22#127×C197—22no—
23#127×C197C2214yescomplemented
24C2×C42×C197C2211yescharacteristic, complemented
25#104×C197C2218yescomplemented
26#100×C197C2214yescomplemented
27C24×C197—22no—
28C24×C197—21no—
29C24×C197D412yes—
30#41×C197—28no—
31#41×C197—28no—
32#41×C197—22no—
33#41×C197—22no—
34#41×C197D414yescomplemented
35#41×C197C2314yes—
36C2×Q8×C197D412yescomplemented
37C2×D4×C197—24no—
38C2×D4×C197—28no—
39C2×D4×C197—28no—
40C2×D4×C197—22no—
41C2×D4×C197—28no—
42C2×D4×C197—28no—
43C2×D4×C197—28no—
44C2×D4×C197—22no—
45C2×D4×C197—22no—
46C2×D4×C197—22no—
47C2×D4×C197—22no—
48C2×D4×C197—21no—
49C2×D4×C197—48no—
50C2×D4×C197D412yes—
51C2×D4×C197D412yescomplemented
52C2×D4×C197C2312yes—
53C2×D4×C197C2312yescomplemented
54C22×C4×C197—22no—
55C22×C4×C197—21no—
56C22×C4×C197—22no—
57C22×C4×C197C2311yescharacteristic
58SD16×C197—28no—
59D8×C197—28no—
60M16×C197—22no—
61M16×C197C2314yescomplemented
62C2×C8×C197—22no—
63C42×C197—21no—
64C42×C197C2312yescomplemented
65C23×C197—28no—
66C23×C197—22no—
67C23×C197—22no—
68C23×C197—22no—
69C23×C197—22no—
70C23×C197—44no—
71C23×C197—44no—
72C23×C197—41no—
73C23×C197—41no—
74C23×C197—84no—
75C23×C197C2×D411yescharacteristic
76C23×C197C2×D414yes—
77Q8×C197—22no—
78Q8×C197C2×D414yescomplemented
79D4×C197—216no—
80D4×C197—24no—
81D4×C197—24no—
82D4×C197—24no—
83D4×C197—22no—
84D4×C197—24no—
85D4×C197—24no—
86D4×C197—416no—
87D4×C197—42no—
88D4×C197C2×D414yescomplemented
89C2×C4×C197—28no—
90C2×C4×C197—24no—
91C2×C4×C197—22no—
92C2×C4×C197—22no—
93C2×C4×C197—24no—
94C2×C4×C197—24no—
95C2×C4×C197C2×D411yescharacteristic
96C2×C4×C197C2×D412yes—
97C2×C4×C197C2×D412yes—
98C2×C4×C197C2411yescharacteristic
99C8×C197—24no—
100C22×C197—22no—
101C22×C197—24no—
102C22×C197—24no—
103C22×C197—44no—
104C22×C197—44no—
105C22×C197—44no—
106C22×C197—44no—
107C22×C197—42no—
108C22×C197—82no—
109C22×C197—82no—
110C22×C197—88no—
111C22×C197—82no—
112C22×C197#12011yescharacteristic
113C22×C197#12011yescharacteristic
114C22×C197C22×D411yescharacteristic
115C4×C197—24no—
116C4×C197—24no—
117C4×C197#12012yes—
118C4×C197C22×D412yes—
119C2×C197—21no—
120C2×C197—21no—
121C2×C197—44no—
122C2×C197—44no—
123C2×C197—82no—
124C2×C197#45312yesdirect product factor
125C2×C197C2×#12011yescharacteristic
126C197C2×#45311yescharacteristic, direct product factor, Sylow
127C2×#453C19711yesmaximal, characteristic, direct product factor, Sylow
128C2×H(5,𝔽2)C2×C19711yescharacteristic, complemented
129C2×AGL(1,ℤ/8)C2×C19712yescomplemented
130C2×#127C2×C19711yescharacteristic, complemented
131#453C2×C19718yesdirect product factor
132C2×#104C2×C19712yescomplemented
133C2×#100C2×C19711yescharacteristic, complemented
134H(5,𝔽2)—24no—
135H(5,𝔽2)—22no—
136H(5,𝔽2)C22×C19714yescomplemented
137C2×#41—22no—
138C2×#41C22×C19712yes—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4C22×C19711yescharacteristic
143C22×D4C22×C19711yescharacteristic, complemented
144AGL(1,ℤ/8)—24no—
145AGL(1,ℤ/8)C22×C19718yescomplemented
146C2×SD16—22no—
147C2×D8—22no—
148C2×M16C22×C19712yescomplemented
149#127—22no—
150#127C22×C19714yescomplemented
151C2×C42C22×C19711yescharacteristic, complemented
152#104C22×C19718yescomplemented
153#100C22×C19714yescomplemented
154C24—22no—
155C24—21no—
156C24D4×C19712yes—
157#41—28no—
158#41—28no—
159#41—22no—
160#41—22no—
161#41D4×C19714yescomplemented
162#41C23×C19714yes—
163C2×Q8D4×C19712yescomplemented
164C2×D4—24no—
165C2×D4—28no—
166C2×D4—28no—
167C2×D4—22no—
168C2×D4—28no—
169C2×D4—28no—
170C2×D4—28no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—21no—
176C2×D4—48no—
177C2×D4D4×C19712yes—
178C2×D4D4×C19712yescomplemented
179C2×D4C23×C19712yes—
180C2×D4C23×C19712yescomplemented
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4C23×C19711yescharacteristic
185SD16—28no—
186D8—28no—
187M16—22no—
188M16C23×C19714yescomplemented
189C2×C8—22no—
190C42—21no—
191C42C23×C19712yescomplemented
192C23—28no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—44no—
198C23—44no—
199C23—41no—
200C23—41no—
201C23—84no—
202C23C2×D4×C19711yescharacteristic
203C23C2×D4×C19714yes—
204Q8—22no—
205Q8C2×D4×C19714yescomplemented
206D4—216no—
207D4—24no—
208D4—24no—
209D4—24no—
210D4—22no—
211D4—24no—
212D4—24no—
213D4—416no—
214D4—42no—
215D4C2×D4×C19714yescomplemented
216C2×C4—28no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4C2×D4×C19711yescharacteristic
223C2×C4C2×D4×C19712yes—
224C2×C4C2×D4×C19712yes—
225C2×C4C24×C19711yescharacteristic
226C8—24no—
227C22—22no—
228C22—24no—
229C22—24no—
230C22—44no—
231C22—44no—
232C22—44no—
233C22—44no—
234C22—42no—
235C22—82no—
236C22—82no—
237C22—88no—
238C22—82no—
239C22#120×C19711yescharacteristic
240C22#120×C19711yescharacteristic
241C22C22×D4×C19711yescharacteristic
242C4—24no—
243C4—24no—
244C4#120×C19712yes—
245C4C22×D4×C19712yes—
246C2—21no—
247C2—21no—
248C2—44no—
249C2—44no—
250C2—82no—
251C2#453×C19712yesdirect product factor
252C2C2×#120×C19711yescharacteristic
subgroupG/H
center, socleC22×C197#120
commutator, Frattini, cosocleC2×C4C24×C197
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C197 ◁ C22×C4×C197 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×H(5,𝔽2)×C197C2semidirect
2. C2×AGL(1,ℤ/8)×C197C2semidirect
3. C2×#127×C197C2semidirect
4. #453×C197C2direct (N × H)
5. C2×#104×C197C2semidirect
6. C2×#100×C197C2semidirect
9. H(5,𝔽2)×C197C22semidirect
16. C22×D4×C197C22semidirect
18. AGL(1,ℤ/8)×C197C22semidirect
21. C2×M16×C197C22semidirect
23. #127×C197C22semidirect
24. C2×C42×C197C22semidirect
25. #104×C197C22semidirect
26. #100×C197C22semidirect
34. #41×C197D4semidirect
36. C2×Q8×C197D4semidirect
51. C2×D4×C197D4semidirect
53. C2×D4×C197C23semidirect
61. M16×C197C23semidirect
64. C42×C197C23semidirect
78. Q8×C197C2×D4semidirect
88. D4×C197C2×D4semidirect
124. C2×C197#453direct (N × H)
126. C197C2×#453direct (N × H)
127. C2×#453C197direct (N × H)
128. C2×H(5,𝔽2)C2×C197semidirect
129. C2×AGL(1,ℤ/8)C2×C197semidirect
130. C2×#127C2×C197semidirect
131. #453C2×C197direct (N × H)
132. C2×#104C2×C197semidirect
133. C2×#100C2×C197semidirect
136. H(5,𝔽2)C22×C197semidirect
143. C22×D4C22×C197semidirect
145. AGL(1,ℤ/8)C22×C197semidirect
148. C2×M16C22×C197semidirect
150. #127C22×C197semidirect
151. C2×C42C22×C197semidirect
152. #104C22×C197semidirect
153. #100C22×C197semidirect
161. #41D4×C197semidirect
163. C2×Q8D4×C197semidirect
178. C2×D4D4×C197semidirect
180. C2×D4C23×C197semidirect
188. M16C23×C197semidirect
191. C42C23×C197semidirect
205. Q8C2×D4×C197semidirect
215. D4C2×D4×C197semidirect
251. C2#453×C197direct (N × H)

Maximal subgroups

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

subgroupindex
C2×H(5,𝔽2)×C1972
C2×AGL(1,ℤ/8)×C1972
C2×#127×C1972
#453×C1972
C2×#104×C1972
C2×#100×C1972
C2×#453197