Finite Group Databasebeta

← C2×#27161256 #38752041

Subgroups — C2×#27161256 #38752041

Order 25216.

760 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20263371C211yesmaximal, characteristic
2C2×#20263357C212yesmaximal, complemented
3#27161256C218yesmaximal, direct product factor
4C2×#20263254C212yesmaximal, complemented
5C2×#20263243C211yesmaximal, characteristic, complemented
6C2×#124×C197C211yesmaximal, characteristic, complemented
7#20263371—24no—
8#20263371—22no—
9#20263371C2214yes—
10C2×#15691358C2212yes—
11C2×#15691357—22no—
12C22×C22⋊D197—22no—
13C2×D4×D197—22no—
14#20263357—24no—
15#20263357C2218yescomplemented
16C2×#15691348—22no—
17C2×#15691345—22no—
18C2×#15691342C2212yescomplemented
19#20263254C2218yescomplemented
20#20263243C2214yescomplemented
21C22×Q8×C197C2211yescharacteristic, complemented
22C22×D4×C197C2211yescharacteristic
23#124×C197—22no—
24#124×C197C2214yescomplemented
25C2×C22⋊C4×C197—22no—
26C2×C42×C197C2211yescharacteristic, complemented
27C23×D197—22no—
28#15691358—22no—
29#15691358—22no—
30#15691358D414yes—
31#15691358C2314yes—
32#15691357—28no—
33#15691357—28no—
34C2×Dic394D412yes—
35C2×D788D412yes—
36C2×C22⋊D197—24no—
37C2×C22⋊D197—28no—
38C2×C22⋊D197—22no—
39C2×C22⋊D197—28no—
40D4×D197—28no—
41D4×D197—28no—
42C22×Dic197—22no—
43C2×C4×D197—22no—
44#15691348—28no—
45#15691345—28no—
46#15691342—22no—
47#15691342C2314yescomplemented
48C2×#13096802—22no—
49C24×C197D412yes—
50C2×Q8×C197—21no—
51C2×Q8×C197—22no—
52C2×Q8×C197—22no—
53C2×Q8×C197C2312yescomplemented
54C2×D4×C197—22no—
55C2×D4×C197—22no—
56C2×D4×C197D412yes—
57C2×D4×C197C2312yes—
58C22×C4×C197—21no—
59C22×C4×C197—21no—
60C22×C4×C197C2311yescharacteristic
61C22⋊C4×C197—28no—
62C42×C197—21no—
63C42×C197C2312yescomplemented
64C22×D197—22no—
65C22×D197—28no—
66C22×D197—84no—
67Dic394—22no—
68Dic394C2×D414yes—
69D788—22no—
70D788C2×D414yes—
71C22⋊D197—216no—
72C22⋊D197—24no—
73C22⋊D197—24no—
74C2×Dic197—28no—
75C2×Dic197—22no—
76C4×D197—24no—
77C4×D197—24no—
78#13096802—24no—
79C23×C197—22no—
80C23×C197—22no—
81C23×C197—44no—
82C23×C197C2×D411yescharacteristic
83C23×C197C2×D414yes—
84Q8×C197—24no—
85Q8×C197—42no—
86D4×C197—24no—
87D4×C197—24no—
88C2×C4×C197—22no—
89C2×C4×C197—22no—
90C2×C4×C197—42no—
91C2×C4×C197—41no—
92C2×C4×C197—41no—
93C2×C4×C197C2×D412yes—
94C2×C4×C197C2×D412yes—
95C2×C4×C197C2×D411yescharacteristic
96C2×C4×C197C2411yescharacteristic
97C22×C197—22no—
98C22×C197—24no—
99C22×C197—44no—
100C22×C197—44no—
101C22×C197—82no—
102C22×C197#12011yescharacteristic
103C22×C197#12011yescharacteristic
104C22×C197C22×D411yescharacteristic
105C2×D197—24no—
106C2×D197—44no—
107C2×D197—88no—
108C4×C197—42no—
109C4×C197—42no—
110C4×C197#12012yes—
111C4×C197C22×D412yes—
112Dic197—24no—
113C2×C197—21no—
114C2×C197—21no—
115C2×C197—44no—
116C2×C197#45512yescomplemented
117C2×C197C2×#12011yescharacteristic
118D197—44no—
119C197C2×#45511yescharacteristic, complemented, Sylow
120C2×#455—1971nomaximal, Sylow
121C2×H(5,𝔽2)—1971no—
122C2×#137—1972no—
123C2×#124D19711yescharacteristic, complemented
124#455—1978no—
125C2×#104—1972no—
126C2×#100—1971no—
127H(5,𝔽2)—3944no—
128H(5,𝔽2)—3942no—
129H(5,𝔽2)—1974no—
130C2×#41—3942no—
131C2×#41—1972no—
132C22×Q8C2×D19711yescharacteristic, complemented
133C22×D4—3942no—
134C22×D4—3942no—
135C22×D4C2×D19711yescharacteristic
136#137—3944no—
137#137—1978no—
138C2×Q16—3942no—
139C2×SD16—3942no—
140C2×M16—1972no—
141#124—22no—
142#124C2×D19714yescomplemented
143C2×C22⋊C4—22no—
144C2×C42C2×D19711yescharacteristic, complemented
145#104—1978no—
146#100—1974no—
147C24—3942no—
148C24C22⋊D19712yes—
149#41—3948no—
150#41—3948no—
151#41—3942no—
152#41—3942no—
153#41—1974no—
154#41—1974no—
155C2×Q8—21no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—1972no—
159C2×Q8C22×D19712yescomplemented
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—3948no—
163C2×D4—3948no—
164C2×D4—3944no—
165C2×D4—3948no—
166C2×D4—3942no—
167C2×D4—3948no—
168C2×D4—1972no—
169C2×D4C22⋊D19712yes—
170C2×D4C22×D19712yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—3942no—
174C22×C4—3942no—
175C22×C4C22×D19711yescharacteristic
176Q16—3948no—
177SD16—3948no—
178M16—3942no—
179M16—1974no—
180C2×C8—3942no—
181C22⋊C4—28no—
182C42—21no—
183C42C22×D19712yescomplemented
184C23—22no—
185C23—22no—
186C23—44no—
187C23—15764no—
188C23—3948no—
189C23—3942no—
190C23C2×C22⋊D19711yescharacteristic
191C23C2×C22⋊D19714yes—
192Q8—24no—
193Q8—42no—
194Q8—3942no—
195Q8—1974no—
196D4—24no—
197D4—24no—
198D4—39416no—
199D4—3944no—
200D4—3944no—
201D4—3942no—
202D4—1974no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—3948no—
209C2×C4—3942no—
210C2×C4—3944no—
211C2×C4—3944no—
212C2×C4D4×D19712yes—
213C2×C4D4×D19712yes—
214C2×C4C2×C22⋊D19711yescharacteristic
215C2×C4C23×D19711yescharacteristic
216C8—3944no—
217C22—22no—
218C22—24no—
219C22—44no—
220C22—44no—
221C22—82no—
222C22—7884no—
223C22—15768no—
224C22—3944no—
225C22#2026329411yescharacteristic
226C22#2026329411yescharacteristic
227C22C22×C22⋊D19711yescharacteristic
228C4—42no—
229C4—42no—
230C4—3944no—
231C4#2026329412yes—
232C4C2×D4×D19712yes—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—7884no—
237C2#2716125612yesdirect product factor
238C2C2×#2026329411yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#124×C197C2
socleC22×C197#120
prime radicalC2×#20263371C2
hypercenterC2×#124D197
hypocenterC197C2×#455
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#20263357C2semidirect
3. #27161256C2direct (N × H)
4. C2×#20263254C2semidirect
5. C2×#20263243C2semidirect
6. C2×#124×C197C2semidirect
15. #20263357C22semidirect
18. C2×#15691342C22semidirect
19. #20263254C22semidirect
20. #20263243C22semidirect
21. C22×Q8×C197C22semidirect
24. #124×C197C22semidirect
26. C2×C42×C197C22semidirect
47. #15691342C23semidirect
53. C2×Q8×C197C23semidirect
63. C42×C197C23semidirect
116. C2×C197#455semidirect
119. C197C2×#455semidirect
123. C2×#124D197semidirect
132. C22×Q8C2×D197semidirect
142. #124C2×D197semidirect
144. C2×C42C2×D197semidirect
159. C2×Q8C22×D197semidirect
183. C42C22×D197semidirect
237. C2#27161256direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#202633712
C2×#202633572
#271612562
C2×#202632542
C2×#202632432
C2×#124×C1972
C2×#455197