Finite Group Databasebeta

← C2×#27161254 #38752050

Subgroups — C2×#27161254 #38752050

Order 25216.

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

normal subgroups only

This group has 728 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×#20263374C211yesmaximal, characteristic, complemented
2C2×#20263354C212yesmaximal, complemented
3C2×#20263315C211yesmaximal, characteristic, complemented
4#27161254C218yesmaximal, direct product factor
5C2×#20263252C212yesmaximal, complemented
6C2×#101×C197C211yesmaximal, characteristic, complemented
7#20263374—24no—
8#20263374—22no—
9#20263374C2214yescomplemented
10C2×#15691359—22no—
11C2×#15691358—22no—
12C2×#15691358C2212yes—
13C2×Q8×D197—22no—
14C22×D788C2211yescharacteristic
15#20263354—24no—
16#20263354C2218yescomplemented
17C2×#15691347—22no—
18C2×D1576—22no—
19#20263315—22no—
20#20263315C2214yescomplemented
21C2×#15691335—22no—
22C2×C4×Dic197C2211yescharacteristic
23#20263252C2218yescomplemented
24C22×Q8×C197C2211yescharacteristic
25C2×M16×C197C2212yes—
26#101×C197C2214yescomplemented
27C23×D197—21no—
28#15691359—28no—
29#15691359—28no—
30#15691358—28no—
31#15691358—28no—
32#15691358—22no—
33#15691358—22no—
34#15691358D414yes—
35#15691358C2314yes—
36C2×Dic394—22no—
37C2×Dic394D412yes—
38Q8×D197—28no—
39Q8×D197—28no—
40C2×D788—22no—
41C2×D788—22no—
42C2×D788—22no—
43C2×D788—21no—
44C2×D788D412yes—
45C2×D788C2312yes—
46C2×C22⋊D197—22no—
47C22×Dic197—21no—
48C2×C4×D197—24no—
49C2×C4×D197—22no—
50#15691347—28no—
51D1576—28no—
52#15691335—28no—
53C4×Dic197—21no—
54C4×Dic197C2312yes—
55C2×Q8×C197—22no—
56C2×Q8×C197—22no—
57C2×Q8×C197D412yes—
58C2×Q8×C197C2312yes—
59C22×C4×C197D412yes—
60C22×C4×C197C2311yescharacteristic
61M16×C197—22no—
62M16×C197C2314yes—
63C2×C8×C197—22no—
64C22×D197—22no—
65C22×D197—22no—
66C22×D197—44no—
67C22×D197—41no—
68C22×D197—41no—
69Dic394—28no—
70Dic394—22no—
71Dic394C2×D414yes—
72D788—28no—
73D788—22no—
74D788—24no—
75D788—42no—
76D788C2×D414yes—
77C22⋊D197—24no—
78C22⋊D197—24no—
79C2×Dic197—22no—
80C2×Dic197—22no—
81C2×Dic197—42no—
82C4×D197—216no—
83C4×D197—24no—
84C4×D197—24no—
85C23×C197C2×D411yescharacteristic
86Q8×C197—24no—
87Q8×C197—24no—
88C2×C4×C197—22no—
89C2×C4×C197—22no—
90C2×C4×C197C2×D412yes—
91C2×C4×C197C2×D412yes—
92C2×C4×C197C2×D414yes—
93C2×C4×C197C2×D411yescharacteristic
94C2×C4×C197C2411yescharacteristic
95C8×C197—24no—
96C22×C197—22no—
97C22×C197#12011yescharacteristic
98C22×C197#12011yescharacteristic
99C22×C197C22×D411yescharacteristic
100C2×D197—24no—
101C2×D197—44no—
102C2×D197—44no—
103C2×D197—42no—
104C2×D197—82no—
105C2×D197—82no—
106C4×C197—24no—
107C4×C197#12012yes—
108C4×C197C22×D412yes—
109Dic197—24no—
110Dic197—42no—
111C2×C197—21no—
112C2×C197—21no—
113C2×C197#45412yescomplemented
114C2×C197C2×#12011yescharacteristic
115D197—44no—
116D197—82no—
117C197C2×#45411yescharacteristic, complemented, Sylow
118C2×#454—1971nomaximal, Sylow
119C2×2−1+4—1971no—
120C2×AGL(1,ℤ/8)—1972no—
121C2×#124—1971no—
122#454—1978no—
123C2×#104—1972no—
124C2×#101D19711yescharacteristic, complemented
1252−1+4—3944no—
1262−1+4—3942no—
1272−1+4—1974no—
128C2×#41—3942no—
129C2×#41—3942no—
130C2×#41—1972no—
131C22×Q8—3942no—
132C22×Q8C2×D19711yescharacteristic
133C22×D4—1971no—
134AGL(1,ℤ/8)—3944no—
135AGL(1,ℤ/8)—1978no—
136C2×SD16—3942no—
137C2×D8—3942no—
138C2×M16C2×D19712yes—
139#124—3942no—
140#124—1974no—
141C2×C22⋊C4—3942no—
142C2×C42—1971no—
143#104—1978no—
144#101C2×D19714yescomplemented
145C24—3941no—
146#41—3948no—
147#41—3948no—
148#41—3948no—
149#41—3942no—
150#41—3948no—
151#41—3942no—
152#41—1974no—
153#41—1974no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—3942no—
157C2×Q8—3948no—
158C2×Q8—3948no—
159C2×Q8—1972no—
160C2×Q8D78812yes—
161C2×Q8C22×D19712yes—
162C2×D4—3942no—
163C2×D4—3942no—
164C2×D4—3942no—
165C2×D4—3942no—
166C2×D4—3941no—
167C2×D4—1972no—
168C2×D4—1972no—
169C22×C4—3944no—
170C22×C4—3941no—
171C22×C4—3942no—
172C22×C4D78812yes—
173C22×C4C22×D19711yescharacteristic
174SD16—3948no—
175D8—3948no—
176M16—22no—
177M16C22×D19714yes—
178C2×C8—22no—
179C22⋊C4—3948no—
180C42—3941no—
181C42—1972no—
182C23—7884no—
183C23—7881no—
184C23—7881no—
185C23—3942no—
186C23—3942no—
187C23C2×D78811yescharacteristic
188Q8—24no—
189Q8—24no—
190Q8—3948no—
191Q8—3942no—
192Q8—1974no—
193D4—7882no—
194D4—3948no—
195D4—3944no—
196D4—3944no—
197D4—3942no—
198D4—3944no—
199D4—1974no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—7882no—
203C2×C4—39416no—
204C2×C4—3942no—
205C2×C4—3942no—
206C2×C4—3944no—
207C2×C4—3944no—
208C2×C4D4×D19712yes—
209C2×C4D4×D19712yes—
210C2×C4C2×D78814yes—
211C2×C4C2×D78811yescharacteristic
212C2×C4C23×D19711yescharacteristic
213C8—24no—
214C22—22no—
215C22—7884no—
216C22—7884no—
217C22—7882no—
218C22—15762no—
219C22—15762no—
220C22—3944no—
221C22#2026329611yescharacteristic
222C22#2026329611yescharacteristic
223C22C22×D78811yescharacteristic
224C4—24no—
225C4—7882no—
226C4—3944no—
227C4#2026329612yes—
228C4C2×D4×D19712yes—
229C2—21no—
230C2—21no—
231C2—7884no—
232C2—15762no—
233C2#2716125412yesdirect product factor
234C2C2×#2026329611yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#101×C197C2
socleC22×C197#120
hypercenterC2×#101D197
hypocenterC197C2×#454
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
1. C2×#20263374C2semidirect
2. C2×#20263354C2semidirect
3. C2×#20263315C2semidirect
4. #27161254C2direct (N × H)
5. C2×#20263252C2semidirect
6. C2×#101×C197C2semidirect
9. #20263374C22semidirect
16. #20263354C22semidirect
20. #20263315C22semidirect
23. #20263252C22semidirect
26. #101×C197C22semidirect
113. C2×C197#454semidirect
117. C197C2×#454semidirect
124. C2×#101D197semidirect
144. #101C2×D197semidirect
233. C2#27161254direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#202633742
C2×#202633542
C2×#202633152
#271612542
C2×#202632522
C2×#101×C1972
C2×#454197