Finite Group Databasebeta

← C2×#27161309 #38752273

Subgroups — C2×#27161309 #38752273

Order 25216.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15691347C211yesmaximal, characteristic, complemented
2C22×C8×D197C211yesmaximal, characteristic, complemented
3C2×#20263306C211yesmaximal, characteristic, complemented
4C2×#20263301C211yesmaximal, characteristic
5#27161309C218yesmaximal, direct product factor
6C2×#20263249C211yesmaximal, characteristic, complemented
7C2×#20263246C211yesmaximal, characteristic, complemented
8C2×#106×C197C211yesmaximal, characteristic, complemented
9C22×Dic394C2211yescharacteristic
10C22×D788—21no—
11C22×D788C2211yescharacteristic
12C22×C4×D197C2211yescharacteristic
13C2×#15691347—24no—
14C2×#15691347C2214yescomplemented
15C22×#13096802C2211yescharacteristic
16C2×C8×D197—24no—
17C2×C8×D197C2214yescomplemented
18#20263306—22no—
19#20263306C2214yescomplemented
20#20263301—22no—
21#20263301C2214yes—
22C2×C4⋊Dic197—21no—
23C2×#15691335—21no—
24C2×#15691335—21no—
25#20263249C2214yescomplemented
26#20263246C2214yescomplemented
27C22×C8×C197C2211yescharacteristic
28C2×C4⋊C4×C197C2211yescharacteristic
29C2×C4⋊C4×C197C2211yescharacteristic
30#106×C197C2214yescomplemented
31C23×D197—21no—
32C23×D197D411yescharacteristic
33C2×Dic394—21no—
34C2×Dic394—22no—
35C2×Dic394—22no—
36C2×Dic394C2312yes—
37C2×D788—24no—
38C2×D788—22no—
39C2×D788—21no—
40C2×D788—22no—
41C2×D788—44no—
42C2×D788C2312yes—
43C22×Dic197—21no—
44C22×Dic197D411yescharacteristic
45C2×C4×D197—22no—
46C2×C4×D197—22no—
47C2×C4×D197D412yes—
48C2×C4×D197C2312yes—
49#15691347—28no—
50C2×#13096802—22no—
51C2×#13096802C2312yes—
52C8×D197—28no—
53C4⋊Dic197—24no—
54#15691335—24no—
55#15691335—24no—
56C22×C4×C197—21no—
57C22×C4×C197—21no—
58C22×C4×C197C2311yescharacteristic
59C2×C8×C197D414yescomplemented
60C2×C8×C197C2312yes—
61C4⋊C4×C197—21no—
62C4⋊C4×C197—21no—
63C4⋊C4×C197C2312yes—
64C4⋊C4×C197C2312yes—
65C22×D197—22no—
66C22×D197—22no—
67C22×D197—22no—
68C22×D197—42no—
69C22×D197—42no—
70C22×D197—42no—
71C22×D197SD1614yescomplemented
72C22×D197C2×D412yes—
73Dic394—22no—
74Dic394—22no—
75Dic394—42no—
76D788—22no—
77D788—22no—
78D788—48no—
79D788—42no—
80C2×Dic197—22no—
81C2×Dic197—22no—
82C2×Dic197—42no—
83C2×Dic197C2×D412yes—
84C4×D197—24no—
85C4×D197—22no—
86C4×D197—22no—
87#13096802—22no—
88C23×C197C2×D411yescharacteristic
89C2×C4×C197—22no—
90C2×C4×C197—22no—
91C2×C4×C197—42no—
92C2×C4×C197—42no—
93C2×C4×C197C2×D411yescharacteristic
94C2×C4×C197C2×D412yes—
95C2×C4×C197#4112yes—
96C2×C4×C197C2411yescharacteristic
97C8×C197C2×D414yescomplemented
98C22×C197#12112yes—
99C22×C197C2×SD1612yes—
100C22×C197#13512yes—
101C22×C197C22×D411yescharacteristic
102C2×D197—22no—
103C2×D197—22no—
104C2×D197—28no—
105C2×D197—42no—
106C2×D197—42no—
107C2×D197—42no—
108C2×D197—84no—
109C2×D197C2×SD1614yescomplemented
110C4×C197—42no—
111C4×C197—42no—
112C4×C197#12112yes—
113C4×C197C22×D411yescharacteristic
114C4×C197C2×#4111yescharacteristic
115Dic197—22no—
116Dic197—42no—
117C2×C197#46514yescomplemented
118C2×C197C2×#12111yescharacteristic
119C2×C197C22×SD1611yescharacteristic
120C2×C197C2×#13511yescharacteristic
121D197—24no—
122D197—82no—
123C197C2×#46511yescharacteristic, complemented, Sylow
124C2×#465—1971nomaximal, Sylow
125C22×SD16—1971no—
126C23×C8—1971no—
127C2×C22⋊Q8—1971no—
128C2×#121—1971no—
129#465—1978no—
130C2×#106D19711yescharacteristic, complemented
131C2×#103—1971no—
132C2×#102—1971no—
133C22×Q8—1971no—
134C22×D4—3941no—
135C22×D4—1971no—
136C23×C4—1971no—
137C2×SD16—3944no—
138C2×SD16—1974no—
139C22×C8—3944no—
140C22×C8—1971no—
141C22×C8—1974no—
142C22×C8C2×D19711yescharacteristic
143C22⋊Q8—3942no—
144C22⋊Q8—1974no—
145#121—3942no—
146#121—1974no—
147C2×C4⋊C4—3941no—
148C2×C4⋊C4C2×D19711yescharacteristic
149C2×C4⋊C4C2×D19711yescharacteristic
150C2×C22⋊C4—3941no—
151C2×C22⋊C4—3941no—
152#106C2×D19714yescomplemented
153#103—1974no—
154#102—1974no—
155C24—3941no—
156C24—1971no—
157C2×Q8—3942no—
158C2×Q8—3942no—
159C2×Q8—3941no—
160C2×Q8—1972no—
161C2×D4—7884no—
162C2×D4—3944no—
163C2×D4—3941no—
164C2×D4—3942no—
165C2×D4—3942no—
166C2×D4—1972no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—3941no—
170C22×C4—3942no—
171C22×C4—3942no—
172C22×C4—1971no—
173C22×C4—1972no—
174C22×C4—1972no—
175C22×C4C22×D19711yescharacteristic
176SD16—3948no—
177C2×C8—3942no—
178C2×C8—3948no—
179C2×C8—1972no—
180C2×C8D78814yescomplemented
181C2×C8C22×D19712yes—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—3944no—
185C4⋊C4C22×D19712yes—
186C4⋊C4C22×D19712yes—
187C22⋊C4—3944no—
188C22⋊C4—3944no—
189C23—7882no—
190C23—7882no—
191C23—7882no—
192C23—3942no—
193C23—3942no—
194C23—3942no—
195C23—1974no—
196C23—1972no—
197C23D4×D19711yescharacteristic
198Q8—7882no—
199Q8—3942no—
200Q8—3942no—
201D4—7888no—
202D4—7882no—
203D4—3942no—
204D4—3942no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—7882no—
210C2×C4—3942no—
211C2×C4—3942no—
212C2×C4—3944no—
213C2×C4—3942no—
214C2×C4—3942no—
215C2×C4—1972no—
216C2×C4D4×D19711yescharacteristic
217C2×C4C2×D78812yes—
218C2×C4#1569135912yes—
219C2×C4C23×D19711yescharacteristic
220C8—3942no—
221C8C2×D78814yescomplemented
222C22—7882no—
223C22—7882no—
224C22—7882no—
225C22—15764no—
226C22—3942no—
227C22—3948no—
228C22—3942no—
229C22—1974no—
230C22#2026330112yes—
231C22SD16×D19712yes—
232C22#2026334712yes—
233C22C2×D4×D19711yescharacteristic
234C4—42no—
235C4—42no—
236C4—7882no—
237C4—3942no—
238C4#2026330112yes—
239C4C22×D78811yescharacteristic
240C4C2×#1569135911yescharacteristic
241C2—15762no—
242C2—3944no—
243C2#2716130914yesdirect product factor
244C2C2×#2026330111yescharacteristic
245C2C2×SD16×D19711yescharacteristic
246C2C2×#2026334711yescharacteristic
subgroupG/H
centerC23D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#106×C197C2
socleC23×C197C2×D4
prime radicalC2×#20263301C2
hypercenterC2×#106D197
hypocenterC197C2×#465
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
1. C22×#15691347C2semidirect
2. C22×C8×D197C2semidirect
3. C2×#20263306C2semidirect
5. #27161309C2direct (N × H)
6. C2×#20263249C2semidirect
7. C2×#20263246C2semidirect
8. C2×#106×C197C2semidirect
14. C2×#15691347C22semidirect
17. C2×C8×D197C22semidirect
19. #20263306C22semidirect
25. #20263249C22semidirect
26. #20263246C22semidirect
30. #106×C197C22semidirect
59. C2×C8×C197D4semidirect
71. C22×D197SD16semidirect
97. C8×C197C2×D4semidirect
109. C2×D197C2×SD16semidirect
117. C2×C197#465semidirect
123. C197C2×#465semidirect
130. C2×#106D197semidirect
152. #106C2×D197semidirect
180. C2×C8D788semidirect
221. C8C2×D788semidirect
243. C2#27161309direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156913472
C22×C8×D1972
C2×#202633062
C2×#202633012
#271613092
C2×#202632492
C2×#202632462
C2×#106×C1972
C2×#465197