Finite Group Databasebeta

← C2×#27161363 #38752476

Subgroups — C2×#27161363 #38752476

Order 25216.

416 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 157 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×#20263325C211yesmaximal, characteristic
2C2×#20263291C211yesmaximal, characteristic
3#27161363C218yesmaximal, direct product factor
4C2×#20263257C211yesmaximal, characteristic
5C2×#20263255C211yesmaximal, characteristic
6C2×#20263251C211yesmaximal, characteristic
7C2×#20263249C211yesmaximal, characteristic
8C2×#106×C197C211yesmaximal, characteristic
9C22×Dic394—21no—
10C22×Dic394C2211yescharacteristic
11C22×#13096802C2211yescharacteristic
12#20263325—22no—
13#20263325C2214yes—
14#20263291—22no—
15#20263291C2214yes—
16C2×C4⋊Dic197C2211yescharacteristic
17C2×#15691337—21no—
18C2×#15691337—21no—
19C2×C4×Dic197—21no—
20C2×C4×Dic197C2211yescharacteristic
21#20263257C2214yes—
22#20263255C2214yes—
23#20263251C2214yes—
24#20263249C2214yes—
25C22×C8×C197C2211yescharacteristic
26C2×C4⋊C4×C197C2211yescharacteristic
27C2×C4⋊C4×C197C2211yescharacteristic
28#106×C197C2214yes—
29C2×Dic394—24no—
30C2×Dic394—21no—
31C2×Dic394—22no—
32C2×Dic394—44no—
33C2×Dic394Q814yescomplemented
34C2×Dic394C2312yes—
35C22×Dic197—21no—
36C22×Dic197—21no—
37C22×Dic197D411yescharacteristic
38C22×Dic197D411yescharacteristic
39C2×#13096802—22no—
40C2×#13096802C2312yes—
41C4⋊Dic197—21no—
42C4⋊Dic197C2312yes—
43#15691337—24no—
44#15691337—24no—
45C4×Dic197—24no—
46C4×Dic197D412yes—
47C4×Dic197C2312yes—
48C22×C4×C197—21no—
49C22×C4×C197—21no—
50C22×C4×C197C2311yescharacteristic
51C2×C8×C197—22no—
52C2×C8×C197C2312yes—
53C4⋊C4×C197—21no—
54C4⋊C4×C197—21no—
55C4⋊C4×C197C2312yes—
56C4⋊C4×C197C2312yes—
57Dic394—24no—
58Dic394—22no—
59Dic394—48no—
60Dic394C2×Q814yescomplemented
61C2×Dic197—24no—
62C2×Dic197—22no—
63C2×Dic197—22no—
64C2×Dic197—22no—
65C2×Dic197—42no—
66C2×Dic197SD1614yes—
67C2×Dic197C2×D412yes—
68C2×Dic197C2×D412yes—
69#13096802—22no—
70C23×C197C2×D411yescharacteristic
71C2×C4×C197—22no—
72C2×C4×C197—22no—
73C2×C4×C197—42no—
74C2×C4×C197—42no—
75C2×C4×C197C2×D411yescharacteristic
76C2×C4×C197C2×Q812yes—
77C2×C4×C197#4112yes—
78C2×C4×C197C2411yescharacteristic
79C8×C197—22no—
80C22×C197C22⋊Q812yes—
81C22×C197C2×SD1612yes—
82C22×C197#13712yes—
83C22×C197C22×D411yescharacteristic
84C4×C197—42no—
85C4×C197—42no—
86C4×C197C22⋊Q812yes—
87C4×C197C22×Q811yescharacteristic
88C4×C197C2×#4111yescharacteristic
89Dic197—24no—
90Dic197—22no—
91Dic197—42no—
92Dic197C2×SD1614yes—
93C2×C197#47514yescomplemented
94C2×C197C2×C22⋊Q811yescharacteristic
95C2×C197C22×SD1611yescharacteristic
96C2×C197C2×#13711yescharacteristic
97C197C2×#47511yescharacteristic, complemented, Sylow
98C2×#475—1971nomaximal, Sylow
99C2×C4⋊Q8—1971no—
100C2×C4×Q8—1971no—
101#475—1978no—
102C2×#106—1971no—
103C2×#106D19711yescharacteristic
104C2×C4⋊C8—1971no—
105C2×#103—1971no—
106C2×#103—1971no—
107C22×Q8—3941no—
108C22×Q8—1971no—
109C22×C8—1971no—
110C22×C8C2×D19711yescharacteristic
111C4⋊Q8—3942no—
112C4⋊Q8—1974no—
113C4×Q8—3942no—
114C4×Q8—1974no—
115C2×C4⋊C4—3941no—
116C2×C4⋊C4—3941no—
117C2×C4⋊C4—1971no—
118C2×C4⋊C4C2×D19711yescharacteristic
119C2×C4⋊C4C2×D19711yescharacteristic
120C2×C42—3941no—
121C2×C42—1971no—
122#106—1974no—
123#106C2×D19714yes—
124C4⋊C8—1974no—
125#103—1974no—
126#103—1974no—
127C2×Q8—7884no—
128C2×Q8—3944no—
129C2×Q8—3941no—
130C2×Q8—3942no—
131C2×Q8—1974no—
132C2×Q8—1972no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—3941no—
136C22×C4—3941no—
137C22×C4—1971no—
138C22×C4—1971no—
139C22×C4C22×D19711yescharacteristic
140C2×C8—22no—
141C2×C8—3942no—
142C2×C8—1972no—
143C2×C8C22×D19712yes—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—3944no—
147C4⋊C4—3944no—
148C4⋊C4—3941no—
149C4⋊C4—1972no—
150C4⋊C4C22×D19712yes—
151C4⋊C4C22×D19712yes—
152C42—3944no—
153C42—1972no—
154C42—1972no—
155C23D4×D19711yescharacteristic
156Q8—7888no—
157Q8—3944no—
158Q8—3942no—
159Q8—1974no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—7882no—
165C2×C4—3942no—
166C2×C4—3944no—
167C2×C4—3942no—
168C2×C4—3942no—
169C2×C4—1972no—
170C2×C4—1974no—
171C2×C4—1972no—
172C2×C4D4×D19711yescharacteristic
173C2×C4Q8×D19712yes—
174C2×C4#1569135812yes—
175C2×C4C23×D19711yescharacteristic
176C8—22no—
177C8—3942no—
178C22#2026330412yes—
179C22SD16×D19712yes—
180C22#2026335512yes—
181C22C2×D4×D19711yescharacteristic
182C4—42no—
183C4—42no—
184C4—7882no—
185C4—3942no—
186C4—3944no—
187C4—1974no—
188C4#2026330412yes—
189C4C2×Q8×D19711yescharacteristic
190C4C2×#1569135811yescharacteristic
191C2#2716136314yesdirect product factor
192C2C2×#2026330411yescharacteristic
193C2C2×SD16×D19711yescharacteristic
194C2C2×#2026335511yescharacteristic
subgroupG/H
centerC23D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#106×C197C2
socle, prime radicalC23×C197C2×D4
hypercenterC2×#106D197
hypocenterC197C2×#475
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
3. #27161363C2direct (N × H)
33. C2×Dic394Q8semidirect
60. Dic394C2×Q8semidirect
93. C2×C197#475semidirect
97. C197C2×#475semidirect
191. C2#27161363direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#202633252
C2×#202632912
#271613632
C2×#202632572
C2×#202632552
C2×#202632512
C2×#202632492
C2×#106×C1972
C2×#475197