Finite Group Databasebeta

← C22×#20263297 #38755931

Subgroups — C22×#20263297 #38755931

Order 25216.

2528 conjugacy classes of subgroups; only the 685 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 685 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
1C23×C22⋊D197C212yesmaximal, complemented
2C23×C4×D197C211yesmaximal, characteristic, complemented
3C2×#20263297C2124yesmaximal, direct product factor
4C22×C4⋊Dic197C211yesmaximal, characteristic, complemented
5C22×C22⋊Dic197C212yesmaximal, complemented
6C23×D4×C197C211yesmaximal, characteristic, complemented
7C24×D197C2211yescharacteristic
8C22×C22⋊D197—216no—
9C22×C22⋊D197C22124yescomplemented
10C23×Dic197C2211yescharacteristic
11C23×Dic197C2212yes—
12C22×C4×D197—28no—
13C22×C4×D197C22112yescomplemented
14#20263297C22164yesdirect product factor
15C2×C4⋊Dic197C22112yescomplemented
16C2×C22⋊Dic197C22124yescomplemented
17C25×C197C2212yes—
18C22×D4×C197—28no—
19C22×D4×C197C22112yescomplemented
20C23×C4×C197C2211yescharacteristic
21C23×D197—24no—
22C23×D197—24no—
23C23×D197D418yescomplemented
24C23×D197C2316yes—
25C2×C22⋊D197—296no—
26C2×C22⋊D197C23132yescomplemented
27C22×Dic197—24no—
28C22×Dic197—28no—
29C22×Dic197C2316yes—
30C22×Dic197C23112yes—
31C2×C4×D197—248no—
32C2×C4×D197C23116yescomplemented
33C4⋊Dic197C23116yescomplemented
34C22⋊Dic197C23132yescomplemented
35C24×C197—28no—
36C24×C197—28no—
37C24×C197—28no—
38C24×C197C23112yes—
39C24×C197C2311yescharacteristic
40C2×D4×C197—248no—
41C2×D4×C197C23116yescomplemented
42C22×C4×C197D418yescomplemented
43C22×C4×C197C2316yes—
44C22×D197—232no—
45C22×D197—212no—
46C22×D197—212no—
47C22×D197C2×D4124yescomplemented
48C22×D197C2414yes—
49C22⋊D197—264no—
50C2×Dic197—212no—
51C2×Dic197—224no—
52C2×Dic197C2414yes—
53C2×Dic197C2418yes—
54C4×D197—232no—
55C23×C197—224no—
56C23×C197—224no—
57C23×C197—224no—
58C23×C197—432no—
59C23×C197C2×D414yes—
60C23×C197C2×D414yes—
61C23×C197#4114yes—
62C23×C197C2418yes—
63C23×C197C2413yes—
64D4×C197—232no—
65C2×C4×C197C2×D4124yescomplemented
66C2×C4×C197C2414yes—
67C22×C197—28no—
68C22×C197—28no—
69C22×C197—28no—
70C22×C197—448no—
71C22×C197#121116yescomplemented
72C22×C197C22×D416yes—
73C22×C197C22×D416yes—
74C22×C197C2×#4116yes—
75C22×C197C2511yescharacteristic
76C2×D197—248no—
77C2×D197—24no—
78C2×D197—24no—
79C2×D197C22×D418yescomplemented
80C4×C197C22×D418yescomplemented
81Dic197—24no—
82Dic197—28no—
83C2×C197—48no—
84C2×C197C2×#121112yescomplemented
85C2×C197C23×D411yescharacteristic
86C2×C197C23×D411yescharacteristic
87C2×C197C22×#4111yescharacteristic
88D197—28no—
89C197C22×#12111yescharacteristic, complemented, Sylow
90C22×#121—1971nomaximal, Sylow
91C23×D4—1972no—
92C23×D4D19711yescharacteristic, complemented
93C24×C4—1971no—
94C2×#121—19724no—
95C22×C4⋊C4—1971no—
96C22×C22⋊C4—1972no—
97C25—1971no—
98C25C2×D19712yes—
99C22×D4—28no—
100C22×D4—39416no—
101C22×D4—19724no—
102C22×D4C2×D197112yescomplemented
103C23×C4—3948no—
104C23×C4—1971no—
105C23×C4—1972no—
106C23×C4—19712no—
107C23×C4C2×D19711yescharacteristic
108#121—19764no—
109C2×C4⋊C4—19712no—
110C2×C22⋊C4—19724no—
111C24—28no—
112C24—28no—
113C24—28no—
114C24—3944no—
115C24—3944no—
116C24—1978no—
117C24—1976no—
118C24C22×D197112yes—
119C24C22×D19711yescharacteristic
120C2×D4—248no—
121C2×D4—39496no—
122C2×D4—19732no—
123C2×D4C22×D197116yescomplemented
124C22×C4—3948no—
125C22×C4—3944no—
126C22×C4—39448no—
127C22×C4—1976no—
128C22×C4—19712no—
129C22×C4—19716no—
130C22×C4C22⋊D19718yescomplemented
131C22×C4C22×D19716yes—
132C4⋊C4—19716no—
133C22⋊C4—19732no—
134C23—224no—
135C23—224no—
136C23—224no—
137C23—432no—
138C23—39432no—
139C23—39412no—
140C23—39412no—
141C23—19724no—
142C23—1974no—
143C23D4×D19714yes—
144C23C2×C22⋊D19714yes—
145C23#1569135714yes—
146C23C23×D19718yes—
147C23C23×D19713yes—
148D4—232no—
149D4—39464no—
150C2×C4—39424no—
151C2×C4—39412no—
152C2×C4—39432no—
153C2×C4—1974no—
154C2×C4—1978no—
155C2×C4C2×C22⋊D197124yescomplemented
156C2×C4C23×D19714yes—
157C22—28no—
158C22—28no—
159C22—28no—
160C22—448no—
161C22—3944no—
162C22—3944no—
163C22—39448no—
164C22—1978no—
165C22#20263297116yesdirect product factor
166C22C2×D4×D19716yes—
167C22C22×C22⋊D19716yes—
168C22C2×#1569135716yes—
169C22C24×D19711yescharacteristic
170C4—3948no—
171C4—3944no—
172C4C22×C22⋊D19718yescomplemented
173C2—48no—
174C2—3948no—
175C2C2×#20263297112yesdirect product factor
176C2C22×D4×D19711yescharacteristic
177C2C23×C22⋊D19711yescharacteristic
178C2C22×#1569135711yescharacteristic
subgroupG/H
centerC24C22×D197
commutator, cosocleC22×C197C25
FrattiniC22C24×D197
FittingC23×D4×C197C2
socleC24×C197C23
hypercenterC23×D4D197
hypocenterC197C22×#121
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D197C2semidirect
2. C23×C4×D197C2semidirect
3. C2×#20263297C2direct (N × H)
4. C22×C4⋊Dic197C2semidirect
5. C22×C22⋊Dic197C2semidirect
6. C23×D4×C197C2semidirect
9. C22×C22⋊D197C22semidirect
13. C22×C4×D197C22semidirect
14. #20263297C22direct (N × H)
15. C2×C4⋊Dic197C22semidirect
16. C2×C22⋊Dic197C22semidirect
19. C22×D4×C197C22semidirect
23. C23×D197D4semidirect
26. C2×C22⋊D197C23semidirect
32. C2×C4×D197C23semidirect
33. C4⋊Dic197C23semidirect
34. C22⋊Dic197C23semidirect
41. C2×D4×C197C23semidirect
42. C22×C4×C197D4semidirect
47. C22×D197C2×D4semidirect
65. C2×C4×C197C2×D4semidirect
71. C22×C197#121semidirect
79. C2×D197C22×D4semidirect
80. C4×C197C22×D4semidirect
84. C2×C197C2×#121semidirect
89. C197C22×#121semidirect
92. C23×D4D197semidirect
102. C22×D4C2×D197semidirect
123. C2×D4C22×D197semidirect
130. C22×C4C22⋊D197semidirect
155. C2×C4C2×C22⋊D197semidirect
165. C22#20263297direct (N × H)
172. C4C22×C22⋊D197semidirect
175. C2C2×#20263297direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D1972
C23×C4×D1972
C2×#202632972
C22×C4⋊Dic1972
C22×C22⋊Dic1972
C23×D4×C1972
C22×#121197