Finite Group Databasebeta

← C22×#20263322 #38755983

Subgroups — C22×#20263322 #38755983

Order 25216.

3000 conjugacy classes of subgroups; only the 725 normal subgroups are shown.

normal subgroups only

This group has 3000 conjugacy classes of subgroups — too many to draw. Only its 725 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×D788C211yesmaximal, characteristic, complemented
2C23×C22⋊D197C214yesmaximal, complemented
3C2×#20263322C2124yesmaximal, direct product factor
4C22×C4×Dic197C211yesmaximal, characteristic, complemented
5C23×D4×C197C211yesmaximal, characteristic, complemented
6C24×D197C2212yes—
7C22×D788—28no—
8C22×D788C22112yescomplemented
9C22×C22⋊D197—232no—
10C22×C22⋊D197C22148yescomplemented
11C23×Dic197C2212yes—
12#20263322C22164yesdirect product factor
13C2×C4×Dic197C22112yescomplemented
14C25×C197C2212yes—
15C22×D4×C197—28no—
16C22×D4×C197C22112yescomplemented
17C23×C4×C197C2211yescharacteristic
18C23×D197—216no—
19C23×D197—28no—
20C23×D197C23112yes—
21C2×D788—248no—
22C2×D788C23116yescomplemented
23C2×C22⋊D197—2192no—
24C2×C22⋊D197C23164yescomplemented
25C22×Dic197D4116yescomplemented
26C22×Dic197C23112yes—
27C4×Dic197C23116yescomplemented
28C24×C197—28no—
29C24×C197—216no—
30C24×C197C23112yes—
31C24×C197C2311yescharacteristic
32C2×D4×C197—248no—
33C2×D4×C197C23116yescomplemented
34C22×C4×C197D418yescomplemented
35C22×C4×C197C2316yes—
36C22×D197—248no—
37C22×D197—224no—
38C22×D197—432no—
39C22×D197C2418yes—
40D788—232no—
41C22⋊D197—2128no—
42C2×Dic197C2×D4148yescomplemented
43C2×Dic197C2418yes—
44C23×C197—224no—
45C23×C197—248no—
46C23×C197—432no—
47C23×C197C2×D418yes—
48C23×C197C2×D414yes—
49C23×C197C2418yes—
50C23×C197C2413yes—
51D4×C197—232no—
52C2×C4×C197C2×D4124yescomplemented
53C2×C4×C197C2414yes—
54C22×C197—28no—
55C22×C197—216no—
56C22×C197—448no—
57C22×C197#127116yescomplemented
58C22×C197C22×D4112yes—
59C22×C197C22×D416yes—
60C22×C197C2511yescharacteristic
61C2×D197—216no—
62C2×D197—28no—
63C2×D197—448no—
64C4×C197C22×D418yescomplemented
65Dic197C22×D4116yescomplemented
66C2×C197—48no—
67C2×C197C2×#127112yescomplemented
68C2×C197C23×D412yes—
69C2×C197C23×D411yescharacteristic
70D197—48no—
71C197C22×#12711yescharacteristic, complemented, Sylow
72C22×#127—1971nomaximal, Sylow
73C23×D4—1974no—
74C23×D4—1971no—
75C23×D4D19711yescharacteristic, complemented
76C2×#127—19724no—
77C22×C42—1971no—
78C25—1972no—
79C25C2×D19712yes—
80C22×D4—28no—
81C22×D4—39432no—
82C22×D4—3948no—
83C22×D4—19748no—
84C22×D4—19712no—
85C22×D4C2×D197112yescomplemented
86C23×C4—1972no—
87C23×C4C2×D19711yescharacteristic
88#127—19764no—
89C2×C42—19712no—
90C24—216no—
91C24—28no—
92C24—39416no—
93C24—3948no—
94C24—19712no—
95C24C22×D197112yes—
96C24C22×D19711yescharacteristic
97C2×D4—248no—
98C2×D4—394192no—
99C2×D4—39448no—
100C2×D4—19764no—
101C2×D4—19716no—
102C2×D4C22×D197116yescomplemented
103C22×C4—19716no—
104C22×C4—19712no—
105C22×C4C22⋊D19718yescomplemented
106C22×C4C22×D19716yes—
107C42—19716no—
108C23—248no—
109C23—224no—
110C23—432no—
111C23—78832no—
112C23—39448no—
113C23—39424no—
114C23—1978no—
115C23D4×D19718yes—
116C23C2×C22⋊D19714yes—
117C23C23×D19718yes—
118C23C23×D19713yes—
119D4—232no—
120D4—394128no—
121D4—39432no—
122C2×C4—19748no—
123C2×C4—1978no—
124C2×C4C2×C22⋊D197124yescomplemented
125C2×C4C23×D19714yes—
126C22—216no—
127C22—28no—
128C22—448no—
129C22—78848no—
130C22—39416no—
131C22—3948no—
132C22#20263322116yesdirect product factor
133C22C2×D4×D197112yes—
134C22C22×C22⋊D19716yes—
135C22C24×D19711yescharacteristic
136C4—19716no—
137C4C22×C22⋊D19718yescomplemented
138C2—48no—
139C2—7888no—
140C2C2×#20263322112yesdirect product factor
141C2C22×D4×D19712yes—
142C2C23×C22⋊D19711yescharacteristic
subgroupG/H
centerC24C22×D197
commutator, cosocleC22×C197C25
FrattiniC22C24×D197
FittingC23×D4×C197C2
socleC24×C197C23
hypercenterC23×D4D197
hypocenterC197C22×#127
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×D788C2semidirect
2. C23×C22⋊D197C2semidirect
3. C2×#20263322C2direct (N × H)
4. C22×C4×Dic197C2semidirect
5. C23×D4×C197C2semidirect
8. C22×D788C22semidirect
10. C22×C22⋊D197C22semidirect
12. #20263322C22direct (N × H)
13. C2×C4×Dic197C22semidirect
16. C22×D4×C197C22semidirect
22. C2×D788C23semidirect
24. C2×C22⋊D197C23semidirect
25. C22×Dic197D4semidirect
27. C4×Dic197C23semidirect
33. C2×D4×C197C23semidirect
34. C22×C4×C197D4semidirect
42. C2×Dic197C2×D4semidirect
52. C2×C4×C197C2×D4semidirect
57. C22×C197#127semidirect
64. C4×C197C22×D4semidirect
65. Dic197C22×D4semidirect
67. C2×C197C2×#127semidirect
71. C197C22×#127semidirect
75. C23×D4D197semidirect
85. C22×D4C2×D197semidirect
102. C2×D4C22×D197semidirect
105. C22×C4C22⋊D197semidirect
124. C2×C4C2×C22⋊D197semidirect
132. C22#20263322direct (N × H)
137. C4C22×C22⋊D197semidirect
140. C2C2×#20263322direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D7882
C23×C22⋊D1972
C2×#202633222
C22×C4×Dic1972
C23×D4×C1972
C22×#127197