Finite Group Databasebeta

← C22×#15690806 #27155007

Subgroups — C22×#15690806 #27155007

Order 12416.

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×D388C211yesmaximal, characteristic, complemented
2C23×C22⋊D97C214yesmaximal, complemented
3C2×#15690806C2124yesmaximal, direct product factor
4C22×C4×Dic97C211yesmaximal, characteristic, complemented
5C23×D4×C97C211yesmaximal, characteristic, complemented
6C24×D97C2212yes—
7C22×D388—28no—
8C22×D388C22112yescomplemented
9C22×C22⋊D97—232no—
10C22×C22⋊D97C22148yescomplemented
11C23×Dic97C2212yes—
12#15690806C22164yesdirect product factor
13C2×C4×Dic97C22112yescomplemented
14C25×C97C2212yes—
15C22×D4×C97—28no—
16C22×D4×C97C22112yescomplemented
17C23×C4×C97C2211yescharacteristic
18C23×D97—216no—
19C23×D97—28no—
20C23×D97C23112yes—
21C2×D388—248no—
22C2×D388C23116yescomplemented
23C2×C22⋊D97—2192no—
24C2×C22⋊D97C23164yescomplemented
25C22×Dic97D4116yescomplemented
26C22×Dic97C23112yes—
27C4×Dic97C23116yescomplemented
28C24×C97—216no—
29C24×C97—28no—
30C24×C97C23112yes—
31C24×C97C2311yescharacteristic
32C2×D4×C97—248no—
33C2×D4×C97C23116yescomplemented
34C22×C4×C97D418yescomplemented
35C22×C4×C97C2316yes—
36C23×C97—224no—
37C23×C97—248no—
38C23×C97—432no—
39C23×C97C2×D418yes—
40C23×C97C2×D414yes—
41C23×C97C2418yes—
42C23×C97C2413yes—
43C22×D97—248no—
44C22×D97—224no—
45C22×D97—432no—
46C22×D97C2418yes—
47D4×C97—232no—
48C2×C4×C97C2×D4124yescomplemented
49C2×C4×C97C2414yes—
50C22⋊D97—2128no—
51C2×Dic97C2×D4148yescomplemented
52C2×Dic97C2418yes—
53D388—232no—
54C22×C97—216no—
55C22×C97—28no—
56C22×C97—448no—
57C22×C97#127116yescomplemented
58C22×C97C22×D4112yes—
59C22×C97C22×D416yes—
60C22×C97C2511yescharacteristic
61C2×D97—28no—
62C2×D97—216no—
63C2×D97—448no—
64C4×C97C22×D418yescomplemented
65Dic97C22×D4116yescomplemented
66C2×C97—48no—
67C2×C97C2×#127112yescomplemented
68C2×C97C23×D412yes—
69C2×C97C23×D411yescharacteristic
70D97—48no—
71C22×#127—971nomaximal, Sylow
72C97C22×#12711yescharacteristic, complemented, Sylow
73C23×D4—974no—
74C23×D4—971no—
75C23×D4D9711yescharacteristic, complemented
76C2×#127—9724no—
77C22×C42—971no—
78C25—972no—
79C25C2×D9712yes—
80C22×D4—28no—
81C22×D4—9748no—
82C22×D4—9712no—
83C22×D4—19432no—
84C22×D4—1948no—
85C22×D4C2×D97112yescomplemented
86C23×C4—972no—
87C23×C4C2×D9711yescharacteristic
88#127—9764no—
89C2×C42—9712no—
90C24—216no—
91C24—28no—
92C24—9712no—
93C24—19416no—
94C24—1948no—
95C24C22×D97112yes—
96C24C22×D9711yescharacteristic
97C2×D4—248no—
98C2×D4—9764no—
99C2×D4—9716no—
100C2×D4—194192no—
101C2×D4—19448no—
102C2×D4C22×D97116yescomplemented
103C22×C4—9716no—
104C22×C4—9712no—
105C22×C4C22⋊D9718yescomplemented
106C22×C4C22×D9716yes—
107C42—9716no—
108C23—248no—
109C23—224no—
110C23—432no—
111C23—978no—
112C23—38832no—
113C23—19448no—
114C23—19424no—
115C23D4×D9718yes—
116C23C2×C22⋊D9714yes—
117C23C23×D9718yes—
118C23C23×D9713yes—
119D4—232no—
120D4—194128no—
121D4—19432no—
122C2×C4—9748no—
123C2×C4—978no—
124C2×C4C2×C22⋊D97124yescomplemented
125C2×C4C23×D9714yes—
126C22—216no—
127C22—28no—
128C22—448no—
129C22—38848no—
130C22—19416no—
131C22—1948no—
132C22#15690806116yesdirect product factor
133C22C2×D4×D97112yes—
134C22C22×C22⋊D9716yes—
135C22C24×D9711yescharacteristic
136C4—9716no—
137C4C22×C22⋊D9718yescomplemented
138C2—48no—
139C2—3888no—
140C2C2×#15690806112yesdirect product factor
141C2C22×D4×D9712yes—
142C2C23×C22⋊D9711yescharacteristic
subgroupG/H
centerC24C22×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC23×D4×C97C2
socleC24×C97C23
hypercenterC23×D4D97
hypocenterC97C22×#127
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×D388C2semidirect
2. C23×C22⋊D97C2semidirect
3. C2×#15690806C2direct (N × H)
4. C22×C4×Dic97C2semidirect
5. C23×D4×C97C2semidirect
8. C22×D388C22semidirect
10. C22×C22⋊D97C22semidirect
12. #15690806C22direct (N × H)
13. C2×C4×Dic97C22semidirect
16. C22×D4×C97C22semidirect
22. C2×D388C23semidirect
24. C2×C22⋊D97C23semidirect
25. C22×Dic97D4semidirect
27. C4×Dic97C23semidirect
33. C2×D4×C97C23semidirect
34. C22×C4×C97D4semidirect
48. C2×C4×C97C2×D4semidirect
51. C2×Dic97C2×D4semidirect
57. C22×C97#127semidirect
64. C4×C97C22×D4semidirect
65. Dic97C22×D4semidirect
67. C2×C97C2×#127semidirect
72. C97C22×#127semidirect
75. C23×D4D97semidirect
85. C22×D4C2×D97semidirect
102. C2×D4C22×D97semidirect
105. C22×C4C22⋊D97semidirect
124. C2×C4C2×C22⋊D97semidirect
132. C22#15690806direct (N × H)
137. C4C22×C22⋊D97semidirect
140. C2C2×#15690806direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D3882
C23×C22⋊D972
C2×#156908062
C22×C4×Dic972
C23×D4×C972
C22×#12797