Finite Group Databasebeta

← C2×#127×D97 #27155006

Subgroups — C2×#127×D97 #27155006

Order 12416.

3000 conjugacy classes of subgroups; only the 661 normal subgroups are shown.

normal subgroups only

This group has 3000 conjugacy classes of subgroups — too many to draw. Only its 661 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×D4×D97C216yesmaximal, complemented
2C2×#15690807C211yesmaximal, characteristic, complemented
3C2×#15690806C216yesmaximal, complemented
4#127×D97C2116yesmaximal, direct product factor
5C2×C42×D97C211yesmaximal, characteristic, complemented
6C2×#127×C97C211yesmaximal, characteristic, complemented
7C24×D97C2214yes—
8C22×D388C2216yescomplemented
9C22×C22⋊D97C22112yescomplemented
10C2×D4×D97—248no—
11C2×D4×D97C22148yescomplemented
12C22×C4×D97C2213yes—
13#15690807C2218yescomplemented
14#15690806C22148yescomplemented
15C2×C4×Dic97C2213yescomplemented
16C42×D97C2218yescomplemented
17C22×D4×C97C2216yescomplemented
18#127×C97C2218yescomplemented
19C2×C42×C97C2211yescharacteristic, complemented
20C23×D97—248no—
21C23×D97C23116yes—
22C23×D97C2314yes—
23C23×D97C2311yescharacteristic
24C2×D388—224no—
25C2×D388C23124yescomplemented
26C2×C22⋊D97—248no—
27C2×C22⋊D97C23148yescomplemented
28D4×D97—2192no—
29C22×Dic97C2313yes—
30C2×C4×D97D4124yescomplemented
31C2×C4×D97C23112yes—
32C4×Dic97C23112yescomplemented
33C24×C97C2314yes—
34C2×D4×C97—224no—
35C2×D4×C97C23124yescomplemented
36C22×C4×C97C2313yes—
37C42×C97C2314yescomplemented
38C23×C97—224no—
39C23×C97C2418yes—
40C23×C97C2411yescharacteristic
41C22×D97—224no—
42C22×D97—296no—
43C22×D97—464no—
44C22×D97C2×D4112yes—
45C22×D97C2418yes—
46C22×D97C2412yes—
47D4×C97—248no—
48C2×C4×C97C2×D4112yescomplemented
49C2×C4×C97C2416yes—
50C22⋊D97—296no—
51C2×Dic97C2×D4112yescomplemented
52C2×Dic97C2416yes—
53D388—248no—
54C4×D97C2×D4148yescomplemented
55C22×C97—224no—
56C22×C97—416no—
57C22×C97C22×D416yes—
58C22×C97C2511yescharacteristic
59C2×D97—224no—
60C2×D97—416no—
61C2×D97—464no—
62C2×D97#127116yesdirect product factor
63C2×D97C22×D4112yes—
64C4×C97C22×D4112yescomplemented
65Dic97C22×D4112yescomplemented
66C2×C97—48no—
67C2×C97C2×#12714yescomplemented
68C2×C97C23×D413yes—
69D97—48no—
70D97C2×#12718yesdirect product factor
71C22×#127—971nomaximal, Sylow
72C97C22×#12711yescharacteristic, complemented, Sylow
73C23×D4—976no—
74C2×#127—976no—
75C2×#127—9716no—
76C2×#127—971no—
77C2×#127D9711yescharacteristic, direct product factor
78C22×C42—971no—
79C25—974no—
80C22×D4—9712no—
81C22×D4—9748no—
82C22×D4—976no—
83C22×D4—19448no—
84C22×D4C2×D9716yescomplemented
85C23×C4—973no—
86#127—9748no—
87#127—978no—
88#127C2×D9718yesdirect product factor
89C2×C42—973no—
90C2×C42—978no—
91C2×C42C2×D9711yescharacteristic, complemented
92C24—971no—
93C24—9716no—
94C24—974no—
95C24—19448no—
96C24C22×D9714yes—
97C2×D4—224no—
98C2×D4—9748no—
99C2×D4—9724no—
100C2×D4—194192no—
101C2×D4—19448no—
102C2×D4—19424no—
103C2×D4C22×D97124yescomplemented
104C22×C4—973no—
105C22×C4—9724no—
106C22×C4—9712no—
107C22×C4C22×D9713yes—
108C42—9712no—
109C42C22×D9714yescomplemented
110C23—224no—
111C23—9712no—
112C23—972no—
113C23—978no—
114C23—38864no—
115C23—19496no—
116C23—19424no—
117C23C23×D9718yes—
118C23C23×D9711yescharacteristic
119D4—248no—
120D4—19496no—
121D4—19448no—
122C2×C4—9712no—
123C2×C4—976no—
124C2×C4—9748no—
125C2×C4D4×D97112yescomplemented
126C2×C4C23×D9716yes—
127C22—224no—
128C22—416no—
129C22—9716no—
130C22—9712no—
131C22—38864no—
132C22—38816no—
133C22—19424no—
134C22C2×D4×D9716yes—
135C22C24×D9711yescharacteristic
136C4—9712no—
137C4C2×D4×D97112yescomplemented
138C2—48no—
139C2—978no—
140C2—3888no—
141C2#127×D9714yesdirect product factor
142C2C22×D4×D9713yes—
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#127×C97C2
socleC23×C97C24
hypercenterC2×#127D97
hypocenterC97C22×#127
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×D4×D97C2semidirect
2. C2×#15690807C2semidirect
3. C2×#15690806C2semidirect
4. #127×D97C2direct (N × H)
5. C2×C42×D97C2semidirect
6. C2×#127×C97C2semidirect
8. C22×D388C22semidirect
9. C22×C22⋊D97C22semidirect
11. C2×D4×D97C22semidirect
13. #15690807C22semidirect
14. #15690806C22semidirect
15. C2×C4×Dic97C22semidirect
16. C42×D97C22semidirect
17. C22×D4×C97C22semidirect
18. #127×C97C22semidirect
19. C2×C42×C97C22semidirect
25. C2×D388C23semidirect
27. C2×C22⋊D97C23semidirect
30. C2×C4×D97D4semidirect
32. C4×Dic97C23semidirect
35. C2×D4×C97C23semidirect
37. C42×C97C23semidirect
48. C2×C4×C97C2×D4semidirect
51. C2×Dic97C2×D4semidirect
54. C4×D97C2×D4semidirect
62. C2×D97#127direct (N × H)
64. C4×C97C22×D4semidirect
65. Dic97C22×D4semidirect
67. C2×C97C2×#127semidirect
70. D97C2×#127direct (N × H)
72. C97C22×#127semidirect
77. C2×#127D97direct (N × H)
84. C22×D4C2×D97semidirect
88. #127C2×D97direct (N × H)
91. C2×C42C2×D97semidirect
103. C2×D4C22×D97semidirect
109. C42C22×D97semidirect
125. C2×C4D4×D97semidirect
137. C4C2×D4×D97semidirect
141. C2#127×D97direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D972
C2×#156908072
C2×#156908062
#127×D972
C2×C42×D972
C2×#127×C972
C22×#12797