Finite Group Databasebeta

← C2×#127×D23 #15686667

Subgroups — C2×#127×D23 #15686667

Order 2944.

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×D23C216yesmaximal, complemented
2C2×#10625045C211yesmaximal, characteristic, complemented
3C2×#10625111C216yesmaximal, complemented
4#127×D23C2116yesmaximal, direct product factor
5C2×C42×D23C211yesmaximal, characteristic, complemented
6C2×C23×#127C211yesmaximal, characteristic, complemented
7C24×D23C2214yes—
8C22×D4×C23C2216yescomplemented
9C22×C22⋊D23C22112yescomplemented
10C2×D4×D23—248no—
11C2×D4×D23C22148yescomplemented
12C22×D92C2216yescomplemented
13C22×C4×D23C2213yes—
14C23×#127C2218yescomplemented
15C2×C42×C23C2211yescharacteristic, complemented
16#10625111C22148yescomplemented
17C2×C4×Dic23C2213yescomplemented
18#10625045C2218yescomplemented
19C42×D23C2218yescomplemented
20C24×C23C2314yes—
21C23×D23—248no—
22C23×D23C2314yes—
23C23×D23C23116yes—
24C23×D23C2311yescharacteristic
25C2×D4×C23—224no—
26C2×D4×C23C23124yescomplemented
27C22×C4×C23C2313yes—
28C2×C22⋊D23—248no—
29C2×C22⋊D23C23148yescomplemented
30C22×Dic23C2313yes—
31D4×D23—2192no—
32C2×D92—224no—
33C2×D92C23124yescomplemented
34C2×C4×D23D4124yescomplemented
35C2×C4×D23C23112yes—
36C42×C23C2314yescomplemented
37C4×Dic23C23112yescomplemented
38C23×C23—224no—
39C23×C23C2418yes—
40C23×C23C2411yescharacteristic
41C22×D23—224no—
42C22×D23—296no—
43C22×D23—464no—
44C22×D23C2×D4112yes—
45C22×D23C2418yes—
46C22×D23C2412yes—
47D4×C23—248no—
48C2×C4×C23C2×D4112yescomplemented
49C2×C4×C23C2416yes—
50C22⋊D23—296no—
51C2×Dic23C2×D4112yescomplemented
52C2×Dic23C2416yes—
53D92—248no—
54C4×D23C2×D4148yescomplemented
55C22×#127—231nomaximal, Sylow
56C22×C23—224no—
57C22×C23—416no—
58C22×C23C22×D416yes—
59C22×C23C2511yescharacteristic
60C2×D23—224no—
61C2×D23—416no—
62C2×D23—464no—
63C2×D23#127116yesdirect product factor
64C2×D23C22×D4112yes—
65C4×C23C22×D4112yescomplemented
66Dic23C22×D4112yescomplemented
67C23×D4—236no—
68C2×#127—236no—
69C2×#127—2316no—
70C2×#127—231no—
71C2×#127D2311yescharacteristic, direct product factor
72C22×C42—231no—
73C2×C23—48no—
74C2×C23C2×#12714yescomplemented
75C2×C23C23×D413yes—
76D23—48no—
77D23C2×#12718yesdirect product factor
78C25—234no—
79C22×D4—2312no—
80C22×D4—2348no—
81C22×D4—236no—
82C22×D4—4648no—
83C22×D4C2×D2316yescomplemented
84C23×C4—233no—
85#127—2348no—
86#127—238no—
87#127C2×D2318yesdirect product factor
88C2×C42—233no—
89C2×C42—238no—
90C2×C42C2×D2311yescharacteristic, complemented
91C23C22×#12711yescharacteristic, complemented, Sylow
92C24—231no—
93C24—2316no—
94C24—234no—
95C24—4648no—
96C24C22×D2314yes—
97C2×D4—224no—
98C2×D4—2348no—
99C2×D4—2324no—
100C2×D4—46192no—
101C2×D4—4648no—
102C2×D4—4624no—
103C2×D4C22×D23124yescomplemented
104C22×C4—233no—
105C22×C4—2324no—
106C22×C4—2312no—
107C22×C4C22×D2313yes—
108C42—2312no—
109C42C22×D2314yescomplemented
110C23—224no—
111C23—2312no—
112C23—232no—
113C23—238no—
114C23—4696no—
115C23—4624no—
116C23—9264no—
117C23C23×D2318yes—
118C23C23×D2311yescharacteristic
119D4—248no—
120D4—4696no—
121D4—4648no—
122C2×C4—2312no—
123C2×C4—236no—
124C2×C4—2348no—
125C2×C4D4×D23112yescomplemented
126C2×C4C23×D2316yes—
127C22—224no—
128C22—416no—
129C22—2316no—
130C22—2312no—
131C22—4624no—
132C22—9264no—
133C22—9216no—
134C22C2×D4×D2316yes—
135C22C24×D2311yescharacteristic
136C4—2312no—
137C4C2×D4×D23112yescomplemented
138C2—48no—
139C2—238no—
140C2—928no—
141C2#127×D2314yesdirect product factor
142C2C22×D4×D2313yes—
subgroupG/H
centerC23C23×D23
commutator, cosocleC22×C23C25
FrattiniC22C24×D23
FittingC2×C23×#127C2
socleC23×C23C24
hypercenterC2×#127D23
hypocenterC23C22×#127
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×D4×D23C2semidirect
2. C2×#10625045C2semidirect
3. C2×#10625111C2semidirect
4. #127×D23C2direct (N × H)
5. C2×C42×D23C2semidirect
6. C2×C23×#127C2semidirect
8. C22×D4×C23C22semidirect
9. C22×C22⋊D23C22semidirect
11. C2×D4×D23C22semidirect
12. C22×D92C22semidirect
14. C23×#127C22semidirect
15. C2×C42×C23C22semidirect
16. #10625111C22semidirect
17. C2×C4×Dic23C22semidirect
18. #10625045C22semidirect
19. C42×D23C22semidirect
26. C2×D4×C23C23semidirect
29. C2×C22⋊D23C23semidirect
33. C2×D92C23semidirect
34. C2×C4×D23D4semidirect
36. C42×C23C23semidirect
37. C4×Dic23C23semidirect
48. C2×C4×C23C2×D4semidirect
51. C2×Dic23C2×D4semidirect
54. C4×D23C2×D4semidirect
63. C2×D23#127direct (N × H)
65. C4×C23C22×D4semidirect
66. Dic23C22×D4semidirect
71. C2×#127D23direct (N × H)
74. C2×C23C2×#127semidirect
77. D23C2×#127direct (N × H)
83. C22×D4C2×D23semidirect
87. #127C2×D23direct (N × H)
90. C2×C42C2×D23semidirect
91. C23C22×#127semidirect
103. C2×D4C22×D23semidirect
109. C42C22×D23semidirect
125. C2×C4D4×D23semidirect
137. C4C2×D4×D23semidirect
141. C2#127×D23direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D232
C2×#106250452
C2×#106251112
#127×D232
C2×C42×D232
C2×C23×#1272
C22×#12723