Finite Group Databasebeta

← C23×#120×C47 #26102353

Subgroups — C23×#120×C47 #26102353

Order 12032.

42862 conjugacy classes of subgroups; only the 7902 normal subgroups are shown.

normal subgroups only

This group has 42862 conjugacy classes of subgroups — too many to draw. Only its 7902 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
1C27×C47C211yesmaximal, characteristic, complemented
2C24×D4×C47C213yesmaximal, complemented
3C22×#120×C47C2156yesmaximal, direct product factor
4C23×C22⋊C4×C47C213yesmaximal, complemented
5C26×C47—248no—
6C26×C47C2213yes—
7C26×C47C22128yescomplemented
8C26×C47C2211yescharacteristic, complemented
9C23×D4×C47—248no—
10C23×D4×C47C22184yescomplemented
11C24×C4×C47C2213yescomplemented
12C2×#120×C47C221448yesdirect product factor
13C22×C22⋊C4×C47C22184yescomplemented
14C25×C47—2512no—
15C25×C47—248no—
16C25×C47—2672no—
17C25×C47—224no—
18C25×C47D4148yescomplemented
19C25×C47C23142yes—
20C25×C47C2311yescharacteristic
21C25×C47C231112yescomplemented
22C25×C47C23114yescomplemented
23C22×D4×C47—2672no—
24C22×D4×C47C231336yescomplemented
25C23×C4×C47—224no—
26C23×C4×C47C23142yescomplemented
27#120×C47C231512yesdirect product factor
28C2×C22⋊C4×C47C231336yescomplemented
29C24×C47—23584no—
30C24×C47—2384no—
31C24×C47—2336no—
32C24×C47—21344no—
33C24×C47—2168no—
34C24×C47—464no—
35C24×C47C2×D4124yes—
36C24×C47C2×D41336yescomplemented
37C24×C47C24184yes—
38C24×C47C2417yes—
39C24×C47C24164yescomplemented
40C24×C47C24128yescomplemented
41C2×D4×C47—21344no—
42C2×D4×C47C241192yescomplemented
43C22×C4×C47—2168no—
44C22×C4×C47C24184yescomplemented
45C22⋊C4×C47C241192yescomplemented
46C23×C47—23584no—
47C23×C47—21344no—
48C23×C47—2336no—
49C23×C47—2384no—
50C23×C47—2168no—
51C23×C47—4224no—
52C23×C47#120164yesdirect product factor
53C23×C47C22×D4184yes—
54C23×C47C22×D41336yescomplemented
55C23×C47C25124yes—
56C23×C47C2517yes—
57C23×C47C2518yescomplemented
58D4×C47—2384no—
59C2×C4×C47—2168no—
60C2×C4×C47C25124yescomplemented
61C23×#120C4711yesmaximal, characteristic, direct product factor, Sylow
62C22×C47—2512no—
63C22×C47—248no—
64C22×C47—2672no—
65C22×C47—224no—
66C22×C47—4112no—
67C22×C47C2×#1201112yesdirect product factor
68C22×C47C23×D4142yes—
69C22×C47C23×D4148yescomplemented
70C22×C47C2611yescharacteristic
71C4×C47—224no—
72C27C2×C4711yescharacteristic, complemented
73C24×D4C2×C4713yescomplemented
74C22×#120C2×C47156yesdirect product factor
75C23×C22⋊C4C2×C4713yescomplemented
76C2×C47—248no—
77C2×C47—48no—
78C2×C47C22×#120128yesdirect product factor
79C2×C47C24×D413yes—
80C26—248no—
81C26C22×C4713yes—
82C26C22×C47128yescomplemented
83C26C22×C4711yescharacteristic, complemented
84C23×D4—248no—
85C23×D4C22×C47184yescomplemented
86C24×C4C22×C4713yescomplemented
87C2×#120C22×C471448yesdirect product factor
88C22×C22⋊C4C22×C47184yescomplemented
89C47C23×#12011yescharacteristic, direct product factor, Sylow
90C25—2512no—
91C25—248no—
92C25—2672no—
93C25—224no—
94C25D4×C47148yescomplemented
95C25C23×C47142yes—
96C25C23×C4711yescharacteristic
97C25C23×C471112yescomplemented
98C25C23×C47114yescomplemented
99C22×D4—2672no—
100C22×D4C23×C471336yescomplemented
101C23×C4—224no—
102C23×C4C23×C47142yescomplemented
103#120C23×C471512yesdirect product factor
104C2×C22⋊C4C23×C471336yescomplemented
105C24—23584no—
106C24—2384no—
107C24—2336no—
108C24—21344no—
109C24—2168no—
110C24—464no—
111C24C2×D4×C47124yes—
112C24C2×D4×C471336yescomplemented
113C24C24×C47184yes—
114C24C24×C4717yes—
115C24C24×C47164yescomplemented
116C24C24×C47128yescomplemented
117C2×D4—21344no—
118C2×D4C24×C471192yescomplemented
119C22×C4—2168no—
120C22×C4C24×C47184yescomplemented
121C22⋊C4C24×C471192yescomplemented
122C23—23584no—
123C23—21344no—
124C23—2336no—
125C23—2384no—
126C23—2168no—
127C23—4224no—
128C23#120×C47164yesdirect product factor
129C23C22×D4×C47184yes—
130C23C22×D4×C471336yescomplemented
131C23C25×C47124yes—
132C23C25×C4717yes—
133C23C25×C4718yescomplemented
134D4—2384no—
135C2×C4—2168no—
136C2×C4C25×C47124yescomplemented
137C22—2512no—
138C22—248no—
139C22—2672no—
140C22—224no—
141C22—4112no—
142C22C2×#120×C471112yesdirect product factor
143C22C23×D4×C47142yes—
144C22C23×D4×C47148yescomplemented
145C22C26×C4711yescharacteristic
146C4—224no—
147C2—248no—
148C2—48no—
149C2C22×#120×C47128yesdirect product factor
150C2C24×D4×C4713yes—
subgroupG/H
center, socleC25×C47C23
commutator, Frattini, cosocleC22C26×C47
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C25×C47 ◁ G

Complements

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

NH ≅ G/Ntype
1. C27×C47C2semidirect
2. C24×D4×C47C2semidirect
3. C22×#120×C47C2direct (N × H)
4. C23×C22⋊C4×C47C2semidirect
7. C26×C47C22semidirect
8. C26×C47C22semidirect
10. C23×D4×C47C22semidirect
11. C24×C4×C47C22semidirect
12. C2×#120×C47C22direct (N × H)
13. C22×C22⋊C4×C47C22semidirect
18. C25×C47D4semidirect
21. C25×C47C23semidirect
22. C25×C47C23semidirect
24. C22×D4×C47C23semidirect
26. C23×C4×C47C23semidirect
27. #120×C47C23direct (N × H)
28. C2×C22⋊C4×C47C23semidirect
36. C24×C47C2×D4semidirect
39. C24×C47C24semidirect
40. C24×C47C24semidirect
42. C2×D4×C47C24semidirect
44. C22×C4×C47C24semidirect
45. C22⋊C4×C47C24semidirect
52. C23×C47#120direct (N × H)
54. C23×C47C22×D4semidirect
57. C23×C47C25semidirect
60. C2×C4×C47C25semidirect
61. C23×#120C47direct (N × H)
67. C22×C47C2×#120direct (N × H)
69. C22×C47C23×D4semidirect
72. C27C2×C47semidirect
73. C24×D4C2×C47semidirect
74. C22×#120C2×C47direct (N × H)
75. C23×C22⋊C4C2×C47semidirect
78. C2×C47C22×#120direct (N × H)
82. C26C22×C47semidirect
83. C26C22×C47semidirect
85. C23×D4C22×C47semidirect
86. C24×C4C22×C47semidirect
87. C2×#120C22×C47direct (N × H)
88. C22×C22⋊C4C22×C47semidirect
89. C47C23×#120direct (N × H)
94. C25D4×C47semidirect
97. C25C23×C47semidirect
98. C25C23×C47semidirect
100. C22×D4C23×C47semidirect
102. C23×C4C23×C47semidirect
103. #120C23×C47direct (N × H)
104. C2×C22⋊C4C23×C47semidirect
112. C24C2×D4×C47semidirect
115. C24C24×C47semidirect
116. C24C24×C47semidirect
118. C2×D4C24×C47semidirect
120. C22×C4C24×C47semidirect
121. C22⋊C4C24×C47semidirect
128. C23#120×C47direct (N × H)
130. C23C22×D4×C47semidirect
133. C23C25×C47semidirect
136. C2×C4C25×C47semidirect
142. C22C2×#120×C47direct (N × H)
144. C22C23×D4×C47semidirect
149. C2C22×#120×C47direct (N × H)

Maximal subgroups

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

subgroupindex
C27×C472
C24×D4×C472
C22×#120×C472
C23×C22⋊C4×C472
C23×#12047