Finite Group Databasebeta

← C23×#120×C73 #32972059

Subgroups — C23×#120×C73 #32972059

Order 18688.

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×C73C211yesmaximal, characteristic, complemented
2C24×D4×C73C213yesmaximal, complemented
3C22×#120×C73C2156yesmaximal, direct product factor
4C23×C22⋊C4×C73C213yesmaximal, complemented
5C26×C73—248no—
6C26×C73C2213yes—
7C26×C73C22128yescomplemented
8C26×C73C2211yescharacteristic, complemented
9C23×D4×C73—248no—
10C23×D4×C73C22184yescomplemented
11C24×C4×C73C2213yescomplemented
12C2×#120×C73C221448yesdirect product factor
13C22×C22⋊C4×C73C22184yescomplemented
14C25×C73—2512no—
15C25×C73—248no—
16C25×C73—2672no—
17C25×C73—224no—
18C25×C73D4148yescomplemented
19C25×C73C23142yes—
20C25×C73C2311yescharacteristic
21C25×C73C231112yescomplemented
22C25×C73C23114yescomplemented
23C22×D4×C73—2672no—
24C22×D4×C73C231336yescomplemented
25C23×C4×C73—224no—
26C23×C4×C73C23142yescomplemented
27#120×C73C231512yesdirect product factor
28C2×C22⋊C4×C73C231336yescomplemented
29C24×C73—23584no—
30C24×C73—2384no—
31C24×C73—2336no—
32C24×C73—21344no—
33C24×C73—2168no—
34C24×C73—464no—
35C24×C73C2×D4124yes—
36C24×C73C2×D41336yescomplemented
37C24×C73C24184yes—
38C24×C73C2417yes—
39C24×C73C24164yescomplemented
40C24×C73C24128yescomplemented
41C2×D4×C73—21344no—
42C2×D4×C73C241192yescomplemented
43C22×C4×C73—2168no—
44C22×C4×C73C24184yescomplemented
45C22⋊C4×C73C241192yescomplemented
46C23×C73—23584no—
47C23×C73—21344no—
48C23×C73—2336no—
49C23×C73—2384no—
50C23×C73—2168no—
51C23×C73—4224no—
52C23×C73#120164yesdirect product factor
53C23×C73C22×D4184yes—
54C23×C73C22×D41336yescomplemented
55C23×C73C25124yes—
56C23×C73C2517yes—
57C23×C73C2518yescomplemented
58D4×C73—2384no—
59C2×C4×C73—2168no—
60C2×C4×C73C25124yescomplemented
61C22×C73—2512no—
62C22×C73—248no—
63C22×C73—2672no—
64C22×C73—224no—
65C22×C73—4112no—
66C22×C73C2×#1201112yesdirect product factor
67C22×C73C23×D4142yes—
68C22×C73C23×D4148yescomplemented
69C22×C73C2611yescharacteristic
70C4×C73—224no—
71C23×#120C7311yesmaximal, characteristic, direct product factor, Sylow
72C2×C73—248no—
73C2×C73—48no—
74C2×C73C22×#120128yesdirect product factor
75C2×C73C24×D413yes—
76C27C2×C7311yescharacteristic, complemented
77C24×D4C2×C7313yescomplemented
78C22×#120C2×C73156yesdirect product factor
79C23×C22⋊C4C2×C7313yescomplemented
80C73C23×#12011yescharacteristic, direct product factor, Sylow
81C26—248no—
82C26C22×C7313yes—
83C26C22×C73128yescomplemented
84C26C22×C7311yescharacteristic, complemented
85C23×D4—248no—
86C23×D4C22×C73184yescomplemented
87C24×C4C22×C7313yescomplemented
88C2×#120C22×C731448yesdirect product factor
89C22×C22⋊C4C22×C73184yescomplemented
90C25—2512no—
91C25—248no—
92C25—2672no—
93C25—224no—
94C25D4×C73148yescomplemented
95C25C23×C73142yes—
96C25C23×C7311yescharacteristic
97C25C23×C731112yescomplemented
98C25C23×C73114yescomplemented
99C22×D4—2672no—
100C22×D4C23×C731336yescomplemented
101C23×C4—224no—
102C23×C4C23×C73142yescomplemented
103#120C23×C731512yesdirect product factor
104C2×C22⋊C4C23×C731336yescomplemented
105C24—23584no—
106C24—2384no—
107C24—2336no—
108C24—21344no—
109C24—2168no—
110C24—464no—
111C24C2×D4×C73124yes—
112C24C2×D4×C731336yescomplemented
113C24C24×C73184yes—
114C24C24×C7317yes—
115C24C24×C73164yescomplemented
116C24C24×C73128yescomplemented
117C2×D4—21344no—
118C2×D4C24×C731192yescomplemented
119C22×C4—2168no—
120C22×C4C24×C73184yescomplemented
121C22⋊C4C24×C731192yescomplemented
122C23—23584no—
123C23—21344no—
124C23—2336no—
125C23—2384no—
126C23—2168no—
127C23—4224no—
128C23#120×C73164yesdirect product factor
129C23C22×D4×C73184yes—
130C23C22×D4×C731336yescomplemented
131C23C25×C73124yes—
132C23C25×C7317yes—
133C23C25×C7318yescomplemented
134D4—2384no—
135C2×C4—2168no—
136C2×C4C25×C73124yescomplemented
137C22—2512no—
138C22—248no—
139C22—2672no—
140C22—224no—
141C22—4112no—
142C22C2×#120×C731112yesdirect product factor
143C22C23×D4×C73142yes—
144C22C23×D4×C73148yescomplemented
145C22C26×C7311yescharacteristic
146C4—224no—
147C2—248no—
148C2—48no—
149C2C22×#120×C73128yesdirect product factor
150C2C24×D4×C7313yes—
subgroupG/H
center, socleC25×C73C23
commutator, Frattini, cosocleC22C26×C73
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C25×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C27×C73C2semidirect
2. C24×D4×C73C2semidirect
3. C22×#120×C73C2direct (N × H)
4. C23×C22⋊C4×C73C2semidirect
7. C26×C73C22semidirect
8. C26×C73C22semidirect
10. C23×D4×C73C22semidirect
11. C24×C4×C73C22semidirect
12. C2×#120×C73C22direct (N × H)
13. C22×C22⋊C4×C73C22semidirect
18. C25×C73D4semidirect
21. C25×C73C23semidirect
22. C25×C73C23semidirect
24. C22×D4×C73C23semidirect
26. C23×C4×C73C23semidirect
27. #120×C73C23direct (N × H)
28. C2×C22⋊C4×C73C23semidirect
36. C24×C73C2×D4semidirect
39. C24×C73C24semidirect
40. C24×C73C24semidirect
42. C2×D4×C73C24semidirect
44. C22×C4×C73C24semidirect
45. C22⋊C4×C73C24semidirect
52. C23×C73#120direct (N × H)
54. C23×C73C22×D4semidirect
57. C23×C73C25semidirect
60. C2×C4×C73C25semidirect
66. C22×C73C2×#120direct (N × H)
68. C22×C73C23×D4semidirect
71. C23×#120C73direct (N × H)
74. C2×C73C22×#120direct (N × H)
76. C27C2×C73semidirect
77. C24×D4C2×C73semidirect
78. C22×#120C2×C73direct (N × H)
79. C23×C22⋊C4C2×C73semidirect
80. C73C23×#120direct (N × H)
83. C26C22×C73semidirect
84. C26C22×C73semidirect
86. C23×D4C22×C73semidirect
87. C24×C4C22×C73semidirect
88. C2×#120C22×C73direct (N × H)
89. C22×C22⋊C4C22×C73semidirect
94. C25D4×C73semidirect
97. C25C23×C73semidirect
98. C25C23×C73semidirect
100. C22×D4C23×C73semidirect
102. C23×C4C23×C73semidirect
103. #120C23×C73direct (N × H)
104. C2×C22⋊C4C23×C73semidirect
112. C24C2×D4×C73semidirect
115. C24C24×C73semidirect
116. C24C24×C73semidirect
118. C2×D4C24×C73semidirect
120. C22×C4C24×C73semidirect
121. C22⋊C4C24×C73semidirect
128. C23#120×C73direct (N × H)
130. C23C22×D4×C73semidirect
133. C23C25×C73semidirect
136. C2×C4C25×C73semidirect
142. C22C2×#120×C73direct (N × H)
144. C22C23×D4×C73semidirect
149. C2C22×#120×C73direct (N × H)

Maximal subgroups

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

subgroupindex
C27×C732
C24×D4×C732
C22×#120×C732
C23×C22⋊C4×C732
C23×#12073