Finite Group Databasebeta

← C23×C7×#120 #13239871

Subgroups — C23×C7×#120 #13239871

Order 1792.

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×C7C211yesmaximal, characteristic, complemented
2C24×C7×D4C213yesmaximal, complemented
3C22×C7×#120C2156yesmaximal, direct product factor
4C23×C7×C22⋊C4C213yesmaximal, complemented
5C26×C7—248no—
6C26×C7C2213yes—
7C26×C7C22128yescomplemented
8C26×C7C2211yescharacteristic, complemented
9C23×C7×D4—248no—
10C23×C7×D4C22184yescomplemented
11C24×C4×C7C2213yescomplemented
12C2×C7×#120C221448yesdirect product factor
13C22×C7×C22⋊C4C22184yescomplemented
14C23×#120C711yesmaximal, characteristic, direct product factor, Sylow
15C25×C7—2512no—
16C25×C7—248no—
17C25×C7—2672no—
18C25×C7—224no—
19C25×C7D4148yescomplemented
20C25×C7C23142yes—
21C25×C7C2311yescharacteristic
22C25×C7C231112yescomplemented
23C25×C7C23114yescomplemented
24C22×C7×D4—2672no—
25C22×C7×D4C231336yescomplemented
26C23×C4×C7—224no—
27C23×C4×C7C23142yescomplemented
28C7×#120C231512yesdirect product factor
29C2×C7×C22⋊C4C231336yescomplemented
30C27C2×C711yescharacteristic, complemented
31C24×D4C2×C713yescomplemented
32C22×#120C2×C7156yesdirect product factor
33C23×C22⋊C4C2×C713yescomplemented
34C24×C7—23584no—
35C24×C7—2384no—
36C24×C7—2336no—
37C24×C7—21344no—
38C24×C7—2168no—
39C24×C7—464no—
40C24×C7C2×D4124yes—
41C24×C7C2×D41336yescomplemented
42C24×C7C24184yes—
43C24×C7C2417yes—
44C24×C7C24164yescomplemented
45C24×C7C24128yescomplemented
46C2×C7×D4—21344no—
47C2×C7×D4C241192yescomplemented
48C22×C4×C7—2168no—
49C22×C4×C7C24184yescomplemented
50C7×C22⋊C4C241192yescomplemented
51C26—248no—
52C26C22×C713yes—
53C26C22×C7128yescomplemented
54C26C22×C711yescharacteristic, complemented
55C23×D4—248no—
56C23×D4C22×C7184yescomplemented
57C24×C4C22×C713yescomplemented
58C2×#120C22×C71448yesdirect product factor
59C22×C22⋊C4C22×C7184yescomplemented
60C23×C7—23584no—
61C23×C7—21344no—
62C23×C7—2336no—
63C23×C7—2384no—
64C23×C7—2168no—
65C23×C7—4224no—
66C23×C7#120164yesdirect product factor
67C23×C7C22×D4184yes—
68C23×C7C22×D41336yescomplemented
69C23×C7C25124yes—
70C23×C7C2517yes—
71C23×C7C2518yescomplemented
72C7×D4—2384no—
73C2×C4×C7—2168no—
74C2×C4×C7C25124yescomplemented
75C25—2512no—
76C25—248no—
77C25—2672no—
78C25—224no—
79C25C7×D4148yescomplemented
80C25C23×C7142yes—
81C25C23×C711yescharacteristic
82C25C23×C71112yescomplemented
83C25C23×C7114yescomplemented
84C22×D4—2672no—
85C22×D4C23×C71336yescomplemented
86C23×C4—224no—
87C23×C4C23×C7142yescomplemented
88#120C23×C71512yesdirect product factor
89C2×C22⋊C4C23×C71336yescomplemented
90C22×C7—2512no—
91C22×C7—248no—
92C22×C7—2672no—
93C22×C7—224no—
94C22×C7—4112no—
95C22×C7C2×#1201112yesdirect product factor
96C22×C7C23×D4142yes—
97C22×C7C23×D4148yescomplemented
98C22×C7C2611yescharacteristic
99C4×C7—224no—
100C24—23584no—
101C24—2384no—
102C24—2336no—
103C24—21344no—
104C24—2168no—
105C24—464no—
106C24C2×C7×D4124yes—
107C24C2×C7×D41336yescomplemented
108C24C24×C7184yes—
109C24C24×C717yes—
110C24C24×C7164yescomplemented
111C24C24×C7128yescomplemented
112C2×D4—21344no—
113C2×D4C24×C71192yescomplemented
114C22×C4—2168no—
115C22×C4C24×C7184yescomplemented
116C22⋊C4C24×C71192yescomplemented
117C2×C7—248no—
118C2×C7—48no—
119C2×C7C22×#120128yesdirect product factor
120C2×C7C24×D413yes—
121C23—23584no—
122C23—2336no—
123C23—2384no—
124C23—21344no—
125C23—2168no—
126C23—4224no—
127C23C7×#120164yesdirect product factor
128C23C22×C7×D4184yes—
129C23C22×C7×D41336yescomplemented
130C23C25×C7124yes—
131C23C25×C717yes—
132C23C25×C718yescomplemented
133D4—2384no—
134C2×C4—2168no—
135C2×C4C25×C7124yescomplemented
136C7C23×#12011yescharacteristic, direct product factor, Sylow
137C22—2512no—
138C22—248no—
139C22—2672no—
140C22—224no—
141C22—4112no—
142C22C2×C7×#1201112yesdirect product factor
143C22C23×C7×D4142yes—
144C22C23×C7×D4148yescomplemented
145C22C26×C711yescharacteristic
146C4—224no—
147C2—248no—
148C2—48no—
149C2C22×C7×#120128yesdirect product factor
150C2C24×C7×D413yes—
subgroupG/H
center, socleC25×C7C23
commutator, Frattini, cosocleC22C26×C7
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C25×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C27×C7C2semidirect
2. C24×C7×D4C2semidirect
3. C22×C7×#120C2direct (N × H)
4. C23×C7×C22⋊C4C2semidirect
7. C26×C7C22semidirect
8. C26×C7C22semidirect
10. C23×C7×D4C22semidirect
11. C24×C4×C7C22semidirect
12. C2×C7×#120C22direct (N × H)
13. C22×C7×C22⋊C4C22semidirect
14. C23×#120C7direct (N × H)
19. C25×C7D4semidirect
22. C25×C7C23semidirect
23. C25×C7C23semidirect
25. C22×C7×D4C23semidirect
27. C23×C4×C7C23semidirect
28. C7×#120C23direct (N × H)
29. C2×C7×C22⋊C4C23semidirect
30. C27C2×C7semidirect
31. C24×D4C2×C7semidirect
32. C22×#120C2×C7direct (N × H)
33. C23×C22⋊C4C2×C7semidirect
41. C24×C7C2×D4semidirect
44. C24×C7C24semidirect
45. C24×C7C24semidirect
47. C2×C7×D4C24semidirect
49. C22×C4×C7C24semidirect
50. C7×C22⋊C4C24semidirect
53. C26C22×C7semidirect
54. C26C22×C7semidirect
56. C23×D4C22×C7semidirect
57. C24×C4C22×C7semidirect
58. C2×#120C22×C7direct (N × H)
59. C22×C22⋊C4C22×C7semidirect
66. C23×C7#120direct (N × H)
68. C23×C7C22×D4semidirect
71. C23×C7C25semidirect
74. C2×C4×C7C25semidirect
79. C25C7×D4semidirect
82. C25C23×C7semidirect
83. C25C23×C7semidirect
85. C22×D4C23×C7semidirect
87. C23×C4C23×C7semidirect
88. #120C23×C7direct (N × H)
89. C2×C22⋊C4C23×C7semidirect
95. C22×C7C2×#120direct (N × H)
97. C22×C7C23×D4semidirect
107. C24C2×C7×D4semidirect
110. C24C24×C7semidirect
111. C24C24×C7semidirect
113. C2×D4C24×C7semidirect
115. C22×C4C24×C7semidirect
116. C22⋊C4C24×C7semidirect
119. C2×C7C22×#120direct (N × H)
127. C23C7×#120direct (N × H)
129. C23C22×C7×D4semidirect
132. C23C25×C7semidirect
135. C2×C4C25×C7semidirect
136. C7C23×#120direct (N × H)
142. C22C2×C7×#120direct (N × H)
144. C22C23×C7×D4semidirect
149. C2C22×C7×#120direct (N × H)

Maximal subgroups

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

subgroupindex
C27×C72
C24×C7×D42
C22×C7×#1202
C23×C7×C22⋊C42
C23×#1207