Finite Group Databasebeta

← C22×C7×#213 #13061514

Subgroups — C22×C7×#213 #13061514

Order 1344.

676 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 254 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×C3×C7×D4C211yesmaximal, characteristic, complemented
2C22×C7×D12C211yesmaximal, characteristic, complemented
3C2×C7×#213C2112yesmaximal, direct product factor
4C22×C7×C3⋊C8C211yesmaximal, characteristic, complemented
5C22×C7×D8—31nomaximal, Hall
6C24×C3×C7—21no—
7C23×D3×C7—21no—
8C2×C3×C7×D4—23no—
9C2×C3×C7×D4C2216yescomplemented
10C22×C3×C4×C7C2211yescharacteristic
11C2×C7×D12—23no—
12C2×C7×D12C2216yescomplemented
13C7×#213C22116yesdirect product factor
14C2×C7×C3⋊C8C2216yescomplemented
15C22×C7×D4—31no—
16C22×C7×D4D311yescharacteristic, complemented
17C2×C7×D8—312no—
18C22×C7×C8—31no—
19C22×#213C711yesmaximal, characteristic, direct product factor, Hall
20C23×C3×C7—26no—
21C23×C3×C7—44no—
22C23×C3×C7D411yescharacteristic
23C22×D3×C7—26no—
24C22×D3×C7—44no—
25C3×C7×D4—26no—
26C3×C7×D4C2314yescomplemented
27C2×C3×C4×C7D413yes—
28C2×C3×C4×C7C2313yes—
29C7×D12—26no—
30C7×D12C2314yescomplemented
31C7×C3⋊C8C2314yescomplemented
32C24×C7—21no—
33C24×C7—61no—
34C2×C7×D4—23no—
35C2×C7×D4—36no—
36C2×C7×D4—63no—
37C2×C7×D4C2×D316yescomplemented
38C22×C4×C7C2×D311yescharacteristic
39C7×D8—316no—
40C2×C7×C8—36no—
41C22×C3×D4C2×C711yescharacteristic, complemented
42C22×D12C2×C711yescharacteristic, complemented
43C2×#213C2×C7112yesdirect product factor
44C22×C3⋊C8C2×C711yescharacteristic, complemented
45C22×C3×C7—24no—
46C22×C3×C7—412no—
47C22×C3×C7D814yescomplemented
48C22×C3×C7C2×D413yes—
49C2×D3×C7—24no—
50C2×D3×C7—412no—
51C3×C4×C7C2×D413yes—
52C3×C4×C7C2411yescharacteristic
53C22×D8—31noSylow
54C23×C7—26no—
55C23×C7—44no—
56C23×C7—66no—
57C23×C7—124no—
58C23×C7#6711yescharacteristic
59C7×D4—26no—
60C7×D4—34no—
61C7×D4—66no—
62C7×D4C22×D314yescomplemented
63C2×C4×C7#6713yes—
64C2×C4×C7C22×D313yes—
65C7×C8—34no—
66C24×C3—21no—
67C23×D3—21no—
68C2×C3×D4—23no—
69C2×C3×D4C22×C716yescomplemented
70C22×C3×C4C22×C711yescharacteristic
71C2×D12—23no—
72C2×D12C22×C716yescomplemented
73#213C22×C7116yesdirect product factor
74C2×C3⋊C8C22×C716yescomplemented
75C2×C3×C7—44no—
76C2×C3×C7C2×D816yescomplemented
77C2×C3×C7C22×D411yescharacteristic
78D3×C7—44no—
79C22×D4—31no—
80C22×D4D3×C711yescharacteristic, complemented
81C2×D8—312no—
82C22×C8—31no—
83C22×C7—24no—
84C22×C7—412no—
85C22×C7—64no—
86C22×C7—1212no—
87C22×C7#21314yesdirect product factor
88C22×C7C2×#6713yes—
89C4×C7C2×#6713yes—
90C4×C7C23×D311yescharacteristic
91C23×C3—26no—
92C23×C3—44no—
93C23×C3C7×D411yescharacteristic
94C22×D3—26no—
95C22×D3—44no—
96C3×D4—26no—
97C3×D4C23×C714yescomplemented
98C2×C3×C4C7×D413yes—
99C2×C3×C4C23×C713yes—
100D12—26no—
101D12C23×C714yescomplemented
102C3⋊C8C23×C714yescomplemented
103C3×C7C22×D811yescharacteristic, complemented, Hall
104C24—21no—
105C24—61no—
106C2×D4—23no—
107C2×D4—36no—
108C2×D4—63no—
109C2×D4C2×D3×C716yescomplemented
110C22×C4C2×D3×C711yescharacteristic
111D8—316no—
112C2×C8—36no—
113C2×C7—44no—
114C2×C7—124no—
115C2×C7C2×#21316yesdirect product factor
116C2×C7C22×#6711yescharacteristic
117C22×C3—24no—
118C22×C3—412no—
119C22×C3C7×D814yescomplemented
120C22×C3C2×C7×D413yes—
121C2×D3—24no—
122C2×D3—412no—
123C3×C4C2×C7×D413yes—
124C3×C4C24×C711yescharacteristic
125C23—26no—
126C23—44no—
127C23—66no—
128C23—124no—
129C23C7×#6711yescharacteristic
130D4—26no—
131D4—34no—
132D4—66no—
133D4C22×D3×C714yescomplemented
134C2×C4C7×#6713yes—
135C2×C4C22×D3×C713yes—
136C8—34no—
137C7C22×#21311yescharacteristic, direct product factor, Sylow
138C2×C3—44no—
139C2×C3C2×C7×D816yescomplemented
140C2×C3C22×C7×D411yescharacteristic
141D3—44no—
142C22—24no—
143C22—412no—
144C22—64no—
145C22—1212no—
146C22C7×#21314yesdirect product factor
147C22C2×C7×#6713yes—
148C4C2×C7×#6713yes—
149C4C23×D3×C711yescharacteristic
150C3C22×C7×D811yescharacteristic, complemented, Sylow
151C2—44no—
152C2—124no—
153C2C2×C7×#21316yesdirect product factor
154C2C22×C7×#6711yescharacteristic
subgroupG/H
centerC23×C7#67
commutator, cosocleC3×C4C24×C7
FrattiniC4C23×D3×C7
FittingC22×C3×C7×D4C2
socleC23×C3×C7D4
hypercenterC22×C7×D4D3
hypocenterC3C22×C7×D8
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C7 ◁ C22×C4×C7 ◁ C22×C7×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C7×D4C2semidirect
2. C22×C7×D12C2semidirect
3. C2×C7×#213C2direct (N × H)
4. C22×C7×C3⋊C8C2semidirect
9. C2×C3×C7×D4C22semidirect
12. C2×C7×D12C22semidirect
13. C7×#213C22direct (N × H)
14. C2×C7×C3⋊C8C22semidirect
16. C22×C7×D4D3semidirect
19. C22×#213C7direct (N × H)
26. C3×C7×D4C23semidirect
30. C7×D12C23semidirect
31. C7×C3⋊C8C23semidirect
37. C2×C7×D4C2×D3semidirect
41. C22×C3×D4C2×C7semidirect
42. C22×D12C2×C7semidirect
43. C2×#213C2×C7direct (N × H)
44. C22×C3⋊C8C2×C7semidirect
47. C22×C3×C7D8semidirect
62. C7×D4C22×D3semidirect
69. C2×C3×D4C22×C7semidirect
72. C2×D12C22×C7semidirect
73. #213C22×C7direct (N × H)
74. C2×C3⋊C8C22×C7semidirect
76. C2×C3×C7C2×D8semidirect
80. C22×D4D3×C7semidirect
87. C22×C7#213direct (N × H)
97. C3×D4C23×C7semidirect
101. D12C23×C7semidirect
102. C3⋊C8C23×C7semidirect
103. C3×C7C22×D8semidirect
109. C2×D4C2×D3×C7semidirect
115. C2×C7C2×#213direct (N × H)
119. C22×C3C7×D8semidirect
133. D4C22×D3×C7semidirect
137. C7C22×#213direct (N × H)
139. C2×C3C2×C7×D8semidirect
146. C22C7×#213direct (N × H)
150. C3C22×C7×D8semidirect
153. C2C2×C7×#213direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C7×D42
C22×C7×D122
C2×C7×#2132
C22×C7×C3⋊C82
C22×C7×D83
C22×#2137