Finite Group Databasebeta

← C3×C7×#407 #13056918

Subgroups — C3×C7×#407 #13056918

Order 1344.

380 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 180 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
1C23×C3×C4×C7C211yesmaximal, characteristic
2C2×C3×C7×M16C212yesmaximal, complemented
3C3×C7×C22⋊C8C214yesmaximal, complemented
4C7×#407C311yesmaximal, characteristic, direct product factor, Hall
5C24×C3×C7C411yescharacteristic
6C22×C3×C4×C7—24no—
7C22×C3×C4×C7C411yescharacteristic
8C22×C3×C4×C7C412yes—
9C22×C3×C4×C7C2212yes—
10C22×C3×C4×C7C2211yescharacteristic
11C3×C7×M16—24no—
12C2×C3×C7×C8C2214yescomplemented
13C23×C4×C7C2×C311yescharacteristic
14C2×C7×M16C2×C312yescomplemented
15C7×C22⋊C8C2×C314yescomplemented
16C3×#407C711yesmaximal, characteristic, direct product factor, Hall
17C23×C3×C7—22no—
18C23×C3×C7—24no—
19C23×C3×C7C2×C411yescharacteristic
20C23×C3×C7C2×C412yes—
21C2×C3×C4×C7—22no—
22C2×C3×C4×C7—24no—
23C2×C3×C4×C7—24no—
24C2×C3×C4×C7C2×C412yes—
25C2×C3×C4×C7C2×C411yescharacteristic
26C2×C3×C4×C7D414yes—
27C2×C3×C4×C7C2311yescharacteristic
28C3×C7×C8—24no—
29C24×C7C3×C411yescharacteristic
30C22×C4×C7—24no—
31C22×C4×C7C3×C411yescharacteristic
32C22×C4×C7C3×C412yes—
33C22×C4×C7C22×C312yes—
34C22×C4×C7C22×C311yescharacteristic
35C7×M16—24no—
36C2×C7×C8C22×C314yescomplemented
37C23×C3×C4C2×C711yescharacteristic
38C2×C3×M16C2×C712yescomplemented
39C3×C22⋊C8C2×C714yescomplemented
40C22×C3×C7—28no—
41C22×C3×C7—22no—
42C22×C3×C7—22no—
43C22×C3×C7—22no—
44C22×C3×C7C22⋊C412yes—
45C22×C3×C7M1614yescomplemented
46C22×C3×C7C22×C411yescharacteristic
47C3×C4×C7—22no—
48C3×C4×C7C22⋊C412yes—
49C3×C4×C7C2×D412yes—
50#407C3×C711yescharacteristic, direct product factor, Sylow
51C23×C7—22no—
52C23×C7—24no—
53C23×C7C2×C3×C411yescharacteristic
54C23×C7C2×C3×C412yes—
55C2×C4×C7—22no—
56C2×C4×C7—24no—
57C2×C4×C7—24no—
58C2×C4×C7C2×C3×C412yes—
59C2×C4×C7C2×C3×C411yescharacteristic
60C2×C4×C7C3×D414yes—
61C2×C4×C7C23×C311yescharacteristic
62C7×C8—24no—
63C24×C3C4×C711yescharacteristic
64C22×C3×C4—24no—
65C22×C3×C4C4×C711yescharacteristic
66C22×C3×C4C4×C712yes—
67C22×C3×C4C22×C712yes—
68C22×C3×C4C22×C711yescharacteristic
69C3×M16—24no—
70C2×C3×C8C22×C714yescomplemented
71C2×C3×C7—22no—
72C2×C3×C7—24no—
73C2×C3×C7C2×C22⋊C411yescharacteristic
74C2×C3×C7C2×M1612yes—
75C23×C4C2×C3×C711yescharacteristic
76C2×M16C2×C3×C712yescomplemented
77C22⋊C8C2×C3×C714yescomplemented
78C22×C7—28no—
79C22×C7—22no—
80C22×C7—22no—
81C22×C7—22no—
82C22×C7C3×C22⋊C412yes—
83C22×C7C3×M1614yescomplemented
84C22×C7C22×C3×C411yescharacteristic
85C4×C7—22no—
86C4×C7C3×C22⋊C412yes—
87C4×C7C2×C3×D412yes—
88C23×C3—22no—
89C23×C3—24no—
90C23×C3C2×C4×C711yescharacteristic
91C23×C3C2×C4×C712yes—
92C2×C3×C4—22no—
93C2×C3×C4—24no—
94C2×C3×C4—24no—
95C2×C3×C4C2×C4×C712yes—
96C2×C3×C4C2×C4×C711yescharacteristic
97C2×C3×C4C7×D414yes—
98C2×C3×C4C23×C711yescharacteristic
99C3×C8—24no—
100C3×C7#40711yescharacteristic, direct product factor, Hall
101C24C3×C4×C711yescharacteristic
102C22×C4—24no—
103C22×C4C3×C4×C711yescharacteristic
104C22×C4C3×C4×C712yes—
105C22×C4C22×C3×C712yes—
106C22×C4C22×C3×C711yescharacteristic
107M16—24no—
108C2×C8C22×C3×C714yescomplemented
109C2×C7—22no—
110C2×C7—24no—
111C2×C7C2×C3×C22⋊C411yescharacteristic
112C2×C7C2×C3×M1612yes—
113C22×C3—28no—
114C22×C3—22no—
115C22×C3—22no—
116C22×C3—22no—
117C22×C3C7×C22⋊C412yes—
118C22×C3C7×M1614yescomplemented
119C22×C3C22×C4×C711yescharacteristic
120C3×C4—22no—
121C3×C4C7×C22⋊C412yes—
122C3×C4C2×C7×D412yes—
123C23—22no—
124C23—24no—
125C23C2×C3×C4×C711yescharacteristic
126C23C2×C3×C4×C712yes—
127C2×C4—22no—
128C2×C4—24no—
129C2×C4—24no—
130C2×C4C2×C3×C4×C712yes—
131C2×C4C2×C3×C4×C711yescharacteristic
132C2×C4C3×C7×D414yes—
133C2×C4C23×C3×C711yescharacteristic
134C8—24no—
135C7C3×#40711yescharacteristic, direct product factor, Sylow
136C2×C3—22no—
137C2×C3—24no—
138C2×C3C2×C7×C22⋊C411yescharacteristic
139C2×C3C2×C7×M1612yes—
140C22—28no—
141C22—22no—
142C22—22no—
143C22—22no—
144C22C3×C7×C22⋊C412yes—
145C22C3×C7×M1614yescomplemented
146C22C22×C3×C4×C711yescharacteristic
147C4—22no—
148C4C3×C7×C22⋊C412yes—
149C4C2×C3×C7×D412yes—
150C3C7×#40711yescharacteristic, direct product factor, Sylow
151C2—22no—
152C2—24no—
153C2C2×C3×C7×C22⋊C411yescharacteristic
154C2C2×C3×C7×M1612yes—
subgroupG/H
centerC2×C3×C4×C7C23
commutatorC22C22×C3×C4×C7
Frattini, cosocleC2×C4C23×C3×C7
socleC22×C3×C7C22×C4
prime radicalC24×C3×C7C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C3×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×C7×M16C2semidirect
3. C3×C7×C22⋊C8C2semidirect
4. C7×#407C3direct (N × H)
12. C2×C3×C7×C8C22semidirect
14. C2×C7×M16C2×C3semidirect
15. C7×C22⋊C8C2×C3semidirect
16. C3×#407C7direct (N × H)
36. C2×C7×C8C22×C3semidirect
38. C2×C3×M16C2×C7semidirect
39. C3×C22⋊C8C2×C7semidirect
45. C22×C3×C7M16semidirect
50. #407C3×C7direct (N × H)
70. C2×C3×C8C22×C7semidirect
76. C2×M16C2×C3×C7semidirect
77. C22⋊C8C2×C3×C7semidirect
83. C22×C7C3×M16semidirect
100. C3×C7#407direct (N × H)
108. C2×C8C22×C3×C7semidirect
118. C22×C3C7×M16semidirect
135. C7C3×#407direct (N × H)
145. C22C3×C7×M16semidirect
150. C3C7×#407direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4×C72
C2×C3×C7×M162
C3×C7×C22⋊C82
C7×#4073
C3×#4077