Finite Group Databasebeta

← C3×C7×#442 #13056953

Subgroups — C3×C7×#442 #13056953

Order 1344.

264 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 140 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
1C2×C3×C7×D8C211yesmaximal, characteristic
2C3×C4×C7×D4C212yesmaximal, complemented
3C3×C7×#106C211yesmaximal, characteristic, complemented
4C3×C7×#102C212yesmaximal, complemented
5C3×C7×#97C211yesmaximal, characteristic, complemented
6C7×#442C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C7×D4C2212yes—
8C22×C3×C4×C7—22no—
9C3×C7×D8C414yescomplemented
10C2×C3×C7×C8C2211yescharacteristic
11C2×C3×C7×C8C2211yescharacteristic
12C3×C7×C4⋊C4C2212yes—
13C3×C7×C22⋊C4—22no—
14C3×C42×C7C2211yescharacteristic
15C2×C7×D8C2×C311yescharacteristic
16C4×C7×D4C2×C312yescomplemented
17C7×#106C2×C311yescharacteristic, complemented
18C7×#102C2×C312yescomplemented
19C7×#97C2×C311yescharacteristic, complemented
20C3×#442C711yesmaximal, characteristic, direct product factor, Hall
21C23×C3×C7—22no—
22C3×C7×D4—22no—
23C3×C7×D4C2×C414yescomplemented
24C2×C3×C4×C7—22no—
25C2×C3×C4×C7—44no—
26C2×C3×C4×C7D412yes—
27C2×C3×C4×C7C2311yescharacteristic
28C3×C7×C8—21no—
29C3×C7×C8C2×C412yescomplemented
30C2×C7×D4C22×C312yes—
31C22×C4×C7—22no—
32C7×D8C3×C414yescomplemented
33C2×C7×C8C22×C311yescharacteristic
34C2×C7×C8C22×C311yescharacteristic
35C7×C4⋊C4C22×C312yes—
36C7×C22⋊C4—22no—
37C42×C7C22×C311yescharacteristic
38C2×C3×D8C2×C711yescharacteristic
39C3×C4×D4C2×C712yescomplemented
40C3×#106C2×C711yescharacteristic, complemented
41C3×#102C2×C712yescomplemented
42C3×#97C2×C711yescharacteristic, complemented
43C22×C3×C7—24no—
44C22×C3×C7—42no—
45C22×C3×C7—42no—
46C22×C3×C7C2×D411yescharacteristic
47C3×C4×C7—22no—
48C3×C4×C7—42no—
49C3×C4×C7C22×C411yescharacteristic
50C3×C4×C7#4111yescharacteristic
51#442C3×C711yescharacteristic, direct product factor, Sylow
52C23×C7—22no—
53C7×D4—22no—
54C7×D4C2×C3×C414yescomplemented
55C2×C4×C7—22no—
56C2×C4×C7—44no—
57C2×C4×C7C3×D412yes—
58C2×C4×C7C23×C311yescharacteristic
59C7×C8—21no—
60C7×C8C2×C3×C412yescomplemented
61C2×C3×D4C22×C712yes—
62C22×C3×C4—22no—
63C3×D8C4×C714yescomplemented
64C2×C3×C8C22×C711yescharacteristic
65C2×C3×C8C22×C711yescharacteristic
66C3×C4⋊C4C22×C712yes—
67C3×C22⋊C4—22no—
68C3×C42C22×C711yescharacteristic
69C2×C3×C7—44no—
70C2×C3×C7C4×D411yescharacteristic
71C2×C3×C7AGL(1,ℤ/8)12yes—
72C2×D8C2×C3×C711yescharacteristic
73C4×D4C2×C3×C712yescomplemented
74#106C2×C3×C711yescharacteristic, complemented
75#102C2×C3×C712yescomplemented
76#97C2×C3×C711yescharacteristic, complemented
77C22×C7—24no—
78C22×C7—42no—
79C22×C7—42no—
80C22×C7C2×C3×D411yescharacteristic
81C4×C7—22no—
82C4×C7—42no—
83C4×C7C22×C3×C411yescharacteristic
84C4×C7C3×#4111yescharacteristic
85C23×C3—22no—
86C3×D4—22no—
87C3×D4C2×C4×C714yescomplemented
88C2×C3×C4—22no—
89C2×C3×C4—44no—
90C2×C3×C4C7×D412yes—
91C2×C3×C4C23×C711yescharacteristic
92C3×C8—21no—
93C3×C8C2×C4×C712yescomplemented
94C3×C7#44211yescharacteristic, direct product factor, Hall
95C2×D4C22×C3×C712yes—
96C22×C4—22no—
97D8C3×C4×C714yescomplemented
98C2×C8C22×C3×C711yescharacteristic
99C2×C8C22×C3×C711yescharacteristic
100C4⋊C4C22×C3×C712yes—
101C22⋊C4—22no—
102C42C22×C3×C711yescharacteristic
103C2×C7—44no—
104C2×C7C3×C4×D411yescharacteristic
105C2×C7C3×AGL(1,ℤ/8)12yes—
106C22×C3—24no—
107C22×C3—42no—
108C22×C3—42no—
109C22×C3C2×C7×D411yescharacteristic
110C3×C4—22no—
111C3×C4—42no—
112C3×C4C22×C4×C711yescharacteristic
113C3×C4C7×#4111yescharacteristic
114C23—22no—
115D4—22no—
116D4C2×C3×C4×C714yescomplemented
117C2×C4—22no—
118C2×C4—44no—
119C2×C4C3×C7×D412yes—
120C2×C4C23×C3×C711yescharacteristic
121C8—21no—
122C8C2×C3×C4×C712yescomplemented
123C7C3×#44211yescharacteristic, direct product factor, Sylow
124C2×C3—44no—
125C2×C3C4×C7×D411yescharacteristic
126C2×C3C7×AGL(1,ℤ/8)12yes—
127C22—24no—
128C22—42no—
129C22—42no—
130C22C2×C3×C7×D411yescharacteristic
131C4—22no—
132C4—42no—
133C4C22×C3×C4×C711yescharacteristic
134C4C3×C7×#4111yescharacteristic
135C3C7×#44211yescharacteristic, direct product factor, Sylow
136C2—44no—
137C2C3×C4×C7×D411yescharacteristic
138C2C3×C7×AGL(1,ℤ/8)12yes—
subgroupG/H
center, socleC22×C3×C7C2×D4
commutatorC4C22×C3×C4×C7
Frattini, cosocleC2×C4C23×C3×C7
prime radicalC2×C3×C7×D8C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C7 ◁ C3×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×C4×C7×D4C2semidirect
3. C3×C7×#106C2semidirect
4. C3×C7×#102C2semidirect
5. C3×C7×#97C2semidirect
6. C7×#442C3direct (N × H)
9. C3×C7×D8C4semidirect
16. C4×C7×D4C2×C3semidirect
17. C7×#106C2×C3semidirect
18. C7×#102C2×C3semidirect
19. C7×#97C2×C3semidirect
20. C3×#442C7direct (N × H)
23. C3×C7×D4C2×C4semidirect
29. C3×C7×C8C2×C4semidirect
32. C7×D8C3×C4semidirect
39. C3×C4×D4C2×C7semidirect
40. C3×#106C2×C7semidirect
41. C3×#102C2×C7semidirect
42. C3×#97C2×C7semidirect
51. #442C3×C7direct (N × H)
54. C7×D4C2×C3×C4semidirect
60. C7×C8C2×C3×C4semidirect
63. C3×D8C4×C7semidirect
73. C4×D4C2×C3×C7semidirect
74. #106C2×C3×C7semidirect
75. #102C2×C3×C7semidirect
76. #97C2×C3×C7semidirect
87. C3×D4C2×C4×C7semidirect
93. C3×C8C2×C4×C7semidirect
94. C3×C7#442direct (N × H)
97. D8C3×C4×C7semidirect
116. D4C2×C3×C4×C7semidirect
122. C8C2×C3×C4×C7semidirect
123. C7C3×#442direct (N × H)
135. C3C7×#442direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C7×D82
C3×C4×C7×D42
C3×C7×#1062
C3×C7×#1022
C3×C7×#972
C7×#4423
C3×#4427