Finite Group Databasebeta

← C23×C3×C22⋊D7 #13062347

Subgroups — C23×C3×C22⋊D7 #13062347

Order 1344.

2724 conjugacy classes of subgroups; only the 1086 normal subgroups are shown.

normal subgroups only

This group has 2724 conjugacy classes of subgroups — too many to draw. Only its 1086 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
1C25×C3×C7C211yesmaximal, characteristic, complemented
2C24×C3×D7C211yesmaximal, characteristic, complemented
3C22×C3×C22⋊D7C2128yesmaximal, direct product factor
4C23×C3×Dic7C211yesmaximal, characteristic, complemented
5C23×C22⋊D7C311yesmaximal, characteristic, direct product factor, Hall
6C24×C3×C7—28no—
7C24×C3×C7C2211yescharacteristic
8C24×C3×C7C22114yescomplemented
9C23×C3×D7—28no—
10C23×C3×D7C22114yescomplemented
11C2×C3×C22⋊D7C221112yesdirect product factor
12C22×C3×Dic7C22114yescomplemented
13C25×C7C2×C311yescharacteristic, complemented
14C24×D7C2×C311yescharacteristic, complemented
15C22×C22⋊D7C2×C3128yesdirect product factor
16C23×Dic7C2×C311yescharacteristic, complemented
17C23×C3×D4—71nomaximal, Hall
18C23×C3×C7—256no—
19C23×C3×C7D418yescomplemented
20C23×C3×C7C2317yes—
21C23×C3×C7C23128yescomplemented
22C22×C3×D7—256no—
23C22×C3×D7C23128yescomplemented
24C3×C22⋊D7C23164yesdirect product factor
25C2×C3×Dic7C23128yescomplemented
26C24×C7—28no—
27C24×C7C22×C311yescharacteristic
28C24×C7C22×C3114yescomplemented
29C23×D7—28no—
30C23×D7C22×C3114yescomplemented
31C2×C22⋊D7C22×C31112yesdirect product factor
32C22×Dic7C22×C3114yescomplemented
33C25×C3—71no—
34C25×C3D711yescharacteristic, complemented
35C22×C3×D4—728no—
36C23×C3×C4—71no—
37C22×C3×C7—256no—
38C22×C3×C7C2×D4128yescomplemented
39C22×C3×C7C2417yes—
40C22×C3×C7C2418yescomplemented
41C2×C3×D7—256no—
42C2×C3×D7C2418yescomplemented
43C3×Dic7C2418yescomplemented
44C23×D4—71noSylow
45C23×C7—256no—
46C23×C7C3×D418yescomplemented
47C23×C7C23×C317yes—
48C23×C7C23×C3128yescomplemented
49C22×D7—256no—
50C22×D7C23×C3128yescomplemented
51C22⋊D7C23×C3164yesdirect product factor
52C2×Dic7C23×C3128yescomplemented
53C24×C3—28no—
54C24×C3—714no—
55C24×C3—148no—
56C24×C3C2×D711yescharacteristic
57C24×C3C2×D7114yescomplemented
58C2×C3×D4—7112no—
59C22×C3×C4—714no—
60C2×C3×C7—28no—
61C2×C3×C7C22×D4114yescomplemented
62C2×C3×C7C2511yescharacteristic
63C3×D7—28no—
64C25—71no—
65C25C3×D711yescharacteristic, complemented
66C22×D4—728no—
67C23×C4—71no—
68C22×C7—256no—
69C22×C7C2×C3×D4128yescomplemented
70C22×C7C24×C317yes—
71C22×C7C24×C318yescomplemented
72C2×D7—256no—
73C2×D7C24×C318yescomplemented
74Dic7C24×C318yescomplemented
75C23×C3—256no—
76C23×C3—728no—
77C23×C3—1456no—
78C23×C3C22⋊D718yesdirect product factor
79C23×C3C22×D717yes—
80C23×C3C22×D7128yescomplemented
81C3×D4—764no—
82C2×C3×C4—728no—
83C3×C7C23×D411yescharacteristic, complemented, Hall
84C24—28no—
85C24—714no—
86C24—148no—
87C24C2×C3×D711yescharacteristic
88C24C2×C3×D7114yescomplemented
89C2×D4—7112no—
90C22×C4—714no—
91C2×C7—28no—
92C2×C7C22×C3×D4114yescomplemented
93C2×C7C25×C311yescharacteristic
94D7—28no—
95C22×C3—256no—
96C22×C3—78no—
97C22×C3—1456no—
98C22×C3C2×C22⋊D7128yesdirect product factor
99C22×C3C23×D717yes—
100C22×C3C23×D718yescomplemented
101C3×C4—78no—
102C23—256no—
103C23—728no—
104C23—1456no—
105C23C3×C22⋊D718yesdirect product factor
106C23C22×C3×D717yes—
107C23C22×C3×D7128yescomplemented
108D4—764no—
109C2×C4—728no—
110C7C23×C3×D411yescharacteristic, complemented, Sylow
111C2×C3—28no—
112C2×C3—148no—
113C2×C3C22×C22⋊D7114yesdirect product factor
114C2×C3C24×D711yescharacteristic
115C22—256no—
116C22—78no—
117C22—1456no—
118C22C2×C3×C22⋊D7128yesdirect product factor
119C22C23×C3×D717yes—
120C22C23×C3×D718yescomplemented
121C4—78no—
122C3C23×C22⋊D711yescharacteristic, direct product factor, Sylow
123C2—28no—
124C2—148no—
125C2C22×C3×C22⋊D7114yesdirect product factor
126C2C24×C3×D711yescharacteristic
subgroupG/H
centerC24×C3C2×D7
commutator, cosocleC2×C7C25×C3
FrattiniC2C24×C3×D7
FittingC25×C3×C7C2
socleC24×C3×C7C22
hypercenterC25×C3D7
hypocenterC7C23×C3×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C24×C3 ◁ C25×C3

Complements

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

NH ≅ G/Ntype
1. C25×C3×C7C2semidirect
2. C24×C3×D7C2semidirect
3. C22×C3×C22⋊D7C2direct (N × H)
4. C23×C3×Dic7C2semidirect
5. C23×C22⋊D7C3direct (N × H)
8. C24×C3×C7C22semidirect
10. C23×C3×D7C22semidirect
11. C2×C3×C22⋊D7C22direct (N × H)
12. C22×C3×Dic7C22semidirect
13. C25×C7C2×C3semidirect
14. C24×D7C2×C3semidirect
15. C22×C22⋊D7C2×C3direct (N × H)
16. C23×Dic7C2×C3semidirect
19. C23×C3×C7D4semidirect
21. C23×C3×C7C23semidirect
23. C22×C3×D7C23semidirect
24. C3×C22⋊D7C23direct (N × H)
25. C2×C3×Dic7C23semidirect
28. C24×C7C22×C3semidirect
30. C23×D7C22×C3semidirect
31. C2×C22⋊D7C22×C3direct (N × H)
32. C22×Dic7C22×C3semidirect
34. C25×C3D7semidirect
38. C22×C3×C7C2×D4semidirect
40. C22×C3×C7C24semidirect
42. C2×C3×D7C24semidirect
43. C3×Dic7C24semidirect
46. C23×C7C3×D4semidirect
48. C23×C7C23×C3semidirect
50. C22×D7C23×C3semidirect
51. C22⋊D7C23×C3direct (N × H)
52. C2×Dic7C23×C3semidirect
57. C24×C3C2×D7semidirect
61. C2×C3×C7C22×D4semidirect
65. C25C3×D7semidirect
69. C22×C7C2×C3×D4semidirect
71. C22×C7C24×C3semidirect
73. C2×D7C24×C3semidirect
74. Dic7C24×C3semidirect
78. C23×C3C22⋊D7direct (N × H)
80. C23×C3C22×D7semidirect
83. C3×C7C23×D4semidirect
88. C24C2×C3×D7semidirect
92. C2×C7C22×C3×D4semidirect
98. C22×C3C2×C22⋊D7direct (N × H)
100. C22×C3C23×D7semidirect
105. C23C3×C22⋊D7direct (N × H)
107. C23C22×C3×D7semidirect
110. C7C23×C3×D4semidirect
113. C2×C3C22×C22⋊D7direct (N × H)
118. C22C2×C3×C22⋊D7direct (N × H)
120. C22C23×C3×D7semidirect
122. C3C23×C22⋊D7direct (N × H)
125. C2C22×C3×C22⋊D7direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C3×C72
C24×C3×D72
C22×C3×C22⋊D72
C23×C3×Dic72
C23×C22⋊D73
C23×C3×D47