Finite Group Databasebeta

← C23×Dic3×D7 #13062279

Subgroups — C23×Dic3×D7 #13062279

Order 1344.

2724 conjugacy classes of subgroups; only the 1240 normal subgroups are shown.

normal subgroups only

This group has 2724 conjugacy classes of subgroups — too many to draw. Only its 1240 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
1C24×C3×D7C211yesmaximal, characteristic
2C23×Dic21C211yesmaximal, characteristic, complemented
3C23×C7×Dic3C211yesmaximal, characteristic, complemented
4C22×Dic3×D7C2128yesmaximal, direct product factor
5C23×C4×D7—31nomaximal, Hall
6C24×C3×C7C2211yescharacteristic
7C23×C3×D7C4116yescomplemented
8C23×C3×D7C22114yes—
9C22×Dic21C22114yescomplemented
10C22×C7×Dic3C22114yescomplemented
11C2×Dic3×D7C221112yesdirect product factor
12C24×D7D311yescharacteristic
13C23×C4×C7—31no—
14C23×Dic7—31no—
15C22×C4×D7—328no—
16C24×Dic3—71nomaximal, Hall
17C23×C3×C7C2×C418yescomplemented
18C23×C3×C7C2317yes—
19C22×C3×D7C2×C41112yescomplemented
20C22×C3×D7C23128yes—
21C2×Dic21C23128yescomplemented
22C2×C7×Dic3C23128yescomplemented
23Dic3×D7C23164yesdirect product factor
24C24×C7C2×D311yescharacteristic
25C23×D7Dic3116yesdirect product factor
26C23×D7C2×D3114yes—
27C22×C4×C7—314no—
28C22×Dic7—314no—
29C2×C4×D7—3112no—
30C25×C3—71no—
31C23×Dic3—71no—
32C23×Dic3—728no—
33C23×Dic3D711yescharacteristic, direct product factor
34C22×C3×C7C22×C4128yescomplemented
35C22×C3×C7C2417yes—
36C2×C3×D7C22×C41112yescomplemented
37C2×C3×D7C2418yes—
38Dic21C2418yescomplemented
39C7×Dic3C2418yescomplemented
40C24×C4—211noSylow
41C23×C7C2×Dic318yescomplemented
42C23×C7C22×D317yes—
43C22×D7C2×Dic31112yesdirect product factor
44C22×D7C22×D3128yes—
45C2×C4×C7—328no—
46C2×Dic7—328no—
47C4×D7—364no—
48C24×C3—716no—
49C24×C3—714no—
50C24×C3C2×D711yescharacteristic
51C22×Dic3—714no—
52C22×Dic3—7112no—
53C22×Dic3C2×D7114yesdirect product factor
54C2×C3×C7C23×C4114yescomplemented
55C2×C3×C7C2511yescharacteristic
56C3×D7C23×C4116yescomplemented
57C25—71no—
58C23×C4—31no—
59C23×C4—2128no—
60C23×C4—211no—
61C22×C7C22×Dic3128yescomplemented
62C22×C7C23×D317yes—
63C2×D7C22×Dic31112yesdirect product factor
64C2×D7C23×D318yes—
65C4×C7—38no—
66Dic7—38no—
67C23×C3—7112no—
68C23×C3—728no—
69C23×C3C4×D718yescomplemented
70C23×C3C22×D717yes—
71C2×Dic3—728no—
72C2×Dic3—764no—
73C2×Dic3C22×D7128yesdirect product factor
74C3×C7C24×C411yescharacteristic, complemented, Hall
75C24—716no—
76C24—714no—
77C24D3×D711yescharacteristic
78C22×C4—314no—
79C22×C4—21112no—
80C22×C4—2114no—
81C2×C7C23×Dic3114yescomplemented
82C2×C7C24×D311yescharacteristic
83D7C23×Dic3116yesdirect product factor
84C22×C3—7112no—
85C22×C3—78no—
86C22×C3C2×C4×D7128yescomplemented
87C22×C3C23×D717yes—
88Dic3—78no—
89Dic3C23×D718yesdirect product factor
90C23—7112no—
91C23—728no—
92C23Dic3×D718yesdirect product factor
93C23C2×D3×D717yes—
94C2×C4—328no—
95C2×C4—2164no—
96C2×C4—2128no—
97C7C24×Dic311yescharacteristic, complemented, Sylow
98C2×C3—716no—
99C2×C3C22×C4×D7114yescomplemented
100C2×C3C24×D711yescharacteristic
101C22—7112no—
102C22—78no—
103C22C2×Dic3×D7128yesdirect product factor
104C22C22×D3×D717yes—
105C4—38no—
106C4—218no—
107C3C23×C4×D711yescharacteristic, complemented, Sylow
108C2—716no—
109C2C22×Dic3×D7114yesdirect product factor
110C2C23×D3×D711yescharacteristic
subgroupG/H
center, hypercenterC24D3×D7
commutator, hypocenterC3×C7C24×C4
FrattiniC2C23×D3×D7
Fitting, socleC24×C3×C7C22
prime radicalC24×C3×D7C2
cosocleC2×C3×C7C25
LCSG ▷ C3×C7
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C23×Dic21C2semidirect
3. C23×C7×Dic3C2semidirect
4. C22×Dic3×D7C2direct (N × H)
7. C23×C3×D7C4semidirect
9. C22×Dic21C22semidirect
10. C22×C7×Dic3C22semidirect
11. C2×Dic3×D7C22direct (N × H)
17. C23×C3×C7C2×C4semidirect
19. C22×C3×D7C2×C4semidirect
21. C2×Dic21C23semidirect
22. C2×C7×Dic3C23semidirect
23. Dic3×D7C23direct (N × H)
25. C23×D7Dic3direct (N × H)
33. C23×Dic3D7direct (N × H)
34. C22×C3×C7C22×C4semidirect
36. C2×C3×D7C22×C4semidirect
38. Dic21C24semidirect
39. C7×Dic3C24semidirect
41. C23×C7C2×Dic3semidirect
43. C22×D7C2×Dic3direct (N × H)
53. C22×Dic3C2×D7direct (N × H)
54. C2×C3×C7C23×C4semidirect
56. C3×D7C23×C4semidirect
61. C22×C7C22×Dic3semidirect
63. C2×D7C22×Dic3direct (N × H)
69. C23×C3C4×D7semidirect
73. C2×Dic3C22×D7direct (N × H)
74. C3×C7C24×C4semidirect
81. C2×C7C23×Dic3semidirect
83. D7C23×Dic3direct (N × H)
86. C22×C3C2×C4×D7semidirect
89. Dic3C23×D7direct (N × H)
92. C23Dic3×D7direct (N × H)
97. C7C24×Dic3semidirect
99. C2×C3C22×C4×D7semidirect
103. C22C2×Dic3×D7direct (N × H)
107. C3C23×C4×D7semidirect
109. C2C22×Dic3×D7direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×D72
C23×Dic212
C23×C7×Dic32
C22×Dic3×D72
C23×C4×D73
C24×Dic37