Finite Group Databasebeta

← C23×C4×D21 #13062369

Subgroups — C23×C4×D21 #13062369

Order 1344.

2724 conjugacy classes of subgroups; only the 1035 normal subgroups are shown.

normal subgroups only

This group has 2724 conjugacy classes of subgroups — too many to draw. Only its 1035 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×D21C211yesmaximal, characteristic
2C23×C3×C4×C7C211yesmaximal, characteristic, complemented
3C23×Dic21C211yesmaximal, characteristic, complemented
4C22×C4×D21C2128yesmaximal, direct product factor
5C23×C4×D7—31nomaximal, Hall
6C24×C3×C7C2211yescharacteristic
7C23×D21C4116yesdirect product factor
8C23×D21C22114yes—
9C22×C3×C4×C7C22114yescomplemented
10C22×Dic21C22114yescomplemented
11C2×C4×D21C221112yesdirect product factor
12C24×D7—31no—
13C23×C4×C7D311yescharacteristic, complemented
14C23×Dic7—31no—
15C22×C4×D7—328no—
16C23×C4×D3—71nomaximal, Hall
17C23×C3×C7C2×C418yescomplemented
18C23×C3×C7C2317yes—
19C22×D21C2×C41112yesdirect product factor
20C22×D21C23128yes—
21C2×C3×C4×C7C23128yescomplemented
22C2×Dic21C23128yescomplemented
23C4×D21C23164yesdirect product factor
24C24×C7C2×D311yescharacteristic
25C23×D7—316no—
26C23×D7—314no—
27C22×C4×C7C2×D3114yescomplemented
28C22×Dic7—314no—
29C2×C4×D7—3112no—
30C24×D3—71no—
31C23×C3×C4D711yescharacteristic, complemented
32C23×Dic3—71no—
33C22×C4×D3—728no—
34C22×C3×C7C22×C4128yescomplemented
35C22×C3×C7C2417yes—
36C2×D21C22×C41112yesdirect product factor
37C2×D21C2418yes—
38C3×C4×C7C2418yescomplemented
39Dic21C2418yescomplemented
40C24×C4—211noSylow
41C23×C7C4×D318yescomplemented
42C23×C7C22×D317yes—
43C22×D7—3112no—
44C22×D7—328no—
45C2×C4×C7C22×D3128yescomplemented
46C2×Dic7—328no—
47C4×D7—364no—
48C24×C3C2×D711yescharacteristic
49C23×D3—716no—
50C23×D3—714no—
51C22×C3×C4C2×D7114yescomplemented
52C22×Dic3—714no—
53C2×C4×D3—7112no—
54C2×C3×C7C23×C4114yescomplemented
55C2×C3×C7C2511yescharacteristic
56D21C23×C4116yesdirect product factor
57C25—211no—
58C23×C4—211no—
59C23×C4—2128no—
60C23×C4D2111yescharacteristic, direct product factor
61C22×C7C2×C4×D3128yescomplemented
62C22×C7C23×D317yes—
63C2×D7—3112no—
64C2×D7—38no—
65C4×C7C23×D318yescomplemented
66Dic7—38no—
67C23×C3C4×D718yescomplemented
68C23×C3C22×D717yes—
69C22×D3—7112no—
70C22×D3—728no—
71C2×C3×C4C22×D7128yescomplemented
72C2×Dic3—728no—
73C4×D3—764no—
74C3×C7C24×C411yescharacteristic, complemented, Hall
75C24—2116no—
76C24—2114no—
77C24C2×D2111yescharacteristic
78C22×C4—2114no—
79C22×C4—21112no—
80C22×C4C2×D21114yesdirect product factor
81C2×C7C22×C4×D3114yescomplemented
82C2×C7C24×D311yescharacteristic
83D7—316no—
84C22×C3C2×C4×D7128yescomplemented
85C22×C3C23×D717yes—
86C2×D3—7112no—
87C2×D3—78no—
88C3×C4C23×D718yescomplemented
89Dic3—78no—
90C23—21112no—
91C23—2128no—
92C23C4×D2118yesdirect product factor
93C23C22×D2117yes—
94C2×C4—2128no—
95C2×C4—2164no—
96C2×C4C22×D21128yesdirect product factor
97C7C23×C4×D311yescharacteristic, complemented, Sylow
98C2×C3C22×C4×D7114yescomplemented
99C2×C3C24×D711yescharacteristic
100D3—716no—
101C22—218no—
102C22—21112no—
103C22C2×C4×D21128yesdirect product factor
104C22C23×D2117yes—
105C4—218no—
106C4C23×D2118yesdirect product factor
107C3C23×C4×D711yescharacteristic, complemented, Sylow
108C2—2116no—
109C2C22×C4×D21114yesdirect product factor
110C2C24×D2111yescharacteristic
subgroupG/H
center, hypercenterC23×C4D21
commutator, hypocenterC3×C7C24×C4
FrattiniC2C24×D21
FittingC23×C3×C4×C7C2
socleC24×C3×C7C22
prime radicalC24×D21C2
cosocleC2×C3×C7C25
LCSG ▷ C3×C7
UCSC1 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C23×C3×C4×C7C2semidirect
3. C23×Dic21C2semidirect
4. C22×C4×D21C2direct (N × H)
7. C23×D21C4direct (N × H)
9. C22×C3×C4×C7C22semidirect
10. C22×Dic21C22semidirect
11. C2×C4×D21C22direct (N × H)
13. C23×C4×C7D3semidirect
17. C23×C3×C7C2×C4semidirect
19. C22×D21C2×C4direct (N × H)
21. C2×C3×C4×C7C23semidirect
22. C2×Dic21C23semidirect
23. C4×D21C23direct (N × H)
27. C22×C4×C7C2×D3semidirect
31. C23×C3×C4D7semidirect
34. C22×C3×C7C22×C4semidirect
36. C2×D21C22×C4direct (N × H)
38. C3×C4×C7C24semidirect
39. Dic21C24semidirect
41. C23×C7C4×D3semidirect
45. C2×C4×C7C22×D3semidirect
51. C22×C3×C4C2×D7semidirect
54. C2×C3×C7C23×C4semidirect
56. D21C23×C4direct (N × H)
60. C23×C4D21direct (N × H)
61. C22×C7C2×C4×D3semidirect
65. C4×C7C23×D3semidirect
67. C23×C3C4×D7semidirect
71. C2×C3×C4C22×D7semidirect
74. C3×C7C24×C4semidirect
80. C22×C4C2×D21direct (N × H)
81. C2×C7C22×C4×D3semidirect
84. C22×C3C2×C4×D7semidirect
88. C3×C4C23×D7semidirect
92. C23C4×D21direct (N × H)
96. C2×C4C22×D21direct (N × H)
97. C7C23×C4×D3semidirect
98. C2×C3C22×C4×D7semidirect
103. C22C2×C4×D21direct (N × H)
106. C4C23×D21direct (N × H)
107. C3C23×C4×D7semidirect
109. C2C22×C4×D21direct (N × H)

Maximal subgroups

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

subgroupindex
C24×D212
C23×C3×C4×C72
C23×Dic212
C22×C4×D212
C23×C4×D73
C23×C4×D37