Finite Group Databasebeta

← C3×C4×C7×#63271 #37524817

Subgroups — C3×C4×C7×#63271 #37524817

Order 22932.

168 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 72 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×#63271C211yesmaximal, characteristic
2C4×C7×#63271C313yesmaximal, direct product factor
3C3×C4×C72×C13C311yesmaximal, characteristic, complemented
4C3×C7×#63271C411yescharacteristic, direct product factor, Hall
5C2×C7×#63271C2×C313yes—
6C2×C3×C72×C13C2×C311yescharacteristic
7C3×C4×#63271C711yesmaximal, characteristic, direct product factor
8C3×C4×C7×C13⋊C3—71nomaximal
9C4×C72×C13C3211yescharacteristic, complemented, Hall
10C3×C72×C13C3×C411yescharacteristic, complemented
11C7×#63271C3×C413yesdirect product factor
12C3×C4×C7×ΓL(1,𝔽8)—131nomaximal, Hall
13C2×C3×#63271C2×C711yescharacteristic
14C2×C3×C7×C13⋊C3—71no—
15C2×C72×C13C2×C3211yescharacteristic
16C3×C4×C7×C13—32no—
17C3×C4×C7×C13ΓL(1,𝔽8)11yescharacteristic, complemented
18C3×C4×C7×C13C3×C711yescharacteristic, complemented
19C4×#63271C3×C713yesdirect product factor
20C4×C7×C13⋊C3—73no—
21C2×C3×C7×ΓL(1,𝔽8)—131no—
22C3×#63271C4×C711yescharacteristic, direct product factor
23C3×C7×C13⋊C3—71no—
24C72×C13C32×C411yescharacteristic, complemented, Hall
25C3×C4×C72C13⋊C311yescharacteristic, complemented
26C4×C7×ΓL(1,𝔽8)—133no—
27C2×C3×C7×C13—32no—
28C2×C3×C7×C13C2×ΓL(1,𝔽8)11yescharacteristic
29C2×C3×C7×C13C2×C3×C711yescharacteristic
30C2×#63271C2×C3×C713yes—
31C2×C7×C13⋊C3—73no—
32C3×C4×C13⋊C3—71noHall
33C3×C7×ΓL(1,𝔽8)—131noHall
34C4×C7×C13—32no—
35C4×C7×C13C3×ΓL(1,𝔽8)11yescharacteristic, complemented
36C4×C7×C13C32×C711yescharacteristic, complemented
37C2×C3×C72C2×C13⋊C311yescharacteristic
38C2×C7×ΓL(1,𝔽8)—133no—
39C3×C7×C13—32no—
40C3×C7×C13C4×ΓL(1,𝔽8)11yescharacteristic, complemented
41C3×C7×C13C3×C4×C711yescharacteristic, complemented
42#63271C3×C4×C713yesdirect product factor
43C7×C13⋊C3—73no—
44C32×C4×C7—911no—
45C3×C4×ΓL(1,𝔽8)—131no—
46C2×C3×C13⋊C3—71no—
47C4×C72C3×C13⋊C311yescharacteristic, complemented, Hall
48C2×C7×C13—32no—
49C2×C7×C13C2×C3×ΓL(1,𝔽8)11yescharacteristic
50C2×C7×C13C2×C32×C711yescharacteristic
51C3×C4×C13C7×ΓL(1,𝔽8)11yescharacteristic, complemented
52C4×C13⋊C3—73no—
53C3×C72C4×C13⋊C311yescharacteristic, complemented
54C7×ΓL(1,𝔽8)—133no—
55C2×C32×C7—911no—
56C2×C3×ΓL(1,𝔽8)—131no—
57C3×C13⋊C3—71noHall
58C2×C72C2×C3×C13⋊C311yescharacteristic
59C7×C13—32no—
60C7×C13C3×C4×ΓL(1,𝔽8)11yescharacteristic, complemented
61C7×C13C32×C4×C711yescharacteristic, complemented
62C3×C4×C7—32no—
63C3×C4×C7—913no—
64C3×C4×C7C7×C13⋊C311yescharacteristic, complemented
65C3×C4×C7#6327111yescharacteristic, direct product factor
66C4×ΓL(1,𝔽8)—133no—
67C2×C3×C13C2×C7×ΓL(1,𝔽8)11yescharacteristic
68C2×C13⋊C3—73no—
69C32×C7—911no—
70C3×ΓL(1,𝔽8)—131no—
71C4×C13C3×C7×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
72C72C3×C4×C13⋊C311yescharacteristic, complemented, Sylow
73C2×C3×C7—32no—
74C2×C3×C7—913no—
75C2×C3×C7C2×C7×C13⋊C311yescharacteristic
76C2×C3×C7C2×#6327111yescharacteristic
77C2×ΓL(1,𝔽8)—133no—
78C3×C13C4×C7×ΓL(1,𝔽8)11yescharacteristic, complemented
79C13⋊C3—73no—
80C32×C4—911noHall
81C4×C7—32no—
82C4×C7C3×C7×C13⋊C311yescharacteristic, complemented
83C4×C7C3×#6327111yescharacteristic, direct product factor
84C2×C13C2×C3×C7×ΓL(1,𝔽8)11yescharacteristic
85C3×C7—32no—
86C3×C7—913no—
87C3×C7C4×C7×C13⋊C311yescharacteristic, complemented
88C3×C7C4×#6327111yescharacteristic, direct product factor
89ΓL(1,𝔽8)—133no—
90C2×C32—911no—
91C2×C7—32no—
92C2×C7C2×C3×C7×C13⋊C311yescharacteristic
93C2×C7C2×C3×#6327111yescharacteristic
94C13C3×C4×C7×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
95C3×C4—913no—
96C3×C4C7×#6327111yescharacteristic, direct product factor
97C32—911noSylow
98C7—32no—
99C7C3×C4×C7×C13⋊C311yescharacteristic, complemented
100C7C3×C4×#6327111yescharacteristic, direct product factor
101C2×C3—913no—
102C2×C3C2×C7×#6327111yescharacteristic
103C4C3×C7×#6327111yescharacteristic, direct product factor, Sylow
104C3—913no—
105C3C4×C7×#6327111yescharacteristic, direct product factor
106C2C2×C3×C7×#6327111yescharacteristic
subgroupG/H
center, hypercenterC3×C4×C7#63271
commutator, hypocenterC7×C13C32×C4×C7
FrattiniC2C2×C3×C7×#63271
FittingC3×C4×C72×C13C3
socleC2×C3×C72×C13C2×C3
prime radicalC2×C3×C7×#63271C2
cosocleC2×C7×C13C2×C32×C7
LCSG ▷ C7×C13
UCSC1 ◁ C3×C4×C7

Complements

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

NH ≅ G/Ntype
2. C4×C7×#63271C3direct (N × H)
3. C3×C4×C72×C13C3semidirect
4. C3×C7×#63271C4direct (N × H)
7. C3×C4×#63271C7direct (N × H)
9. C4×C72×C13C32semidirect
10. C3×C72×C13C3×C4semidirect
11. C7×#63271C3×C4direct (N × H)
17. C3×C4×C7×C13ΓL(1,𝔽8)semidirect
18. C3×C4×C7×C13C3×C7semidirect
19. C4×#63271C3×C7direct (N × H)
22. C3×#63271C4×C7direct (N × H)
24. C72×C13C32×C4semidirect
25. C3×C4×C72C13⋊C3semidirect
35. C4×C7×C13C3×ΓL(1,𝔽8)semidirect
36. C4×C7×C13C32×C7semidirect
40. C3×C7×C13C4×ΓL(1,𝔽8)semidirect
41. C3×C7×C13C3×C4×C7semidirect
42. #63271C3×C4×C7direct (N × H)
47. C4×C72C3×C13⋊C3semidirect
51. C3×C4×C13C7×ΓL(1,𝔽8)semidirect
53. C3×C72C4×C13⋊C3semidirect
60. C7×C13C3×C4×ΓL(1,𝔽8)semidirect
61. C7×C13C32×C4×C7semidirect
64. C3×C4×C7C7×C13⋊C3semidirect
65. C3×C4×C7#63271direct (N × H)
71. C4×C13C3×C7×ΓL(1,𝔽8)semidirect
72. C72C3×C4×C13⋊C3semidirect
78. C3×C13C4×C7×ΓL(1,𝔽8)semidirect
82. C4×C7C3×C7×C13⋊C3semidirect
83. C4×C7C3×#63271direct (N × H)
87. C3×C7C4×C7×C13⋊C3semidirect
88. C3×C7C4×#63271direct (N × H)
94. C13C3×C4×C7×ΓL(1,𝔽8)semidirect
96. C3×C4C7×#63271direct (N × H)
99. C7C3×C4×C7×C13⋊C3semidirect
100. C7C3×C4×#63271direct (N × H)
103. C4C3×C7×#63271direct (N × H)
105. C3C4×C7×#63271direct (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×#632712
C4×C7×#632713
C3×C4×C72×C133
C3×C4×#632717
C3×C4×C7×C13⋊C37
C3×C4×C7×ΓL(1,𝔽8)13