Finite Group Databasebeta

← ΓL(1,𝔽8)×C3⋊(C7⋊#261) #37525237

Subgroups — ΓL(1,𝔽8)×C3⋊(C7⋊#261) #37525237

Order 22932.

120 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 39 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
1C3×C7×ΓL(1,𝔽8)×D13C211yesmaximal, characteristic
2C7×C3⋊(C7⋊#261)C311yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)×C7⋊#261—31nomaximal
4C3×C7×C13×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
5C3×C72×D13C2×C311yescharacteristic
6C7×ΓL(1,𝔽8)×D13—21no—
7C7×ΓL(1,𝔽8)×D13D311yescharacteristic
8C3×C3⋊(C7⋊#261)—71nomaximal
9ΓL(1,𝔽8)×C3⋊#261—71nomaximal
10C7×C7⋊#261—31noHall
11C3×C72×C13C3×C411yescharacteristic, complemented
12C7×C13×ΓL(1,𝔽8)—21no—
13C7×C13×ΓL(1,𝔽8)Dic311yescharacteristic, complemented
14ΓL(1,𝔽8)×Dic21—131nomaximal, Hall
15C32×C7×D13—71no—
16C3×ΓL(1,𝔽8)×D13D711yescharacteristic
17C72×D13C3×D311yescharacteristic
18C3⋊(C7⋊#261)ΓL(1,𝔽8)11yescharacteristic, direct product factor
19C7×C3⋊#261—71no—
20C3×C7⋊#261—211no—
21ΓL(1,𝔽8)×#261—211no—
22C2×C3×C7×ΓL(1,𝔽8)—131no—
23C32×C7×C13—71no—
24C3×C13×ΓL(1,𝔽8)Dic711yescharacteristic, complemented
25C72×C13C3×Dic311yescharacteristic, complemented, Hall
26C7×Dic21—131no—
27ΓL(1,𝔽8)×Dic7—391no—
28C3×C7×D13—61no—
29C3×C7×D13—71no—
30C3×C7×D13—141no—
31C3×C7×D13C2×ΓL(1,𝔽8)11yescharacteristic
32C3×C7×D13C3×D711yescharacteristic
33ΓL(1,𝔽8)×D13—21no—
34ΓL(1,𝔽8)×D13D2111yescharacteristic
35C3×C3⋊#261—491noHall
36C3×C7×ΓL(1,𝔽8)#26111yescharacteristic, complemented, Hall
37C7⋊#261—31no—
38C7×#261—211no—
39C2×C3×C72—131no—
40C2×C7×ΓL(1,𝔽8)—131no—
41C2×C7×ΓL(1,𝔽8)—261no—
42C3×C7×C13—61no—
43C3×C7×C13—71no—
44C3×C7×C13—141no—
45C3×C7×C13C4×ΓL(1,𝔽8)11yescharacteristic, complemented
46C3×C7×C13C3×Dic711yescharacteristic, complemented
47C13×ΓL(1,𝔽8)—21no—
48C13×ΓL(1,𝔽8)Dic2111yescharacteristic, complemented
49C3×Dic21—911no—
50Dic3×ΓL(1,𝔽8)—911no—
51C32×D13—71no—
52C7×Dic7—391noHall
53C7×D13—61no—
54C7×D13D3×ΓL(1,𝔽8)11yescharacteristic
55C7×D13C3×D2111yescharacteristic
56C3⋊#261—71no—
57C3×#261—1471no—
58C3×C72C3×#26111yescharacteristic, complemented
59C7×ΓL(1,𝔽8)—21no—
60C7×ΓL(1,𝔽8)C3⋊#26111yescharacteristic, complemented
61C2×C32×C7—911no—
62C2×C3×ΓL(1,𝔽8)—131no—
63C32×C13—71noHall
64C2×C72—131no—
65C7×C13—61no—
66C7×C13Dic3×ΓL(1,𝔽8)11yescharacteristic, complemented
67C7×C13C3×Dic2111yescharacteristic, complemented
68Dic21—131no—
69C3×Dic7—2731no—
70C7×Dic3—911no—
71C4×ΓL(1,𝔽8)—2731no—
72C3×D13—71no—
73C3×D13—141no—
74C3×D13D7×ΓL(1,𝔽8)11yescharacteristic
75C32×C7—71no—
76C3×ΓL(1,𝔽8)C7⋊#26111yescharacteristic, complemented
77#261—211noHall
78C72C3×C3⋊#26111yescharacteristic, complemented, Sylow
79C2×C3×C7—131no—
80C2×C3×C7—131no—
81C2×C3×C7—781no—
82C2×C3×C7—911no—
83C2×C3×C7—1821no—
84C2×ΓL(1,𝔽8)—131no—
85C2×ΓL(1,𝔽8)—261no—
86C3×C13—71no—
87C3×C13—141no—
88C3×C13ΓL(1,𝔽8)×Dic711yescharacteristic, complemented
89C3×Dic3—6371noHall
90C4×C7—2731no—
91Dic7—391no—
92D13ΓL(1,𝔽8)×D2111yescharacteristic
93C3×C7—61no—
94C3×C7—71no—
95C3×C7—141no—
96C3×C7ΓL(1,𝔽8)×#26111yescharacteristic, complemented
97C3×C7C3×C7⋊#26111yescharacteristic, complemented
98ΓL(1,𝔽8)—21no—
99ΓL(1,𝔽8)C3⋊(C7⋊#261)11yescharacteristic, direct product factor
100C2×C32—911no—
101C2×C7—131no—
102C2×C7—131no—
103C2×C7—781no—
104C13ΓL(1,𝔽8)×Dic2111yescharacteristic, complemented, Sylow
105C3×C4—19111no—
106Dic3—911no—
107C32—71noSylow
108C7—61no—
109C7ΓL(1,𝔽8)×C3⋊#26111yescharacteristic, complemented
110C7C3×C3⋊(C7⋊#261)11yescharacteristic, complemented
111C2×C3—131no—
112C2×C3—911no—
113C2×C3—1821no—
114C4—2731noSylow
115C3—71no—
116C3—141no—
117C3ΓL(1,𝔽8)×C7⋊#26111yescharacteristic, complemented
118C2—131no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C72×C13C3×C4
prime radicalC3×C7×ΓL(1,𝔽8)×D13C2
cosocleC3×C72×D13C2×C3
LCSG ▷ C3×C72×C13
UCSC1

Complements

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

NH ≅ G/Ntype
2. C7×C3⋊(C7⋊#261)C3semidirect
4. C3×C7×C13×ΓL(1,𝔽8)C4semidirect
11. C3×C72×C13C3×C4semidirect
13. C7×C13×ΓL(1,𝔽8)Dic3semidirect
18. C3⋊(C7⋊#261)ΓL(1,𝔽8)direct (N × H)
24. C3×C13×ΓL(1,𝔽8)Dic7semidirect
25. C72×C13C3×Dic3semidirect
36. C3×C7×ΓL(1,𝔽8)#261semidirect
45. C3×C7×C13C4×ΓL(1,𝔽8)semidirect
46. C3×C7×C13C3×Dic7semidirect
48. C13×ΓL(1,𝔽8)Dic21semidirect
58. C3×C72C3×#261semidirect
60. C7×ΓL(1,𝔽8)C3⋊#261semidirect
66. C7×C13Dic3×ΓL(1,𝔽8)semidirect
67. C7×C13C3×Dic21semidirect
76. C3×ΓL(1,𝔽8)C7⋊#261semidirect
78. C72C3×C3⋊#261semidirect
88. C3×C13ΓL(1,𝔽8)×Dic7semidirect
96. C3×C7ΓL(1,𝔽8)×#261semidirect
97. C3×C7C3×C7⋊#261semidirect
99. ΓL(1,𝔽8)C3⋊(C7⋊#261)direct (N × H)
104. C13ΓL(1,𝔽8)×Dic21semidirect
109. C7ΓL(1,𝔽8)×C3⋊#261semidirect
110. C7C3×C3⋊(C7⋊#261)semidirect
117. C3ΓL(1,𝔽8)×C7⋊#261semidirect

Maximal subgroups

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

subgroupindex
C3×C7×ΓL(1,𝔽8)×D132
C7×C3⋊(C7⋊#261)3
ΓL(1,𝔽8)×C7⋊#2613
C3×C3⋊(C7⋊#261)7
ΓL(1,𝔽8)×C3⋊#2617
ΓL(1,𝔽8)×Dic2113