Finite Group Databasebeta

← ΓL(1,𝔽8)⋊(C7⋊#63118) #50304014

Subgroups — ΓL(1,𝔽8)⋊(C7⋊#63118) #50304014

Order 38220.

120 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 38 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
1C5×C7×ΓL(1,𝔽8)×D13C211yesmaximal, characteristic
2C7⋊(C7⋊#63118)C311yesmaximal, characteristic, complemented, Hall
3C5×C7×C13×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
4ΓL(1,𝔽8)⋊(C7⋊#261)—51nomaximal, Hall
5C5×C72×D13C2×C311yescharacteristic
6C3×C7⋊#63118—71nomaximal
7ΓL(1,𝔽8)⋊#63118—71nomaximal
8C7×ΓL(1,𝔽8)×D13D511yescharacteristic
9C5×C72×C13C3×C411yescharacteristic, complemented, Hall
10ΓL(1,𝔽8)⋊Dic35—131nomaximal, Hall
11C5×ΓL(1,𝔽8)×D13D711yescharacteristic
12C3×C5×C7×D13—71no—
13C7⋊(C7⋊#261)—51noHall
14C7×C13×ΓL(1,𝔽8)Dic511yescharacteristic, complemented, Hall
15C7⋊#63118—71no—
16C7⋊#63118—71no—
17C7⋊#63118—212no—
18C2×C5×C7×ΓL(1,𝔽8)—131no—
19C3×C5×C7×C13—71no—
20C5×C13×ΓL(1,𝔽8)Dic711yescharacteristic, complemented
21C72×D13C3×D511yescharacteristic
22C3×C7⋊#261—351no—
23ΓL(1,𝔽8)⋊#261—351no—
24C7⋊Dic35—131noHall
25C5×C7×D13—32no—
26C5×C7×D13AGL(1,𝔽7)11yescharacteristic
27C5×C7×D13C3×D711yescharacteristic
28C3×#63118—491noHall
29C5×C7×ΓL(1,𝔽8)#26111yescharacteristic, complemented, Hall
30C72×C13C3×Dic511yescharacteristic, complemented, Hall
31ΓL(1,𝔽8)⋊Dic7—651noHall
32C3×C7×D13—71no—
33ΓL(1,𝔽8)×D13D3511yescharacteristic
34C2×C5×C72—131no—
35C5×C7×C13—32no—
36C5×C7×C13ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
37C5×C7×C13C3×Dic711yescharacteristic, complemented
38C3×Dic35—911no—
39ΓL(1,𝔽8)⋊Dic5—911no—
40C3×C5×D13—71no—
41C7⋊#261—351no—
42C7⋊#261—351no—
43C7⋊#261—1052no—
44C2×C7×ΓL(1,𝔽8)—131no—
45C3×C7×C13—71no—
46C13×ΓL(1,𝔽8)Dic3511yescharacteristic, complemented
47#63118—491noHall
48C5×C72C3×#26111yescharacteristic, complemented, Hall
49C2×C3×C5×C7—911no—
50C2×C5×ΓL(1,𝔽8)—131no—
51C7⋊Dic7—651noHall
52C3×C5×C13—71noHall
53C7×D13—32no—
54C7×D13C5⋊AGL(1,𝔽7)11yescharacteristic
55C7×D13C3×D3511yescharacteristic
56C3×#261—2451noHall
57C7×ΓL(1,𝔽8)#6311811yescharacteristic, complemented, Hall
58Dic35—2732no—
59Dic35—911no—
60Dic35—911no—
61C5×D13ΓL(1,𝔽8)⋊D711yescharacteristic
62C3×C5×C7—71no—
63C5×ΓL(1,𝔽8)C7⋊#26111yescharacteristic, complemented
64C2×C72—131no—
65C7×C13—32no—
66C7×C13ΓL(1,𝔽8)⋊Dic511yescharacteristic, complemented
67C7×C13C3×Dic3511yescharacteristic, complemented
68C3×Dic7—4551no—
69ΓL(1,𝔽8)⋊C4—4551no—
70C3×D13—71no—
71C2×C5×C7—131no—
72C2×C5×C7—131no—
73C2×C5×C7—392no—
74C5×C13ΓL(1,𝔽8)⋊Dic711yescharacteristic, complemented, Hall
75C3×Dic5—6371noHall
76#261—2451noHall
77C72C3×#6311811yescharacteristic, complemented, Sylow
78C2×C3×C7—911no—
79C2×ΓL(1,𝔽8)—131no—
80C3×C13—71noHall
81C5×C7—32no—
82C5×C7ΓL(1,𝔽8)⋊#26111yescharacteristic, complemented
83C5×C7C3×C7⋊#26111yescharacteristic, complemented
84C2×C3×C5—911no—
85Dic7—13652no—
86Dic7—4551no—
87Dic7—4551no—
88D13C5⋊(ΓL(1,𝔽8)⋊D7)11yescharacteristic
89C3×C7—71no—
90ΓL(1,𝔽8)C7⋊#6311811yescharacteristic, complemented
91Dic5—6371noHall
92C3×C5—71noHall
93C2×C7—131no—
94C2×C7—131no—
95C2×C7—392no—
96C13ΓL(1,𝔽8)⋊Dic3511yescharacteristic, complemented, Sylow
97C3×C4—31851noHall
98C2×C5—131no—
99C7—32no—
100C7ΓL(1,𝔽8)⋊#6311811yescharacteristic, complemented
101C7C3×C7⋊#6311811yescharacteristic, complemented
102C2×C3—911no—
103C5ΓL(1,𝔽8)⋊(C7⋊#261)11yescharacteristic, complemented, Sylow
104C4—31851noSylow
105C3—71noSylow
106C2—131no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C72×C13C3×C4
prime radicalC5×C7×ΓL(1,𝔽8)×D13C2
cosocleC5×C72×D13C2×C3
LCSG ▷ C5×C72×C13
UCSC1

Complements

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

NH ≅ G/Ntype
2. C7⋊(C7⋊#63118)C3semidirect
3. C5×C7×C13×ΓL(1,𝔽8)C4semidirect
9. C5×C72×C13C3×C4semidirect
14. C7×C13×ΓL(1,𝔽8)Dic5semidirect
20. C5×C13×ΓL(1,𝔽8)Dic7semidirect
29. C5×C7×ΓL(1,𝔽8)#261semidirect
30. C72×C13C3×Dic5semidirect
36. C5×C7×C13ΓL(1,𝔽8)⋊C4semidirect
37. C5×C7×C13C3×Dic7semidirect
46. C13×ΓL(1,𝔽8)Dic35semidirect
48. C5×C72C3×#261semidirect
57. C7×ΓL(1,𝔽8)#63118semidirect
63. C5×ΓL(1,𝔽8)C7⋊#261semidirect
66. C7×C13ΓL(1,𝔽8)⋊Dic5semidirect
67. C7×C13C3×Dic35semidirect
74. C5×C13ΓL(1,𝔽8)⋊Dic7semidirect
77. C72C3×#63118semidirect
82. C5×C7ΓL(1,𝔽8)⋊#261semidirect
83. C5×C7C3×C7⋊#261semidirect
90. ΓL(1,𝔽8)C7⋊#63118semidirect
96. C13ΓL(1,𝔽8)⋊Dic35semidirect
100. C7ΓL(1,𝔽8)⋊#63118semidirect
101. C7C3×C7⋊#63118semidirect
103. C5ΓL(1,𝔽8)⋊(C7⋊#261)semidirect

Maximal subgroups

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

subgroupindex
C5×C7×ΓL(1,𝔽8)×D132
C7⋊(C7⋊#63118)3
ΓL(1,𝔽8)⋊(C7⋊#261)5
C3×C7⋊#631187
ΓL(1,𝔽8)⋊#631187
ΓL(1,𝔽8)⋊Dic3513