Finite Group Databasebeta

← ΓL(1,𝔽8)⋊Dic455 #50303650

Subgroups — ΓL(1,𝔽8)⋊Dic455 #50303650

Order 38220.

120 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 50 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×C5×C7×C13×ΓL(1,𝔽8)C211yesmaximal, characteristic
2C7⋊Dic455C311yesmaximal, characteristic, complemented, Hall
3C5×C7×C13×ΓL(1,𝔽8)C411yescharacteristic, complemented, Hall
4ΓL(1,𝔽8)⋊Dic91—51nomaximal, Hall
5C2×C5×C72×C13C2×C311yescharacteristic
6ΓL(1,𝔽8)⋊Dic65—71nomaximal
7C3×Dic455—71nomaximal
8C2×C7×C13×ΓL(1,𝔽8)D511yescharacteristic
9C5×C72×C13C3×C411yescharacteristic, complemented, Hall
10ΓL(1,𝔽8)⋊Dic35—131nomaximal, Hall
11C2×C5×C13×ΓL(1,𝔽8)D711yescharacteristic
12C2×C3×C5×C7×C13—71no—
13C7⋊Dic91—51noHall
14C7×C13×ΓL(1,𝔽8)Dic511yescharacteristic, complemented, Hall
15Dic455—71no—
16Dic455—71no—
17Dic455—212no—
18C2×C5×C7×ΓL(1,𝔽8)D1311yescharacteristic
19C3×C5×C7×C13—71no—
20C5×C13×ΓL(1,𝔽8)Dic711yescharacteristic, complemented
21C2×C72×C13C3×D511yescharacteristic
22C3×Dic91—351no—
23ΓL(1,𝔽8)⋊Dic13—351no—
24C7⋊Dic35—131noHall
25C2×C5×C7×C13—32no—
26C2×C5×C7×C13AGL(1,𝔽7)11yescharacteristic
27C2×C5×C7×C13C3×D711yescharacteristic
28C3×Dic65—491noHall
29C5×C7×ΓL(1,𝔽8)Dic1311yescharacteristic, complemented, Hall
30C72×C13C3×Dic511yescharacteristic, complemented, Hall
31ΓL(1,𝔽8)⋊Dic7—651noHall
32C2×C3×C7×C13—71no—
33C2×C13×ΓL(1,𝔽8)D3511yescharacteristic
34C2×C5×C72C3×D1311yescharacteristic
35C5×C7×C13—32no—
36C5×C7×C13ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
37C5×C7×C13C3×Dic711yescharacteristic, complemented
38C3×Dic35—911no—
39ΓL(1,𝔽8)⋊Dic5—911no—
40C2×C3×C5×C13—71no—
41Dic91—351no—
42Dic91—351no—
43Dic91—1052no—
44C2×C7×ΓL(1,𝔽8)D6511yescharacteristic
45C3×C7×C13—71no—
46C13×ΓL(1,𝔽8)Dic3511yescharacteristic, complemented
47Dic65—491noHall
48C5×C72C3×Dic1311yescharacteristic, complemented, Hall
49C2×C3×C5×C7—71no—
50C2×C5×ΓL(1,𝔽8)D9111yescharacteristic
51C7⋊Dic7—651noHall
52C3×C5×C13—71noHall
53C2×C7×C13—32no—
54C2×C7×C13C5⋊AGL(1,𝔽7)11yescharacteristic
55C2×C7×C13C3×D3511yescharacteristic
56C3×Dic13—2451noHall
57C7×ΓL(1,𝔽8)Dic6511yescharacteristic, complemented, Hall
58Dic35—2732no—
59Dic35—911no—
60Dic35—911no—
61C2×C5×C13ΓL(1,𝔽8)⋊D711yescharacteristic
62C3×C5×C7—71no—
63C5×ΓL(1,𝔽8)Dic9111yescharacteristic, complemented
64C2×C72C3×D6511yescharacteristic
65C7×C13—32no—
66C7×C13ΓL(1,𝔽8)⋊Dic511yescharacteristic, complemented
67C7×C13C3×Dic3511yescharacteristic, complemented
68C3×Dic7—4551no—
69ΓL(1,𝔽8)⋊C4—4551no—
70C2×C3×C13—71no—
71C2×C5×C7—32no—
72C2×C5×C7ΓL(1,𝔽8)⋊D1311yescharacteristic
73C2×C5×C7C3×D9111yescharacteristic
74C5×C13ΓL(1,𝔽8)⋊Dic711yescharacteristic, complemented, Hall
75C3×Dic5—6371noHall
76Dic13—2451noHall
77C72C3×Dic6511yescharacteristic, complemented, Sylow
78C2×C3×C7—71no—
79C2×ΓL(1,𝔽8)D45511yescharacteristic
80C3×C13—71noHall
81C5×C7—32no—
82C5×C7ΓL(1,𝔽8)⋊Dic1311yescharacteristic, complemented
83C5×C7C3×Dic9111yescharacteristic, complemented
84C2×C3×C5—71no—
85Dic7—13652no—
86Dic7—4551no—
87Dic7—4551no—
88C2×C13C5⋊(ΓL(1,𝔽8)⋊D7)11yescharacteristic
89C3×C7—71no—
90ΓL(1,𝔽8)Dic45511yescharacteristic, complemented
91Dic5—6371noHall
92C3×C5—71noHall
93C2×C7—32no—
94C2×C7C3×D45511yescharacteristic
95C2×C7C5⋊(ΓL(1,𝔽8)⋊D13)11yescharacteristic
96C13ΓL(1,𝔽8)⋊Dic3511yescharacteristic, complemented, Sylow
97C3×C4—31851noHall
98C2×C5ΓL(1,𝔽8)⋊D9111yescharacteristic
99C7—32no—
100C7C3×Dic45511yescharacteristic, complemented
101C7ΓL(1,𝔽8)⋊Dic6511yescharacteristic, complemented
102C2×C3—71no—
103C5ΓL(1,𝔽8)⋊Dic9111yescharacteristic, complemented, Sylow
104C4—31851noSylow
105C3—71noSylow
106C2C5⋊(ΓL(1,𝔽8)⋊D91)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2C5⋊(ΓL(1,𝔽8)⋊D91)
commutator, hypocenterC5×C72×C13C3×C4
Fitting, socle, cosocleC2×C5×C72×C13C2×C3
prime radicalC2×C5×C7×C13×ΓL(1,𝔽8)C2
LCSG ▷ C5×C72×C13
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C7⋊Dic455C3semidirect
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)Dic13semidirect
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×Dic13semidirect
57. C7×ΓL(1,𝔽8)Dic65semidirect
63. C5×ΓL(1,𝔽8)Dic91semidirect
66. C7×C13ΓL(1,𝔽8)⋊Dic5semidirect
67. C7×C13C3×Dic35semidirect
74. C5×C13ΓL(1,𝔽8)⋊Dic7semidirect
77. C72C3×Dic65semidirect
82. C5×C7ΓL(1,𝔽8)⋊Dic13semidirect
83. C5×C7C3×Dic91semidirect
90. ΓL(1,𝔽8)Dic455semidirect
96. C13ΓL(1,𝔽8)⋊Dic35semidirect
100. C7C3×Dic455semidirect
101. C7ΓL(1,𝔽8)⋊Dic65semidirect
103. C5ΓL(1,𝔽8)⋊Dic91semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C7×C13×ΓL(1,𝔽8)2
C7⋊Dic4553
ΓL(1,𝔽8)⋊Dic915
ΓL(1,𝔽8)⋊Dic657
C3×Dic4557
ΓL(1,𝔽8)⋊Dic3513