Finite Group Databasebeta

← ΓL(1,𝔽8)×C22⋊#63271 #37525396

Subgroups — ΓL(1,𝔽8)×C22⋊#63271 #37525396

Order 22932.

132 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 29 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
1C22×C7×C13×ΓL(1,𝔽8)C311yesmaximal, characteristic, complemented
2C22⋊#14274262C311yesmaximal, characteristic, complemented
3C7×C22⋊#63271C311yesmaximal, characteristic, complemented
4C22⋊(C13⋊#3862)C311yesmaximal, characteristic, complemented
5ΓL(1,𝔽8)×#63271—41nomaximal, Hall
6C2×C7×C13×ΓL(1,𝔽8)—31no—
7C3×C22⋊#63271—71nomaximal
8ΓL(1,𝔽8)×C22⋊(C13⋊C3)—71nomaximal
9C22×C72×C13C3211yescharacteristic, complemented, Hall
10C13⋊#3862—41no—
11#14274262—41no—
12C7×#63271—41no—
13C7×C13×ΓL(1,𝔽8)A411yescharacteristic, complemented
14ΓL(1,𝔽8)×C22⋊ΓL(1,𝔽8)—131nomaximal, Hall
15C2×C72×C13—31no—
16C22×C3×C7×C13—71no—
17C22⋊#63270—71no—
18C22⋊#63271—71no—
19C22⋊#63271—71no—
20C22⋊#63271—71no—
21C22⋊#63271—212no—
22C22⋊#63271ΓL(1,𝔽8)11yescharacteristic, direct product factor
23C7×C22⋊(C13⋊C3)—71no—
24C22×C13×ΓL(1,𝔽8)ΓL(1,𝔽8)11yescharacteristic, complemented
25C3×#63271—281no—
26ΓL(1,𝔽8)×C13⋊C3—281no—
27C72×C13C3×A411yescharacteristic, complemented, Hall
28C22⋊#3862—131no—
29C22⋊#3861—131no—
30C7×C22⋊ΓL(1,𝔽8)—131no—
31C22×C7×ΓL(1,𝔽8)C13⋊C311yescharacteristic, complemented
32C2×C3×C7×C13—211no—
33C2×C13×ΓL(1,𝔽8)—31no—
34C3×C22⋊(C13⋊C3)—491noHall
35ΓL(1,𝔽8)2—521noHall
36C22×C7×C13—32no—
37C22×C7×C13C3×ΓL(1,𝔽8)11yescharacteristic, complemented
38C22×C7×C13C3×ΓL(1,𝔽8)11yescharacteristic, complemented
39C2×C7×ΓL(1,𝔽8)—31no—
40C3×C7×C13—71no—
41#63271—41no—
42#63271—281no—
43#63271—281no—
44#63271—281no—
45#63271—842no—
46#63270—281no—
47C7×C13⋊C3—281no—
48C13×ΓL(1,𝔽8)C22⋊ΓL(1,𝔽8)11yescharacteristic, complemented
49C3×C22⋊ΓL(1,𝔽8)—911no—
50A4×ΓL(1,𝔽8)—911no—
51C22×C72C3×C13⋊C311yescharacteristic, complemented, Hall
52C2×C7×C13—31no—
53C2×C7×C13—31no—
54C2×C7×C13—92no—
55C22×C3×C13—71no—
56C22⋊(C13⋊C3)—71no—
57C22⋊(C13⋊C3)—491no—
58C22⋊(C13⋊C3)—491no—
59#3862—521no—
60#3861—521no—
61C7×ΓL(1,𝔽8)—521no—
62C7×ΓL(1,𝔽8)C22⋊(C13⋊C3)11yescharacteristic, complemented
63C3×C13⋊C3—1961noHall
64C2×C72—31no—
65C7×C13—32no—
66C7×C13A4×ΓL(1,𝔽8)11yescharacteristic, complemented
67C7×C13C3×C22⋊ΓL(1,𝔽8)11yescharacteristic, complemented
68C22×C3×C7—71no—
69C22⋊ΓL(1,𝔽8)—131no—
70C22⋊ΓL(1,𝔽8)—2732no—
71C22⋊ΓL(1,𝔽8)—911no—
72C22⋊ΓL(1,𝔽8)—911no—
73C22⋊ΓL(1,𝔽8)—911no—
74C22⋊ΓL(1,𝔽8)—911no—
75C7×A4—911no—
76C22×ΓL(1,𝔽8)#6327111yescharacteristic, complemented
77C2×C3×C13—211no—
78C3×ΓL(1,𝔽8)—3641no—
79C3×ΓL(1,𝔽8)—3641no—
80C22×C13ΓL(1,𝔽8)211yescharacteristic, complemented, Hall
81C72C3×C22⋊(C13⋊C3)11yescharacteristic, complemented, Sylow
82C2×C3×C7—211no—
83C2×ΓL(1,𝔽8)—31no—
84C3×C13—71no—
85C13⋊C3—281no—
86C13⋊C3—1961no—
87C13⋊C3—1961no—
88C3×A4—6371noHall
89C22×C7—32no—
90C22×C7ΓL(1,𝔽8)×C13⋊C311yescharacteristic, complemented
91C22×C7C3×#6327111yescharacteristic, complemented
92C2×C13—31no—
93C3×C7—71no—
94C3×C7—3641no—
95ΓL(1,𝔽8)—521no—
96ΓL(1,𝔽8)—10922no—
97ΓL(1,𝔽8)—3641no—
98ΓL(1,𝔽8)—3641no—
99ΓL(1,𝔽8)—3641no—
100ΓL(1,𝔽8)—3641no—
101ΓL(1,𝔽8)C22⋊#6327111yescharacteristic, direct product factor
102C2×C7—31no—
103C2×C7—31no—
104C2×C7—92no—
105C13ΓL(1,𝔽8)×C22⋊ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
106C22×C3—71no—
107A4—911no—
108A4—6371no—
109A4—6371no—
110C32—25481noSylow
111C7—32no—
112C7ΓL(1,𝔽8)×C22⋊(C13⋊C3)11yescharacteristic, complemented
113C7C3×C22⋊#6327111yescharacteristic, complemented
114C2×C3—211no—
115C22ΓL(1,𝔽8)×#6327111yescharacteristic, complemented, Sylow
116C3—71no—
117C3—3641no—
118C3—25481no—
119C3—25481no—
120C2—31no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC22×C72×C13C32
LCSG ▷ C22×C72×C13
UCSC1

Complements

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

NH ≅ G/Ntype
1. C22×C7×C13×ΓL(1,𝔽8)C3semidirect
2. C22⋊#14274262C3semidirect
3. C7×C22⋊#63271C3semidirect
4. C22⋊(C13⋊#3862)C3semidirect
9. C22×C72×C13C32semidirect
13. C7×C13×ΓL(1,𝔽8)A4semidirect
22. C22⋊#63271ΓL(1,𝔽8)direct (N × H)
24. C22×C13×ΓL(1,𝔽8)ΓL(1,𝔽8)semidirect
27. C72×C13C3×A4semidirect
31. C22×C7×ΓL(1,𝔽8)C13⋊C3semidirect
37. C22×C7×C13C3×ΓL(1,𝔽8)semidirect
38. C22×C7×C13C3×ΓL(1,𝔽8)semidirect
48. C13×ΓL(1,𝔽8)C22⋊ΓL(1,𝔽8)semidirect
51. C22×C72C3×C13⋊C3semidirect
62. C7×ΓL(1,𝔽8)C22⋊(C13⋊C3)semidirect
66. C7×C13A4×ΓL(1,𝔽8)semidirect
67. C7×C13C3×C22⋊ΓL(1,𝔽8)semidirect
76. C22×ΓL(1,𝔽8)#63271semidirect
80. C22×C13ΓL(1,𝔽8)2semidirect
81. C72C3×C22⋊(C13⋊C3)semidirect
90. C22×C7ΓL(1,𝔽8)×C13⋊C3semidirect
91. C22×C7C3×#63271semidirect
101. ΓL(1,𝔽8)C22⋊#63271direct (N × H)
105. C13ΓL(1,𝔽8)×C22⋊ΓL(1,𝔽8)semidirect
112. C7ΓL(1,𝔽8)×C22⋊(C13⋊C3)semidirect
113. C7C3×C22⋊#63271semidirect
115. C22ΓL(1,𝔽8)×#63271semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C13×ΓL(1,𝔽8)3
C22⋊#142742623
C7×C22⋊#632713
C22⋊(C13⋊#3862)3
ΓL(1,𝔽8)×#632714
C3×C22⋊#632717
ΓL(1,𝔽8)×C22⋊(C13⋊C3)7
ΓL(1,𝔽8)×C22⋊ΓL(1,𝔽8)13