Finite Group Databasebeta

← ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊#261 #37524903

Subgroups — ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊#261 #37524903

Order 22932.

152 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 41 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
1ΓL(1,𝔽8)2×D13C211yesmaximal, characteristic
2ΓL(1,𝔽8)×C7⋊#261C311yesmaximal, characteristic, complemented
3#21439633C311yesmaximal, characteristic, complemented
4#21439627C311yesmaximal, characteristic, complemented
5C7×ΓL(1,𝔽8)⋊#261C311yesmaximal, characteristic, complemented
6C13×ΓL(1,𝔽8)2C411yescharacteristic, complemented, Hall
7D13×#3862C2×C311yescharacteristic
8D13×#3861C2×C311yescharacteristic
9C7×ΓL(1,𝔽8)×D13C2×C311yescharacteristic
10C7×ΓL(1,𝔽8)×D13C2×C311yescharacteristic
11C3×ΓL(1,𝔽8)×#261—71nomaximal
12C3×ΓL(1,𝔽8)⋊#261—71nomaximal
13C7×C7⋊#261C3211yescharacteristic, complemented, Hall
14C13×#3862C3×C411yescharacteristic, complemented
15C13×#3861C3×C411yescharacteristic, complemented
16C7×C13×ΓL(1,𝔽8)C3×C411yescharacteristic, complemented
17C7×C13×ΓL(1,𝔽8)C3×C411yescharacteristic, complemented
18ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C4—131nomaximal, Hall
19C3×ΓL(1,𝔽8)×D13—71no—
20C3×ΓL(1,𝔽8)×D13—71no—
21C72×D13C2×C3211yescharacteristic
22C3×C7⋊#261—71no—
23C3×C7×#261—71no—
24ΓL(1,𝔽8)⋊#261—71no—
25ΓL(1,𝔽8)⋊#261—71no—
26ΓL(1,𝔽8)⋊#261ΓL(1,𝔽8)11yescharacteristic, direct product factor
27ΓL(1,𝔽8)×#261—71no—
28ΓL(1,𝔽8)×#261—71no—
29ΓL(1,𝔽8)×#261—71no—
30C2×ΓL(1,𝔽8)2—131no—
31C3×C13×ΓL(1,𝔽8)—71no—
32C3×C13×ΓL(1,𝔽8)—71no—
33C72×C13C32×C411yescharacteristic, complemented, Hall
34#10596873—131no—
35#10596872—131no—
36ΓL(1,𝔽8)×Dic7—131no—
37C7×ΓL(1,𝔽8)⋊C4—131no—
38C3×C7×D13—71no—
39C3×C7×D13—71no—
40ΓL(1,𝔽8)×D13—71no—
41ΓL(1,𝔽8)×D13—71no—
42ΓL(1,𝔽8)×D13—71no—
43ΓL(1,𝔽8)×D13—71no—
44ΓL(1,𝔽8)×D13—421no—
45ΓL(1,𝔽8)×D13AGL(1,𝔽7)11yescharacteristic
46ΓL(1,𝔽8)×D13C2×ΓL(1,𝔽8)11yescharacteristic
47C32×#261—491noHall
48ΓL(1,𝔽8)2#26111yescharacteristic, complemented, Hall
49C7⋊#261C3×ΓL(1,𝔽8)11yescharacteristic, complemented
50C7×#261—71no—
51C2×#3862—131no—
52C2×#3861—131no—
53C2×C7×ΓL(1,𝔽8)—131no—
54C2×C7×ΓL(1,𝔽8)—131no—
55C3×C7×C13—71no—
56C3×C7×C13—71no—
57C13×ΓL(1,𝔽8)—71no—
58C13×ΓL(1,𝔽8)—71no—
59C13×ΓL(1,𝔽8)—71no—
60C13×ΓL(1,𝔽8)—71no—
61C13×ΓL(1,𝔽8)—421no—
62C13×ΓL(1,𝔽8)ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
63C13×ΓL(1,𝔽8)C4×ΓL(1,𝔽8)11yescharacteristic, complemented
64C3×C4×ΓL(1,𝔽8)—911no—
65C3×ΓL(1,𝔽8)⋊C4—911no—
66C32×D13—491no—
67C7×Dic7—131noHall
68C7×D13—61no—
69C7×D13C3×AGL(1,𝔽7)11yescharacteristic
70C7×D13C2×C3×ΓL(1,𝔽8)11yescharacteristic
71C3×#261—71no—
72C3×#261—491no—
73C3×#261—491no—
74C3×#261—1471no—
75#3862C3×#26111yescharacteristic, complemented
76#3861C3×#26111yescharacteristic, complemented
77C7×ΓL(1,𝔽8)C3×#26111yescharacteristic, complemented
78C7×ΓL(1,𝔽8)C3×#26111yescharacteristic, complemented
79C2×C3×ΓL(1,𝔽8)—911no—
80C2×C3×ΓL(1,𝔽8)—911no—
81C32×C13—491noHall
82C2×C72—131no—
83C7×C13—61no—
84C7×C13C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
85C7×C13C3×C4×ΓL(1,𝔽8)11yescharacteristic, complemented
86C3×C4×C7—911no—
87C3×Dic7—911no—
88C4×ΓL(1,𝔽8)—2731no—
89C4×ΓL(1,𝔽8)—911no—
90C4×ΓL(1,𝔽8)—911no—
91ΓL(1,𝔽8)⋊C4—131no—
92ΓL(1,𝔽8)⋊C4—911no—
93ΓL(1,𝔽8)⋊C4—911no—
94C3×D13—71no—
95C3×D13—71no—
96C3×D13—491no—
97C3×D13—491no—
98C3×ΓL(1,𝔽8)—71no—
99C3×ΓL(1,𝔽8)—71no—
100#261—211noHall
101C72C32×#26111yescharacteristic, complemented, Sylow
102C2×C3×C7—911no—
103C2×C3×C7—911no—
104C2×ΓL(1,𝔽8)—131no—
105C2×ΓL(1,𝔽8)—131no—
106C2×ΓL(1,𝔽8)—5461no—
107C2×ΓL(1,𝔽8)—911no—
108C2×ΓL(1,𝔽8)—911no—
109C2×ΓL(1,𝔽8)—911no—
110C2×ΓL(1,𝔽8)—911no—
111C3×C13—71no—
112C3×C13—71no—
113C3×C13—491no—
114C3×C13—491no—
115C32×C4—6371noHall
116C4×C7—2731no—
117Dic7—131no—
118D13ΓL(1,𝔽8)×AGL(1,𝔽7)11yescharacteristic
119C3×C7—71no—
120C3×C7—71no—
121ΓL(1,𝔽8)—71no—
122ΓL(1,𝔽8)—71no—
123ΓL(1,𝔽8)—71no—
124ΓL(1,𝔽8)—71no—
125ΓL(1,𝔽8)—421no—
126ΓL(1,𝔽8)ΓL(1,𝔽8)×#26111yescharacteristic, complemented
127ΓL(1,𝔽8)ΓL(1,𝔽8)⋊#26111yescharacteristic, direct product factor
128C2×C32—6371no—
129C2×C7—131no—
130C2×C7—131no—
131C2×C7—781no—
132C13ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented, Sylow
133C3×C4—911no—
134C3×C4—19111no—
135C3×C4—6371no—
136C3×C4—6371no—
137C32—491noSylow
138C7—61no—
139C7C3×ΓL(1,𝔽8)⋊#26111yescharacteristic, complemented
140C7C3×ΓL(1,𝔽8)×#26111yescharacteristic, complemented
141C2×C3—911no—
142C2×C3—911no—
143C2×C3—6371no—
144C2×C3—6371no—
145C4—2731noSylow
146C3—71no—
147C3—71no—
148C3—491no—
149C3—491no—
150C2—131no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC72×C13C32×C4
prime radicalΓL(1,𝔽8)2×D13C2
cosocleC72×D13C2×C32
LCSG ▷ C72×C13
UCSC1

Complements

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

NH ≅ G/Ntype
2. ΓL(1,𝔽8)×C7⋊#261C3semidirect
3. #21439633C3semidirect
4. #21439627C3semidirect
5. C7×ΓL(1,𝔽8)⋊#261C3semidirect
6. C13×ΓL(1,𝔽8)2C4semidirect
13. C7×C7⋊#261C32semidirect
14. C13×#3862C3×C4semidirect
15. C13×#3861C3×C4semidirect
16. C7×C13×ΓL(1,𝔽8)C3×C4semidirect
17. C7×C13×ΓL(1,𝔽8)C3×C4semidirect
26. ΓL(1,𝔽8)⋊#261ΓL(1,𝔽8)direct (N × H)
33. C72×C13C32×C4semidirect
48. ΓL(1,𝔽8)2#261semidirect
49. C7⋊#261C3×ΓL(1,𝔽8)semidirect
62. C13×ΓL(1,𝔽8)ΓL(1,𝔽8)⋊C4semidirect
63. C13×ΓL(1,𝔽8)C4×ΓL(1,𝔽8)semidirect
75. #3862C3×#261semidirect
76. #3861C3×#261semidirect
77. C7×ΓL(1,𝔽8)C3×#261semidirect
78. C7×ΓL(1,𝔽8)C3×#261semidirect
84. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
85. C7×C13C3×C4×ΓL(1,𝔽8)semidirect
101. C72C32×#261semidirect
126. ΓL(1,𝔽8)ΓL(1,𝔽8)×#261semidirect
127. ΓL(1,𝔽8)ΓL(1,𝔽8)⋊#261direct (N × H)
132. C13ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C4semidirect
139. C7C3×ΓL(1,𝔽8)⋊#261semidirect
140. C7C3×ΓL(1,𝔽8)×#261semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)2×D132
ΓL(1,𝔽8)×C7⋊#2613
#214396333
#214396273
C7×ΓL(1,𝔽8)⋊#2613
C3×ΓL(1,𝔽8)×#2617
C3×ΓL(1,𝔽8)⋊#2617
ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C413