Finite Group Databasebeta

← ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊#261) #37524915

Subgroups — ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊#261) #37524915

Order 22932.

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

normal subgroups only

This group has 168 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
1ΓL(1,𝔽8)2×D13C211yesmaximal, characteristic
2ΓL(1,𝔽8)⋊(C7⋊#261)C311yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)⋊(C7⋊#261)C311yesmaximal, characteristic, complemented
4#21439605C311yesmaximal, characteristic, complemented
5#21439601C311yesmaximal, 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⋊#261)C3211yescharacteristic, 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)⋊#261—71no—
27ΓL(1,𝔽8)⋊#261—71no—
28ΓL(1,𝔽8)⋊#261—71no—
29ΓL(1,𝔽8)⋊#261—211no—
30ΓL(1,𝔽8)⋊#261—211no—
31ΓL(1,𝔽8)⋊#261—211no—
32C2×ΓL(1,𝔽8)2—131no—
33C3×C13×ΓL(1,𝔽8)—71no—
34C3×C13×ΓL(1,𝔽8)—71no—
35C72×C13C32×C411yescharacteristic, complemented, Hall
36#10596875—131no—
37#10596874—131no—
38ΓL(1,𝔽8)⋊Dic7—131no—
39ΓL(1,𝔽8)⋊Dic7—131no—
40C3×C7×D13—71no—
41C3×C7×D13—71no—
42ΓL(1,𝔽8)×D13—71no—
43ΓL(1,𝔽8)×D13—71no—
44ΓL(1,𝔽8)×D13—71no—
45ΓL(1,𝔽8)×D13—71no—
46ΓL(1,𝔽8)×D13—211no—
47ΓL(1,𝔽8)×D13—211no—
48ΓL(1,𝔽8)×D13AGL(1,𝔽7)11yescharacteristic
49ΓL(1,𝔽8)×D13AGL(1,𝔽7)11yescharacteristic
50C32×#261—491noHall
51ΓL(1,𝔽8)2#26111yescharacteristic, complemented, Hall
52C7⋊#261—71no—
53C7⋊#261—211no—
54C7⋊#261—211no—
55C7⋊#261—631no—
56C2×#3862—131no—
57C2×#3861—131no—
58C2×C7×ΓL(1,𝔽8)—131no—
59C2×C7×ΓL(1,𝔽8)—131no—
60C3×C7×C13—71no—
61C3×C7×C13—71no—
62C13×ΓL(1,𝔽8)—71no—
63C13×ΓL(1,𝔽8)—71no—
64C13×ΓL(1,𝔽8)—71no—
65C13×ΓL(1,𝔽8)—71no—
66C13×ΓL(1,𝔽8)—211no—
67C13×ΓL(1,𝔽8)—211no—
68C13×ΓL(1,𝔽8)ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
69C13×ΓL(1,𝔽8)ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
70C3×ΓL(1,𝔽8)⋊C4—911no—
71C3×ΓL(1,𝔽8)⋊C4—911no—
72C32×D13—491no—
73C7⋊Dic7—131noHall
74C7×D13—31no—
75C7×D13—31no—
76C7×D13C3×AGL(1,𝔽7)11yescharacteristic
77C7×D13C3×AGL(1,𝔽7)11yescharacteristic
78C3×#261—491no—
79C3×#261—491no—
80C3×#261—491no—
81C3×#261—491no—
82#3862C3×#26111yescharacteristic, complemented
83#3861C3×#26111yescharacteristic, complemented
84C7×ΓL(1,𝔽8)C3×#26111yescharacteristic, complemented
85C7×ΓL(1,𝔽8)C3×#26111yescharacteristic, complemented
86C2×C3×ΓL(1,𝔽8)—911no—
87C2×C3×ΓL(1,𝔽8)—911no—
88C32×C13—491noHall
89C2×C72—131no—
90C7×C13—31no—
91C7×C13—31no—
92C7×C13C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
93C7×C13C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
94C3×Dic7—911no—
95C3×Dic7—911no—
96ΓL(1,𝔽8)⋊C4—2731no—
97ΓL(1,𝔽8)⋊C4—2731no—
98ΓL(1,𝔽8)⋊C4—2731no—
99ΓL(1,𝔽8)⋊C4—2731no—
100ΓL(1,𝔽8)⋊C4—911no—
101ΓL(1,𝔽8)⋊C4—911no—
102ΓL(1,𝔽8)⋊C4—911no—
103ΓL(1,𝔽8)⋊C4—911no—
104C3×D13—71no—
105C3×D13—71no—
106C3×D13—491no—
107C3×D13—491no—
108C3×ΓL(1,𝔽8)—71no—
109C3×ΓL(1,𝔽8)—71no—
110#261—4411noHall
111C72C32×#26111yescharacteristic, complemented, Sylow
112C2×C3×C7—911no—
113C2×C3×C7—911no—
114C2×ΓL(1,𝔽8)—131no—
115C2×ΓL(1,𝔽8)—131no—
116C2×ΓL(1,𝔽8)—2731no—
117C2×ΓL(1,𝔽8)—2731no—
118C2×ΓL(1,𝔽8)—911no—
119C2×ΓL(1,𝔽8)—911no—
120C2×ΓL(1,𝔽8)—911no—
121C2×ΓL(1,𝔽8)—911no—
122C3×C13—71no—
123C3×C13—71no—
124C3×C13—491no—
125C3×C13—491no—
126C32×C4—6371noHall
127Dic7—2731no—
128Dic7—2731no—
129Dic7—2731no—
130Dic7—8191no—
131D13ΓL(1,𝔽8)⋊AGL(1,𝔽7)11yescharacteristic
132C3×C7—71no—
133C3×C7—71no—
134ΓL(1,𝔽8)—71no—
135ΓL(1,𝔽8)—71no—
136ΓL(1,𝔽8)—71no—
137ΓL(1,𝔽8)—71no—
138ΓL(1,𝔽8)—211no—
139ΓL(1,𝔽8)—211no—
140ΓL(1,𝔽8)ΓL(1,𝔽8)⋊#26111yescharacteristic, complemented
141ΓL(1,𝔽8)ΓL(1,𝔽8)⋊#26111yescharacteristic, complemented
142C2×C32—6371no—
143C2×C7—131no—
144C2×C7—131no—
145C2×C7—391no—
146C2×C7—391no—
147C13ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)11yescharacteristic, complemented, Sylow
148C3×C4—19111no—
149C3×C4—19111no—
150C3×C4—6371no—
151C3×C4—6371no—
152C32—491noSylow
153C7—31no—
154C7—31no—
155C7C3×ΓL(1,𝔽8)⋊#26111yescharacteristic, complemented
156C7C3×ΓL(1,𝔽8)⋊#26111yescharacteristic, complemented
157C2×C3—911no—
158C2×C3—911no—
159C2×C3—6371no—
160C2×C3—6371no—
161C4—57331noSylow
162C3—71no—
163C3—71no—
164C3—491no—
165C3—491no—
166C2—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⋊#261)C3semidirect
3. ΓL(1,𝔽8)⋊(C7⋊#261)C3semidirect
4. #21439605C3semidirect
5. #21439601C3semidirect
6. C13×ΓL(1,𝔽8)2C4semidirect
13. C7⋊(C7⋊#261)C32semidirect
14. C13×#3862C3×C4semidirect
15. C13×#3861C3×C4semidirect
16. C7×C13×ΓL(1,𝔽8)C3×C4semidirect
17. C7×C13×ΓL(1,𝔽8)C3×C4semidirect
35. C72×C13C32×C4semidirect
51. ΓL(1,𝔽8)2#261semidirect
68. C13×ΓL(1,𝔽8)ΓL(1,𝔽8)⋊C4semidirect
69. C13×ΓL(1,𝔽8)ΓL(1,𝔽8)⋊C4semidirect
82. #3862C3×#261semidirect
83. #3861C3×#261semidirect
84. C7×ΓL(1,𝔽8)C3×#261semidirect
85. C7×ΓL(1,𝔽8)C3×#261semidirect
92. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
93. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
111. C72C32×#261semidirect
140. ΓL(1,𝔽8)ΓL(1,𝔽8)⋊#261semidirect
141. ΓL(1,𝔽8)ΓL(1,𝔽8)⋊#261semidirect
147. C13ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)semidirect
155. C7C3×ΓL(1,𝔽8)⋊#261semidirect
156. 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⋊#261)3
#214396053
#214396013
C3×ΓL(1,𝔽8)⋊#2617
ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)13