Finite Group Databasebeta

← ΓL(1,𝔽8)⋊#11763984 #37524906

Subgroups — ΓL(1,𝔽8)⋊#11763984 #37524906

Order 22932.

168 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 34 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)×#10587717C211yesmaximal, characteristic
2ΓL(1,𝔽8)⋊(C7⋊#261)C311yesmaximal, characteristic, complemented
3#21439648C311yesmaximal, characteristic, complemented
4#21439603C311yesmaximal, characteristic, complemented
5#21439598C311yesmaximal, characteristic, complemented
6ΓL(1,𝔽8)×#63271C411yescharacteristic, complemented, Hall
7D13⋊#3862C2×C311yescharacteristic
8#16838474C2×C311yescharacteristic
9C7×ΓL(1,𝔽8)×D13C2×C311yescharacteristic
10C7×#10587717C2×C311yescharacteristic
11C3×#11763984—71nomaximal
12ΓL(1,𝔽8)⋊AGL(1,𝔽13)—71nomaximal
13C7⋊(C7⋊#261)C3211yescharacteristic, complemented, Hall
14C13⋊#3862C3×C411yescharacteristic, complemented
15#14274262C3×C411yescharacteristic, complemented
16C7×#63271C3×C411yescharacteristic, complemented
17C7×C13×ΓL(1,𝔽8)C3×C411yescharacteristic, complemented
18ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)—131nomaximal, Hall
19C3×#10587717—71no—
20ΓL(1,𝔽8)×D13⋊C3—71no—
21C72×D13C2×C3211yescharacteristic
22C3×C7⋊#261—71no—
23#11763985—71no—
24#11763984—71no—
25#11763984—71no—
26#11763984—71no—
27#11763984—211no—
28#11763984—211no—
29#11763984—211no—
30ΓL(1,𝔽8)⋊#261—211no—
31#11763979—71no—
32C2×ΓL(1,𝔽8)2—131no—
33C3×#63271—71no—
34ΓL(1,𝔽8)×C13⋊C3—71no—
35C72×C13C32×C411yescharacteristic, complemented, Hall
36#10596875—131no—
37#10596874—131no—
38ΓL(1,𝔽8)⋊Dic7—131no—
39ΓL(1,𝔽8)⋊Dic7—391no—
40C3×C7×D13—71no—
41#10587717—71no—
42#10587717—71no—
43#10587717—71no—
44#10587717—211no—
45#10587717—211no—
46#10587717AGL(1,𝔽7)11yescharacteristic
47#10587716—71no—
48ΓL(1,𝔽8)×D13AGL(1,𝔽7)11yescharacteristic
49C7×D13⋊C3—71no—
50C3×AGL(1,𝔽13)—491noHall
51ΓL(1,𝔽8)2—131noHall
52C7⋊#261—211no—
53C7⋊#261—211no—
54C7⋊#261—631no—
55C7⋊#261—631no—
56C2×#3862—131no—
57C2×#3861—131no—
58C2×C7×ΓL(1,𝔽8)—131no—
59C2×C7×ΓL(1,𝔽8)—391no—
60C3×C7×C13—71no—
61#63271—71no—
62#63271—71no—
63#63271—71no—
64#63271—211no—
65#63271—211no—
66#63271ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
67#63270—71no—
68C7×C13⋊C3—71no—
69C13×ΓL(1,𝔽8)ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
70C3×ΓL(1,𝔽8)⋊C4—911no—
71C3×ΓL(1,𝔽8)⋊C4—911no—
72C3×D13⋊C3—491no—
73C7⋊Dic7—391noHall
74C7×D13—31no—
75C7×D13—31no—
76C7×D13C3×AGL(1,𝔽7)11yescharacteristic
77C7×D13C3×AGL(1,𝔽7)11yescharacteristic
78C3×#261—1471no—
79AGL(1,𝔽13)—491no—
80AGL(1,𝔽13)—491no—
81AGL(1,𝔽13)—1471no—
82#3862—131no—
83#3861—131no—
84C7×ΓL(1,𝔽8)—131no—
85C7×ΓL(1,𝔽8)AGL(1,𝔽13)11yescharacteristic, complemented
86C2×C3×ΓL(1,𝔽8)—911no—
87C2×C3×ΓL(1,𝔽8)—911no—
88C3×C13⋊C3—491noHall
89C2×C72—391no—
90C7×C13—31no—
91C7×C13—31no—
92C7×C13C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
93C7×C13C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
94C3×Dic7—2731no—
95C3×Dic7—911no—
96ΓL(1,𝔽8)⋊C4—2731no—
97ΓL(1,𝔽8)⋊C4—2731no—
98ΓL(1,𝔽8)⋊C4—2731no—
99ΓL(1,𝔽8)⋊C4—911no—
100ΓL(1,𝔽8)⋊C4—911no—
101ΓL(1,𝔽8)⋊C4—911no—
102ΓL(1,𝔽8)⋊C4—911no—
103ΓL(1,𝔽8)⋊C4—911no—
104C3×D13—71no—
105D13⋊C3—71no—
106D13⋊C3—491no—
107D13⋊C3—491no—
108C3×ΓL(1,𝔽8)—911no—
109C3×ΓL(1,𝔽8)—911no—
110#261—4411noHall
111C72C3×AGL(1,𝔽13)11yescharacteristic, complemented, Sylow
112C2×C3×C7—2731no—
113C2×C3×C7—911no—
114C2×ΓL(1,𝔽8)—131no—
115C2×ΓL(1,𝔽8)—2731no—
116C2×ΓL(1,𝔽8)—2731no—
117C2×ΓL(1,𝔽8)—391no—
118C2×ΓL(1,𝔽8)—911no—
119C2×ΓL(1,𝔽8)—911no—
120C2×ΓL(1,𝔽8)—911no—
121C2×ΓL(1,𝔽8)—911no—
122C3×C13—71no—
123C13⋊C3—71no—
124C13⋊C3—491no—
125C13⋊C3—491no—
126C32×C4—6371noHall
127Dic7—2731no—
128Dic7—8191no—
129Dic7—8191no—
130Dic7—8191no—
131D13ΓL(1,𝔽8)⋊AGL(1,𝔽7)11yescharacteristic
132C3×C7—71no—
133C3×C7—911no—
134ΓL(1,𝔽8)—131no—
135ΓL(1,𝔽8)—2731no—
136ΓL(1,𝔽8)—2731no—
137ΓL(1,𝔽8)—911no—
138ΓL(1,𝔽8)—911no—
139ΓL(1,𝔽8)—911no—
140ΓL(1,𝔽8)—911no—
141ΓL(1,𝔽8)#1176398411yescharacteristic, complemented
142C2×C32—6371no—
143C2×C7—391no—
144C2×C7—391no—
145C2×C7—391no—
146C2×C7—1171no—
147C13ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)11yescharacteristic, complemented, Sylow
148C3×C4—19111no—
149C3×C4—19111no—
150C3×C4—6371no—
151C3×C4—6371no—
152C32—6371noSylow
153C7—31no—
154C7—31no—
155C7ΓL(1,𝔽8)⋊AGL(1,𝔽13)11yescharacteristic, complemented
156C7C3×#1176398411yescharacteristic, complemented
157C2×C3—2731no—
158C2×C3—911no—
159C2×C3—6371no—
160C2×C3—6371no—
161C4—57331noSylow
162C3—71no—
163C3—911no—
164C3—6371no—
165C3—6371no—
166C2—391no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC72×C13C32×C4
prime radicalΓL(1,𝔽8)×#10587717C2
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. #21439648C3semidirect
4. #21439603C3semidirect
5. #21439598C3semidirect
6. ΓL(1,𝔽8)×#63271C4semidirect
13. C7⋊(C7⋊#261)C32semidirect
14. C13⋊#3862C3×C4semidirect
15. #14274262C3×C4semidirect
16. C7×#63271C3×C4semidirect
17. C7×C13×ΓL(1,𝔽8)C3×C4semidirect
35. C72×C13C32×C4semidirect
66. #63271ΓL(1,𝔽8)⋊C4semidirect
69. C13×ΓL(1,𝔽8)ΓL(1,𝔽8)⋊C4semidirect
85. C7×ΓL(1,𝔽8)AGL(1,𝔽13)semidirect
92. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
93. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
111. C72C3×AGL(1,𝔽13)semidirect
141. ΓL(1,𝔽8)#11763984semidirect
147. C13ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)semidirect
155. C7ΓL(1,𝔽8)⋊AGL(1,𝔽13)semidirect
156. C7C3×#11763984semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#105877172
ΓL(1,𝔽8)⋊(C7⋊#261)3
#214396483
#214396033
#214395983
C3×#117639847
ΓL(1,𝔽8)⋊AGL(1,𝔽13)7
ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)13