Finite Group Databasebeta

← ΓL(1,𝔽8)⋊#11763979 #37524879

Subgroups — ΓL(1,𝔽8)⋊#11763979 #37524879

Order 22932.

168 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 44 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
1C7×ΓL(1,𝔽8)×D13⋊C3C211yesmaximal, characteristic
2#21439664C311yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)⋊(C7⋊#261)C311yesmaximal, characteristic, complemented
4#21439648C311yesmaximal, characteristic, complemented
5#21439647C311yesmaximal, characteristic, complemented
6C7×ΓL(1,𝔽8)×C13⋊C3C411yescharacteristic, complemented, Hall
7C7×ΓL(1,𝔽8)×D13C2×C311yescharacteristic
8C7×#10587717C2×C311yescharacteristic
9C7×#10587716C2×C311yescharacteristic
10C72×D13⋊C3C2×C311yescharacteristic
11C3×#11763979—71nomaximal
12ΓL(1,𝔽8)⋊AGL(1,𝔽13)—71nomaximal
13C7⋊(C7⋊#261)C3211yescharacteristic, complemented, Hall
14C7×#63270C3×C411yescharacteristic, complemented
15C7×#63271C3×C411yescharacteristic, complemented
16C7×C13×ΓL(1,𝔽8)C3×C411yescharacteristic, complemented
17C72×C13⋊C3C3×C411yescharacteristic, complemented
18C3×ΓL(1,𝔽8)⋊Dic7—131nomaximal, Hall
19C3×C7×D13⋊C3—71no—
20ΓL(1,𝔽8)×D13⋊C3D711yescharacteristic
21C72×D13C2×C3211yescharacteristic
22C3×C7⋊#261—71no—
23#11763985—71no—
24#11763984—71no—
25ΓL(1,𝔽8)⋊#261—71no—
26#11763979—71no—
27#11763979—71no—
28#11763979—71no—
29#11763979—212no—
30#11763979—211no—
31C2×C3×C7×ΓL(1,𝔽8)—131no—
32C3×C7×C13⋊C3—71no—
33ΓL(1,𝔽8)×C13⋊C3Dic711yescharacteristic, complemented
34C72×C13C32×C411yescharacteristic, complemented, Hall
35C3×C7⋊Dic7—131no—
36ΓL(1,𝔽8)⋊Dic7—131no—
37ΓL(1,𝔽8)⋊Dic7—131no—
38ΓL(1,𝔽8)⋊Dic7—391no—
39C3×C7×D13—71no—
40#10587717C3×D711yescharacteristic
41#10587716C3×D711yescharacteristic
42ΓL(1,𝔽8)×D13C3×D711yescharacteristic
43C7×D13⋊C3—32no—
44C7×D13⋊C3—71no—
45C7×D13⋊C3—71no—
46C7×D13⋊C3AGL(1,𝔽7)11yescharacteristic
47C7×D13⋊C3C3×D711yescharacteristic
48C3×AGL(1,𝔽13)—491noHall
49C3×C7×ΓL(1,𝔽8)—131noHall
50C7⋊#261—71no—
51C7⋊#261—212no—
52C7⋊#261—211no—
53C2×C3×C72—131no—
54C2×C7×ΓL(1,𝔽8)—131no—
55C2×C7×ΓL(1,𝔽8)—131no—
56C2×C7×ΓL(1,𝔽8)—391no—
57C3×C7×C13—71no—
58#63271C3×Dic711yescharacteristic, complemented
59#63270C3×Dic711yescharacteristic, complemented
60C7×C13⋊C3—32no—
61C7×C13⋊C3—71no—
62C7×C13⋊C3—71no—
63C7×C13⋊C3ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
64C7×C13⋊C3C3×Dic711yescharacteristic, complemented
65C13×ΓL(1,𝔽8)C3×Dic711yescharacteristic, complemented
66C32×Dic7—911no—
67C3×ΓL(1,𝔽8)⋊C4—911no—
68C3×D13⋊C3—71no—
69C7⋊Dic7—391noHall
70C7×D13—32no—
71C7×D13C3×AGL(1,𝔽7)11yescharacteristic
72C7×D13C32×D711yescharacteristic
73C3×#261—491no—
74AGL(1,𝔽13)—491no—
75AGL(1,𝔽13)—491no—
76AGL(1,𝔽13)—1471no—
77C3×C72—131no—
78C7×ΓL(1,𝔽8)—131no—
79C7×ΓL(1,𝔽8)—131no—
80C7×ΓL(1,𝔽8)AGL(1,𝔽13)11yescharacteristic, complemented
81C2×C32×C7—911no—
82C2×C3×ΓL(1,𝔽8)—131no—
83C3×C13⋊C3—71noHall
84C2×C72—391no—
85C7×C13—32no—
86C7×C13C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
87C7×C13C32×Dic711yescharacteristic, complemented
88C3×Dic7—2731no—
89C3×Dic7—2731no—
90C3×Dic7—2732no—
91C3×Dic7—911no—
92C3×Dic7—911no—
93C3×Dic7—911no—
94ΓL(1,𝔽8)⋊C4—2731no—
95ΓL(1,𝔽8)⋊C4—911no—
96ΓL(1,𝔽8)⋊C4—911no—
97C3×D13—71no—
98D13⋊C3—71no—
99D13⋊C3—71no—
100D13⋊C3ΓL(1,𝔽8)⋊D711yescharacteristic
101C32×C7—911no—
102C3×ΓL(1,𝔽8)—131no—
103#261—1471noHall
104C72C3×AGL(1,𝔽13)11yescharacteristic, complemented, Sylow
105C2×C3×C7—131no—
106C2×C3×C7—131no—
107C2×C3×C7—2731no—
108C2×C3×C7—392no—
109C2×C3×C7—911no—
110C2×C3×C7—911no—
111C2×ΓL(1,𝔽8)—131no—
112C2×ΓL(1,𝔽8)—131no—
113C2×ΓL(1,𝔽8)—391no—
114C3×C13—71no—
115C13⋊C3—71no—
116C13⋊C3—71no—
117C13⋊C3ΓL(1,𝔽8)⋊Dic711yescharacteristic, complemented
118C32×C4—6371noHall
119Dic7—2731no—
120Dic7—8191no—
121Dic7—8192no—
122D13C3×ΓL(1,𝔽8)⋊D711yescharacteristic
123C3×C7—71no—
124C3×C7—131no—
125C3×C7—131no—
126C3×C7—392no—
127C3×C7—911no—
128C3×C7—911no—
129ΓL(1,𝔽8)—131no—
130ΓL(1,𝔽8)—131no—
131ΓL(1,𝔽8)#1176397911yescharacteristic, complemented
132C2×C32—911no—
133C2×C7—391no—
134C2×C7—391no—
135C2×C7—1172no—
136C13C3×ΓL(1,𝔽8)⋊Dic711yescharacteristic, complemented, Sylow
137C3×C4—19111no—
138C3×C4—19111no—
139C3×C4—6371no—
140C3×C4—6371no—
141C32—911noSylow
142C7—32no—
143C7ΓL(1,𝔽8)⋊AGL(1,𝔽13)11yescharacteristic, complemented
144C7C3×#1176397911yescharacteristic, complemented
145C2×C3—131no—
146C2×C3—2731no—
147C2×C3—911no—
148C2×C3—911no—
149C4—57331noSylow
150C3—71no—
151C3—131no—
152C3—911no—
153C3—911no—
154C2—391no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC72×C13C32×C4
prime radicalC7×ΓL(1,𝔽8)×D13⋊C3C2
cosocleC72×D13C2×C32
LCSG ▷ C72×C13
UCSC1

Complements

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

NH ≅ G/Ntype
2. #21439664C3semidirect
3. ΓL(1,𝔽8)⋊(C7⋊#261)C3semidirect
4. #21439648C3semidirect
5. #21439647C3semidirect
6. C7×ΓL(1,𝔽8)×C13⋊C3C4semidirect
13. C7⋊(C7⋊#261)C32semidirect
14. C7×#63270C3×C4semidirect
15. C7×#63271C3×C4semidirect
16. C7×C13×ΓL(1,𝔽8)C3×C4semidirect
17. C72×C13⋊C3C3×C4semidirect
33. ΓL(1,𝔽8)×C13⋊C3Dic7semidirect
34. C72×C13C32×C4semidirect
58. #63271C3×Dic7semidirect
59. #63270C3×Dic7semidirect
63. C7×C13⋊C3ΓL(1,𝔽8)⋊C4semidirect
64. C7×C13⋊C3C3×Dic7semidirect
65. C13×ΓL(1,𝔽8)C3×Dic7semidirect
80. C7×ΓL(1,𝔽8)AGL(1,𝔽13)semidirect
86. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
87. C7×C13C32×Dic7semidirect
104. C72C3×AGL(1,𝔽13)semidirect
117. C13⋊C3ΓL(1,𝔽8)⋊Dic7semidirect
131. ΓL(1,𝔽8)#11763979semidirect
136. C13C3×ΓL(1,𝔽8)⋊Dic7semidirect
143. C7ΓL(1,𝔽8)⋊AGL(1,𝔽13)semidirect
144. C7C3×#11763979semidirect

Maximal subgroups

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

subgroupindex
C7×ΓL(1,𝔽8)×D13⋊C32
#214396643
ΓL(1,𝔽8)⋊(C7⋊#261)3
#214396483
#214396473
C3×#117639797
ΓL(1,𝔽8)⋊AGL(1,𝔽13)7
C3×ΓL(1,𝔽8)⋊Dic713