Finite Group Databasebeta

← ΓL(1,𝔽8)⋊#11763985 #37524905

Subgroups — ΓL(1,𝔽8)⋊#11763985 #37524905

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)×#10587716C211yesmaximal, characteristic
2ΓL(1,𝔽8)⋊(C7⋊#261)C311yesmaximal, characteristic, complemented
3#21439647C311yesmaximal, characteristic, complemented
4#21439603C311yesmaximal, characteristic, complemented
5#21439597C311yesmaximal, characteristic, complemented
6ΓL(1,𝔽8)×#63270C411yescharacteristic, complemented, Hall
7D13⋊#3862C2×C311yescharacteristic
8#16838473C2×C311yescharacteristic
9C7×ΓL(1,𝔽8)×D13C2×C311yescharacteristic
10C7×#10587716C2×C311yescharacteristic
11C3×#11763985—71nomaximal
12ΓL(1,𝔽8)⋊AGL(1,𝔽13)—71nomaximal
13C7⋊(C7⋊#261)C3211yescharacteristic, complemented, Hall
14C13⋊#3862C3×C411yescharacteristic, complemented
15#14274263C3×C411yescharacteristic, complemented
16C7×#63270C3×C411yescharacteristic, complemented
17C7×C13×ΓL(1,𝔽8)C3×C411yescharacteristic, complemented
18ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)—131nomaximal, Hall
19C3×#10587716—71no—
20ΓL(1,𝔽8)×D13⋊C3—71no—
21C72×D13C2×C3211yescharacteristic
22C3×C7⋊#261—71no—
23#11763985—71no—
24#11763985—71no—
25#11763985—71no—
26#11763985—212no—
27#11763985—211no—
28#11763984—71no—
29ΓL(1,𝔽8)⋊#261—211no—
30#11763979—71no—
31C2×ΓL(1,𝔽8)2—131no—
32C3×#63270—71no—
33ΓL(1,𝔽8)×C13⋊C3—71no—
34C72×C13C32×C411yescharacteristic, complemented, Hall
35#10596875—131no—
36#10596874—131no—
37ΓL(1,𝔽8)⋊Dic7—131no—
38ΓL(1,𝔽8)⋊Dic7—391no—
39C3×C7×D13—71no—
40#10587717—71no—
41#10587716—71no—
42#10587716—71no—
43#10587716—71no—
44#10587716—212no—
45#10587716AGL(1,𝔽7)11yescharacteristic
46ΓL(1,𝔽8)×D13AGL(1,𝔽7)11yescharacteristic
47C7×D13⋊C3—71no—
48C3×AGL(1,𝔽13)—491noHall
49ΓL(1,𝔽8)2—131noHall
50C7⋊#261—211no—
51C7⋊#261—211no—
52C7⋊#261—632no—
53C2×#3862—131no—
54C2×#3861—131no—
55C2×C7×ΓL(1,𝔽8)—131no—
56C2×C7×ΓL(1,𝔽8)—391no—
57C3×C7×C13—71no—
58#63271—71no—
59#63270—71no—
60#63270—71no—
61#63270—71no—
62#63270—212no—
63#63270ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
64C7×C13⋊C3—71no—
65C13×ΓL(1,𝔽8)ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
66C3×ΓL(1,𝔽8)⋊C4—911no—
67C3×ΓL(1,𝔽8)⋊C4—911no—
68C3×D13⋊C3—491no—
69C7⋊Dic7—391noHall
70C7×D13—32no—
71C7×D13C3×AGL(1,𝔽7)11yescharacteristic
72C7×D13C3×AGL(1,𝔽7)11yescharacteristic
73C3×#261—1471no—
74AGL(1,𝔽13)—491no—
75AGL(1,𝔽13)—491no—
76AGL(1,𝔽13)—1471no—
77#3862—131no—
78#3861—131no—
79C7×ΓL(1,𝔽8)—131no—
80C7×ΓL(1,𝔽8)AGL(1,𝔽13)11yescharacteristic, complemented
81C2×C3×ΓL(1,𝔽8)—911no—
82C2×C3×ΓL(1,𝔽8)—911no—
83C3×C13⋊C3—491noHall
84C2×C72—391no—
85C7×C13—32no—
86C7×C13C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
87C7×C13C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
88C3×Dic7—2731no—
89C3×Dic7—911no—
90ΓL(1,𝔽8)⋊C4—2731no—
91ΓL(1,𝔽8)⋊C4—2731no—
92ΓL(1,𝔽8)⋊C4—2732no—
93ΓL(1,𝔽8)⋊C4—911no—
94ΓL(1,𝔽8)⋊C4—911no—
95ΓL(1,𝔽8)⋊C4—911no—
96ΓL(1,𝔽8)⋊C4—911no—
97C3×D13—71no—
98D13⋊C3—71no—
99D13⋊C3—491no—
100D13⋊C3—491no—
101C3×ΓL(1,𝔽8)—911no—
102C3×ΓL(1,𝔽8)—911no—
103#261—4411noHall
104C72C3×AGL(1,𝔽13)11yescharacteristic, complemented, Sylow
105C2×C3×C7—2731no—
106C2×C3×C7—911no—
107C2×ΓL(1,𝔽8)—131no—
108C2×ΓL(1,𝔽8)—2732no—
109C2×ΓL(1,𝔽8)—391no—
110C2×ΓL(1,𝔽8)—911no—
111C2×ΓL(1,𝔽8)—911no—
112C2×ΓL(1,𝔽8)—911no—
113C2×ΓL(1,𝔽8)—911no—
114C3×C13—71no—
115C13⋊C3—71no—
116C13⋊C3—491no—
117C13⋊C3—491no—
118C32×C4—6371noHall
119Dic7—2731no—
120Dic7—2731no—
121Dic7—8192no—
122D13ΓL(1,𝔽8)⋊AGL(1,𝔽7)11yescharacteristic
123C3×C7—71no—
124C3×C7—911no—
125ΓL(1,𝔽8)—131no—
126ΓL(1,𝔽8)—2732no—
127ΓL(1,𝔽8)—911no—
128ΓL(1,𝔽8)—911no—
129ΓL(1,𝔽8)—911no—
130ΓL(1,𝔽8)—911no—
131ΓL(1,𝔽8)#1176398511yescharacteristic, complemented
132C2×C32—6371no—
133C2×C7—391no—
134C2×C7—391no—
135C2×C7—1172no—
136C13ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)11yescharacteristic, complemented, Sylow
137C3×C4—19111no—
138C3×C4—19111no—
139C3×C4—6371no—
140C3×C4—6371no—
141C32—6371noSylow
142C7—32no—
143C7ΓL(1,𝔽8)⋊AGL(1,𝔽13)11yescharacteristic, complemented
144C7C3×#1176398511yescharacteristic, complemented
145C2×C3—2731no—
146C2×C3—911no—
147C2×C3—6371no—
148C2×C3—6371no—
149C4—57331noSylow
150C3—71no—
151C3—911no—
152C3—6371no—
153C3—6371no—
154C2—391no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC72×C13C32×C4
prime radicalΓL(1,𝔽8)×#10587716C2
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. #21439647C3semidirect
4. #21439603C3semidirect
5. #21439597C3semidirect
6. ΓL(1,𝔽8)×#63270C4semidirect
13. C7⋊(C7⋊#261)C32semidirect
14. C13⋊#3862C3×C4semidirect
15. #14274263C3×C4semidirect
16. C7×#63270C3×C4semidirect
17. C7×C13×ΓL(1,𝔽8)C3×C4semidirect
34. C72×C13C32×C4semidirect
63. #63270ΓL(1,𝔽8)⋊C4semidirect
65. C13×ΓL(1,𝔽8)ΓL(1,𝔽8)⋊C4semidirect
80. C7×ΓL(1,𝔽8)AGL(1,𝔽13)semidirect
86. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
87. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
104. C72C3×AGL(1,𝔽13)semidirect
131. ΓL(1,𝔽8)#11763985semidirect
136. C13ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)semidirect
143. C7ΓL(1,𝔽8)⋊AGL(1,𝔽13)semidirect
144. C7C3×#11763985semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#105877162
ΓL(1,𝔽8)⋊(C7⋊#261)3
#214396473
#214396033
#214395973
C3×#117639857
ΓL(1,𝔽8)⋊AGL(1,𝔽13)7
ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)13