Finite Group Databasebeta

← ΓL(1,𝔽8)×#11763985 #37524889

Subgroups — ΓL(1,𝔽8)×#11763985 #37524889

Order 22932.

152 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 36 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⋊#261C311yesmaximal, characteristic, complemented
3#21439629C311yesmaximal, characteristic, complemented
4#21439623C311yesmaximal, characteristic, complemented
5C7×#11763985C311yesmaximal, 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⋊#261C3211yescharacteristic, 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#11763985ΓL(1,𝔽8)11yescharacteristic, direct product factor
26#11763982—71no—
27#11763981—71no—
28ΓL(1,𝔽8)×#261—211no—
29C7×AGL(1,𝔽13)—71no—
30C2×ΓL(1,𝔽8)2—131no—
31C3×#63270—71no—
32ΓL(1,𝔽8)×C13⋊C3—71no—
33C72×C13C32×C411yescharacteristic, complemented, Hall
34#10596873—131no—
35#10596872—131no—
36ΓL(1,𝔽8)×Dic7—391no—
37C7×ΓL(1,𝔽8)⋊C4—131no—
38C3×C7×D13—71no—
39#10587717—71no—
40#10587716—71no—
41#10587716—71no—
42#10587716—71no—
43#10587716—421no—
44#10587716C2×ΓL(1,𝔽8)11yescharacteristic
45ΓL(1,𝔽8)×D13AGL(1,𝔽7)11yescharacteristic
46C7×D13⋊C3—71no—
47C3×AGL(1,𝔽13)—491noHall
48ΓL(1,𝔽8)2—131noHall
49C7⋊#261C3×ΓL(1,𝔽8)11yescharacteristic, complemented
50C7×#261—211no—
51C2×#3862—131no—
52C2×#3861—131no—
53C2×C7×ΓL(1,𝔽8)—131no—
54C2×C7×ΓL(1,𝔽8)—391no—
55C3×C7×C13—71no—
56#63271—71no—
57#63270—71no—
58#63270—71no—
59#63270—71no—
60#63270—421no—
61#63270C4×ΓL(1,𝔽8)11yescharacteristic, complemented
62C7×C13⋊C3—71no—
63C13×ΓL(1,𝔽8)ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
64C3×C4×ΓL(1,𝔽8)—911no—
65C3×ΓL(1,𝔽8)⋊C4—911no—
66C3×D13⋊C3—491no—
67C7×Dic7—391noHall
68C7×D13—61no—
69C7×D13C3×AGL(1,𝔽7)11yescharacteristic
70C7×D13C2×C3×ΓL(1,𝔽8)11yescharacteristic
71C3×#261—1471no—
72AGL(1,𝔽13)—71no—
73AGL(1,𝔽13)—491no—
74AGL(1,𝔽13)—491no—
75#3862—131no—
76#3861—131no—
77C7×ΓL(1,𝔽8)—131no—
78C7×ΓL(1,𝔽8)AGL(1,𝔽13)11yescharacteristic, complemented
79C2×C3×ΓL(1,𝔽8)—911no—
80C2×C3×ΓL(1,𝔽8)—911no—
81C3×C13⋊C3—491noHall
82C2×C72—391no—
83C7×C13—61no—
84C7×C13C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
85C7×C13C3×C4×ΓL(1,𝔽8)11yescharacteristic, complemented
86C3×C4×C7—911no—
87C3×Dic7—2731no—
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—
95D13⋊C3—71no—
96D13⋊C3—491no—
97D13⋊C3—491no—
98C3×ΓL(1,𝔽8)—911no—
99C3×ΓL(1,𝔽8)—911no—
100#261—211noHall
101C72C3×AGL(1,𝔽13)11yescharacteristic, complemented, Sylow
102C2×C3×C7—2731no—
103C2×C3×C7—911no—
104C2×ΓL(1,𝔽8)—131no—
105C2×ΓL(1,𝔽8)—5461no—
106C2×ΓL(1,𝔽8)—391no—
107C2×ΓL(1,𝔽8)—911no—
108C2×ΓL(1,𝔽8)—911no—
109C2×ΓL(1,𝔽8)—911no—
110C2×ΓL(1,𝔽8)—911no—
111C3×C13—71no—
112C13⋊C3—71no—
113C13⋊C3—491no—
114C13⋊C3—491no—
115C32×C4—6371noHall
116C4×C7—2731no—
117Dic7—391no—
118D13ΓL(1,𝔽8)×AGL(1,𝔽7)11yescharacteristic
119C3×C7—71no—
120C3×C7—911no—
121ΓL(1,𝔽8)—131no—
122ΓL(1,𝔽8)—5461no—
123ΓL(1,𝔽8)—911no—
124ΓL(1,𝔽8)—911no—
125ΓL(1,𝔽8)—911no—
126ΓL(1,𝔽8)—911no—
127ΓL(1,𝔽8)#1176398511yescharacteristic, direct product factor
128C2×C32—6371no—
129C2×C7—391no—
130C2×C7—391no—
131C2×C7—2341no—
132C13ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented, Sylow
133C3×C4—911no—
134C3×C4—19111no—
135C3×C4—6371no—
136C3×C4—6371no—
137C32—6371noSylow
138C7—61no—
139C7ΓL(1,𝔽8)×AGL(1,𝔽13)11yescharacteristic, complemented
140C7C3×#1176398511yescharacteristic, complemented
141C2×C3—2731no—
142C2×C3—911no—
143C2×C3—6371no—
144C2×C3—6371no—
145C4—2731noSylow
146C3—71no—
147C3—911no—
148C3—6371no—
149C3—6371no—
150C2—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⋊#261C3semidirect
3. #21439629C3semidirect
4. #21439623C3semidirect
5. C7×#11763985C3semidirect
6. ΓL(1,𝔽8)×#63270C4semidirect
13. C7×C7⋊#261C32semidirect
14. C13⋊#3862C3×C4semidirect
15. #14274263C3×C4semidirect
16. C7×#63270C3×C4semidirect
17. C7×C13×ΓL(1,𝔽8)C3×C4semidirect
25. #11763985ΓL(1,𝔽8)direct (N × H)
33. C72×C13C32×C4semidirect
49. C7⋊#261C3×ΓL(1,𝔽8)semidirect
61. #63270C4×ΓL(1,𝔽8)semidirect
63. C13×ΓL(1,𝔽8)ΓL(1,𝔽8)⋊C4semidirect
78. C7×ΓL(1,𝔽8)AGL(1,𝔽13)semidirect
84. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
85. C7×C13C3×C4×ΓL(1,𝔽8)semidirect
101. C72C3×AGL(1,𝔽13)semidirect
127. ΓL(1,𝔽8)#11763985direct (N × H)
132. C13ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C4semidirect
139. C7ΓL(1,𝔽8)×AGL(1,𝔽13)semidirect
140. 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⋊#2613
#214396293
#214396233
C7×#117639853
C3×#117639857
ΓL(1,𝔽8)×AGL(1,𝔽13)7
ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C413