Finite Group Databasebeta

← ΓL(1,𝔽8)×#5450 #72277356

Subgroups — ΓL(1,𝔽8)×#5450 #72277356

Order 4032.

344 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 58 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)×#120⋊C3C213yesmaximal, complemented
2#13060786C312yesmaximal, complemented
3C7×#5450C311yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×H(3,𝔽4)C311yesmaximal, characteristic, complemented
5ΓL(1,𝔽8)×C22⋊A4C2211yescharacteristic, complemented
6C22×A4×ΓL(1,𝔽8)—41nomaximal
7#120⋊ΓL(1,𝔽8)C2×C316yescomplemented
8C7×#120⋊C3C2×C313yescomplemented
9ΓL(1,𝔽8)×#124—33no—
10ΓL(1,𝔽8)×#120—31no—
11ΓL(1,𝔽8)×#120C2×C313yescomplemented
12C3×#5450—71nomaximal, Hall
13C2×A4×ΓL(1,𝔽8)—43no—
14C7×H(3,𝔽4)C3211yescharacteristic, complemented, Hall
15C22⋊(C22⋊ΓL(1,𝔽8))C22×C312yescomplemented
16C7×C22⋊A4C22×C311yescharacteristic, complemented
17C22×C22⋊ΓL(1,𝔽8)—42no—
18C22×C7×A4—41no—
19C24×ΓL(1,𝔽8)A411yescharacteristic, complemented
20C24×ΓL(1,𝔽8)C22×C311yescharacteristic, complemented
21C2×Q8×ΓL(1,𝔽8)—31no—
22C2×D4×ΓL(1,𝔽8)—33no—
23C22⋊C4×ΓL(1,𝔽8)—33no—
24C22⋊C4×ΓL(1,𝔽8)—33no—
25C42×ΓL(1,𝔽8)—31no—
26C3×#120⋊C3—73no—
27C22×C3×ΓL(1,𝔽8)—161no—
28A4×ΓL(1,𝔽8)—41no—
29A4×ΓL(1,𝔽8)—161no—
30C7×#124—33no—
31C7×#120—31no—
32C7×#120C2×C3213yescomplemented
33C3×H(3,𝔽4)—71no—
34#5450—72no—
35#5450ΓL(1,𝔽8)11yescharacteristic, direct product factor
36C2×C22⋊ΓL(1,𝔽8)—46no—
37C2×C7×A4—43no—
38C23×ΓL(1,𝔽8)—31no—
39C23×ΓL(1,𝔽8)—121no—
40C23×ΓL(1,𝔽8)—121no—
41C23×ΓL(1,𝔽8)C2×A413yes—
42Q8×ΓL(1,𝔽8)—121no—
43D4×ΓL(1,𝔽8)—123no—
44C2×C4×ΓL(1,𝔽8)—33no—
45C3×C22⋊A4—71no—
46C22×C3×A4—281no—
47C2×C3×ΓL(1,𝔽8)—163no—
48C24×C7C3×A411yescharacteristic, complemented
49C24×C7C22×C3211yescharacteristic, complemented
50C2×C7×Q8—31no—
51C2×C7×D4—33no—
52C7×C22⋊C4—33no—
53C7×C22⋊C4—33no—
54C42×C7—31no—
55C3×#124—213no—
56C3×#120—73no—
57C3×#120—211no—
58#120⋊C3—76no—
59#120⋊C3C2×ΓL(1,𝔽8)13yescomplemented
60C22×C3×C7—161no—
61C22⋊ΓL(1,𝔽8)—42no—
62C22⋊ΓL(1,𝔽8)—162no—
63C7×A4—41no—
64C7×A4—161no—
65C22×ΓL(1,𝔽8)—41no—
66C22×ΓL(1,𝔽8)—41no—
67C22×ΓL(1,𝔽8)—63no—
68C22×ΓL(1,𝔽8)—63no—
69C22×ΓL(1,𝔽8)—121no—
70C22×ΓL(1,𝔽8)—121no—
71C22×ΓL(1,𝔽8)—162no—
72C22×ΓL(1,𝔽8)C22×A411yescharacteristic
73C4×ΓL(1,𝔽8)—63no—
74C2×C3×A4—283no—
75H(3,𝔽4)C3×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
76C3×ΓL(1,𝔽8)—161noHall
77C23×C7—31no—
78C23×C7—121no—
79C23×C7—121no—
80C23×C7C2×C3×A413yes—
81C7×Q8—121no—
82C7×D4—123no—
83C2×C4×C7—33no—
84C24×C3—71no—
85C24×C3—71no—
86C22⋊A4—72no—
87C22⋊A4C22×ΓL(1,𝔽8)11yescharacteristic, complemented
88C22×A4—41no—
89C22×A4—282no—
90C2×C3×Q8—211no—
91C2×C3×D4—213no—
92C3×C22⋊C4—213no—
93C3×C22⋊C4—213no—
94C3×C42—211no—
95C2×C3×C7—163no—
96C2×ΓL(1,𝔽8)—31no—
97C2×ΓL(1,𝔽8)—43no—
98C2×ΓL(1,𝔽8)—121no—
99C2×ΓL(1,𝔽8)—166no—
100C22×C32—1121no—
101C3×A4—281no—
102C3×A4—1121no—
103#124—33no—
104#120—31no—
105#120C2×C3×ΓL(1,𝔽8)13yescomplemented
106C22×C7—41no—
107C22×C7—41no—
108C22×C7—63no—
109C22×C7—63no—
110C22×C7—121no—
111C22×C7—121no—
112C22×C7C22×C3×A411yescharacteristic
113C4×C7—63no—
114C23×C3—73no—
115C23×C3—211no—
116C23×C3—841no—
117C23×C3—841no—
118C2×A4—43no—
119C2×A4—286no—
120C3×Q8—841no—
121C3×D4—843no—
122C2×C3×C4—213no—
123C3×C7—161no—
124ΓL(1,𝔽8)—162no—
125ΓL(1,𝔽8)#545011yescharacteristic, direct product factor
126C2×C32—1123no—
127C24A4×ΓL(1,𝔽8)11yescharacteristic, complemented
128C24C22×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
129C2×Q8—31no—
130C2×D4—33no—
131C22⋊C4—33no—
132C22⋊C4—33no—
133C42—31no—
134C2×C7—31no—
135C2×C7—43no—
136C2×C7—121no—
137C22×C3—71no—
138C22×C3—161no—
139C22×C3—281no—
140C22×C3—281no—
141C22×C3—423no—
142C22×C3—423no—
143C22×C3—841no—
144C22×C3—841no—
145C22×C3—1122no—
146A4—41no—
147A4—161no—
148A4—282no—
149A4—1122no—
150C3×C4—423no—
151C32—1121noSylow
152C23—31no—
153C23—121no—
154C23—121no—
155C23C2×A4×ΓL(1,𝔽8)13yes—
156Q8—121no—
157D4—123no—
158C2×C4—33no—
159C7C3×#545011yescharacteristic, complemented, Sylow
160C2×C3—163no—
161C2×C3—211no—
162C2×C3—283no—
163C2×C3—841no—
164C2×C3—1126no—
165C22—41no—
166C22—41no—
167C22—63no—
168C22—63no—
169C22—121no—
170C22—121no—
171C22C22×A4×ΓL(1,𝔽8)11yescharacteristic
172C4—63no—
173C3—71no—
174C3—161no—
175C3—1122no—
176C2—31no—
177C2—43no—
178C2—121no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC24×C7C22×C32
FrattiniC22C22×A4×ΓL(1,𝔽8)
FittingC7×H(3,𝔽4)C32
socleC22×C7C22×C3×A4
LCSG ▷ C24×C7
UCSC1

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×#120⋊C3C2semidirect
2. #13060786C3semidirect
3. C7×#5450C3semidirect
4. ΓL(1,𝔽8)×H(3,𝔽4)C3semidirect
5. ΓL(1,𝔽8)×C22⋊A4C22semidirect
7. #120⋊ΓL(1,𝔽8)C2×C3semidirect
8. C7×#120⋊C3C2×C3semidirect
11. ΓL(1,𝔽8)×#120C2×C3semidirect
14. C7×H(3,𝔽4)C32semidirect
15. C22⋊(C22⋊ΓL(1,𝔽8))C22×C3semidirect
16. C7×C22⋊A4C22×C3semidirect
19. C24×ΓL(1,𝔽8)A4semidirect
20. C24×ΓL(1,𝔽8)C22×C3semidirect
32. C7×#120C2×C32semidirect
35. #5450ΓL(1,𝔽8)direct (N × H)
48. C24×C7C3×A4semidirect
49. C24×C7C22×C32semidirect
59. #120⋊C3C2×ΓL(1,𝔽8)semidirect
75. H(3,𝔽4)C3×ΓL(1,𝔽8)semidirect
87. C22⋊A4C22×ΓL(1,𝔽8)semidirect
105. #120C2×C3×ΓL(1,𝔽8)semidirect
125. ΓL(1,𝔽8)#5450direct (N × H)
127. C24A4×ΓL(1,𝔽8)semidirect
128. C24C22×C3×ΓL(1,𝔽8)semidirect
159. C7C3×#5450semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#120⋊C32
#130607863
C7×#54503
ΓL(1,𝔽8)×H(3,𝔽4)3
C22×A4×ΓL(1,𝔽8)4
C3×#54507