Finite Group Databasebeta

← #13057651

Subgroups — #13057651

Order 1344.

134 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 21 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
1#127⋊ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2#10620799C211yesmaximal, characteristic, complemented
3#10620798C211yesmaximal, characteristic, complemented
4#90411C311yesmaximal, characteristic, complemented, Hall
5C2×D7⋊A4—41nomaximal
6C7⋊(C42⋊C3)C2211yescharacteristic, complemented
7C7×#127C2×C311yescharacteristic, complemented
8#6502—31no—
9#6502—31no—
10#6500—31no—
11#6498—31no—
12#6439C2×C311yescharacteristic, complemented
13#6439C2×C311yescharacteristic, complemented
14#560⋊C3—71nomaximal, Hall
15C2×C22⋊ΓL(1,𝔽8)—41no—
16D7⋊A4—41no—
17D7⋊A4—41no—
18C23×D7A411yescharacteristic
19C2×C7×D4—31no—
20C2×C7×D4—31no—
21C2×C22⋊D7—31no—
22C2×C22⋊D7—31no—
23C22×Dic7—31no—
24C42×C7C22×C311yescharacteristic, complemented
25C22⋊Dic7—31no—
26C22⋊Dic7—31no—
27(C2×D7)⋊C4—31no—
28(C2×D7)⋊C4—31no—
29Dic7⋊C4—31no—
30Dic7⋊C4—31no—
31#127⋊C3—71no—
32#126⋊C3—71no—
33#126⋊C3—71no—
34C22⋊ΓL(1,𝔽8)—41no—
35C2×AGL(1,𝔽7)—161no—
36#560—71noSylow
37C23×C7—31no—
38C23×C7C2×A411yescharacteristic
39C22×D7—121no—
40C22×D7C2×A411yescharacteristic
41C22×D7C2×A411yescharacteristic
42C7×D4—121no—
43C7×D4—121no—
44C2×C4×C7—31no—
45C22⋊D7—121no—
46C22⋊D7—121no—
47C2×Dic7—31no—
48C2×Dic7—31no—
49C2×Dic7—121no—
50C22×A4—281no—
51C42⋊C3—71no—
52C2×ΓL(1,𝔽8)—161no—
53AGL(1,𝔽7)—161no—
54AGL(1,𝔽7)—161no—
55#127AGL(1,𝔽7)11yescharacteristic, complemented
56#126—71no—
57#126—71no—
58#123—211no—
59#121—211no—
60#121—211no—
61#120—211no—
62C22×C7—61no—
63C22×C7—61no—
64C22×C7—61no—
65C22×C7—61no—
66C22×C7C22×A411yescharacteristic
67C2×D7—41no—
68C2×D7—61no—
69C2×D7—61no—
70C2×D7—121no—
71C4×C7—61no—
72Dic7—61no—
73Dic7—61no—
74C2×A4—281no—
75C2×A4—281no—
76C2×A4—281no—
77ΓL(1,𝔽8)—161noHall
78C24—71no—
79C2×D4—31no—
80C2×D4—31no—
81C2×D4—211no—
82C2×D4—211no—
83C22×C4—211no—
84C4⋊C4—211no—
85C4⋊C4—211no—
86C22⋊C4—211no—
87C22⋊C4—211no—
88C22⋊C4—211no—
89C22⋊C4—211no—
90C42C2×AGL(1,𝔽7)11yescharacteristic, complemented
91C2×C7—31no—
92C2×C7—41no—
93C2×C7—121no—
94D7—41no—
95D7—41no—
96C22×C3—1121no—
97A4—281no—
98C23—31no—
99C23—71no—
100C23—71no—
101C23—841no—
102C23D7⋊A411yescharacteristic
103D4—121no—
104D4—121no—
105D4—841no—
106D4—841no—
107C2×C4—31no—
108C2×C4—211no—
109C2×C4—211no—
110C2×C4—841no—
111C7#560⋊C311yescharacteristic, complemented, Sylow
112C2×C3—1121no—
113C2×C3—1121no—
114C2×C3—1121no—
115C22—61no—
116C22—61no—
117C22—61no—
118C22—61no—
119C22—281no—
120C22—421no—
121C22—421no—
122C22—841no—
123C22C2×D7⋊A411yescharacteristic
124C4—61no—
125C4—421no—
126C4—421no—
127C3—1121noSylow
128C2—31no—
129C2—41no—
130C2—121no—
131C2—281no—
132C2—281no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC42×C7C22×C3
FrattiniC22C2×D7⋊A4
FittingC7×#127C2×C3
socleC22×C7C22×A4
LCSG ▷ C42×C7
UCSC1

Complements

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

NH ≅ G/Ntype
1. #127⋊ΓL(1,𝔽8)C2semidirect
2. #10620799C2semidirect
3. #10620798C2semidirect
4. #90411C3semidirect
6. C7⋊(C42⋊C3)C22semidirect
7. C7×#127C2×C3semidirect
12. #6439C2×C3semidirect
13. #6439C2×C3semidirect
24. C42×C7C22×C3semidirect
55. #127AGL(1,𝔽7)semidirect
90. C42C2×AGL(1,𝔽7)semidirect
111. C7#560⋊C3semidirect

Maximal subgroups

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

subgroupindex
#127⋊ΓL(1,𝔽8)2
#106207992
#106207982
#904113
C2×D7⋊A44
#560⋊C37