Finite Group Databasebeta

← #13057677

Subgroups — #13057677

Order 1344.

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

normal subgroups only

This group has 146 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
1C2×#41⋊ΓL(1,𝔽8)C211yesmaximal, characteristic
2#10620803C212yesmaximal, complemented
3#90510C311yesmaximal, characteristic, complemented, Hall
4#41⋊ΓL(1,𝔽8)C411yescharacteristic
5#41⋊ΓL(1,𝔽8)C2211yescharacteristic
6C2×C7⋊SL(2,𝔽3)C411yescharacteristic
7#1136⋊C3—41nomaximal
8C2×C7×#41C2×C311yescharacteristic
9#6511—32no—
10#6511C2×C312yescomplemented
11C2×#1136—31no—
12C2×#1136—31no—
13#568⋊C3—71nomaximal, Hall
14C7⋊SL(2,𝔽3)C2×C411yescharacteristic
15C2×C4×ΓL(1,𝔽8)—41no—
16ΓL(1,𝔽8)⋊C8—42no—
17C7×#41—31no—
18C7×#41—31no—
19C7×#41C3×C411yescharacteristic
20C7×#41C22×C311yescharacteristic
21C2×C7×Q8C3×C411yescharacteristic
22C2×C7×D4—31no—
23C22×C4×C7—31no—
24#1136—31no—
25#1136—32no—
26#1136—32no—
27#1136A411yescharacteristic
28C2×C7⋊C8—32no—
29C2×C7⋊C8—32no—
30C2×#41⋊C3—71no—
31#131⋊C3—72no—
32C22×ΓL(1,𝔽8)—41no—
33C4×ΓL(1,𝔽8)—41no—
34C4×ΓL(1,𝔽8)—41no—
35#568—71noSylow
36C23×C7—31no—
37C7×Q8—31no—
38C7×Q8C2×C3×C411yescharacteristic
39C7×D4—31no—
40C7×D4—31no—
41C7×D4—32no—
42C2×C4×C7—31no—
43C2×C4×C7—31no—
44C2×C4×C7—31no—
45C2×C4×C7—31no—
46C2×C4×C7—31no—
47C2×C4×C7C2×A411yescharacteristic
48C7⋊C8—32no—
49C7⋊C8C2×A412yes—
50#41⋊C3—71no—
51#41⋊C3—71no—
52C2×SL(2,𝔽3)—71no—
53C3×M16—281no—
54C2×ΓL(1,𝔽8)—41no—
55C2×ΓL(1,𝔽8)—81no—
56C2×#41AGL(1,𝔽7)11yescharacteristic
57#131—72no—
58#131—212no—
59C2×M16—211no—
60C2×M16—211no—
61C22×C7—31no—
62C22×C7—31no—
63C22×C7—121no—
64C22×C7C4×A411yescharacteristic
65C4×C7—31no—
66C4×C7—31no—
67C4×C7C4×A411yescharacteristic
68C4×C7C22×A411yescharacteristic
69C2×C3×C4—281no—
70SL(2,𝔽3)—71no—
71C3×C8—282no—
72ΓL(1,𝔽8)—41noHall
73#41—31no—
74#41—31no—
75#41ΓL(1,𝔽8)⋊C411yescharacteristic
76#41C2×AGL(1,𝔽7)11yescharacteristic
77C2×Q8ΓL(1,𝔽8)⋊C411yescharacteristic
78C2×D4—31no—
79C22×C4—31no—
80M16—71no—
81M16—212no—
82M16—211no—
83M16—212no—
84C2×C8—212no—
85C2×C8—212no—
86C2×C7—21no—
87C2×C7—61no—
88C2×C7—61no—
89C2×C7C2×C4×A411yescharacteristic
90C22×C3—281no—
91C3×C4—281no—
92C3×C4—281no—
93C23—31no—
94Q8—31no—
95Q8C2×ΓL(1,𝔽8)⋊C411yescharacteristic
96D4—31no—
97D4—31no—
98D4—32no—
99C2×C4—31no—
100C2×C4—31no—
101C2×C4—31no—
102C2×C4—31no—
103C2×C4—31no—
104C2×C4D7⋊A411yescharacteristic
105C8—72no—
106C8—212no—
107C7#568⋊C311yescharacteristic, complemented, Sylow
108C2×C3—281no—
109C2×C3—561no—
110C22—31no—
111C22—31no—
112C22—121no—
113C22Dic7⋊A411yescharacteristic
114C4—31no—
115C4—31no—
116C4Dic7⋊A411yescharacteristic
117C4C2×D7⋊A411yescharacteristic
118C3—281noSylow
119C2—21no—
120C2—61no—
121C2—61no—
122C2C2×Dic7⋊A411yescharacteristic
subgroupG/H
center, FrattiniC4C2×D7⋊A4
commutator, hypocenterC7×Q8C2×C3×C4
FittingC2×C7×#41C2×C3
socleC2×C7C2×C4×A4
prime radicalC2×#41⋊ΓL(1,𝔽8)C2
cosocleC7×#41C22×C3
hypercenterC2×C4D7⋊A4
LCSG ▷ C7×Q8
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #10620803C2semidirect
3. #90510C3semidirect
10. #6511C2×C3semidirect
107. C7#568⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#41⋊ΓL(1,𝔽8)2
#106208032
#905103
#1136⋊C34
#568⋊C37