Finite Group Databasebeta

← #13057683

Subgroups — #13057683

Order 1344.

146 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 26 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#10621615C211yesmaximal, characteristic, complemented
2#10621539C211yesmaximal, characteristic, complemented
3#10620803C211yesmaximal, characteristic, complemented
4#90526C311yesmaximal, characteristic, complemented, Hall
5#41⋊ΓL(1,𝔽8)—21no—
6#41⋊ΓL(1,𝔽8)C2211yescharacteristic
7#66665—21no—
8#1144⋊C3—41nomaximal
9C7×H(5,𝔽2)C2×C311yescharacteristic
10#6553C2×C311yescharacteristic
11#6513—31no—
12#6512—31no—
13#6511C2×C311yescharacteristic
14#6495—31no—
15C2×#1141—31no—
16#576⋊C3—71nomaximal, Hall
17C7⋊SL(2,𝔽3)D411yescharacteristic
18Q8×ΓL(1,𝔽8)—41no—
19ΓL(1,𝔽8)⋊Q8—41no—
20ΓL(1,𝔽8)⋊C8—41no—
21C7×#41—21no—
22C7×#41—31no—
23C7×#41C22×C311yescharacteristic
24C2×C7×D4—31no—
25C2×C7×D4—61no—
26#1161—21no—
27D4×D7—61no—
28#1157—31no—
29C2×D28—31no—
30#1144A411yescharacteristic
31#1143—31no—
32#1142—31no—
33#1141—32no—
34#1141—31no—
35#1136—31no—
36C2×C7⋊C8—31no—
37#994—71no—
38#994—71no—
39#131⋊C3—71no—
40C4×ΓL(1,𝔽8)—41no—
41C4×ΓL(1,𝔽8)—81no—
42ΓL(1,𝔽8)⋊C4—81no—
43#576—71noSylow
44C23×C7—61no—
45C22×D7—61no—
46C7×Q8C3×D411yescharacteristic
47C7×Q8C2×A411yescharacteristic
48C7×D4—31no—
49C7×D4—31no—
50C7×D4—61no—
51C7×D4—61no—
52C2×C4×C7—31no—
53C2×C4×C7—61no—
54C22⋊D7—61no—
55D28—31no—
56D28—61no—
57C4×D7—61no—
58Dic14C2×A411yescharacteristic
59C7⋊C8—31no—
60C7⋊C8C2×A411yescharacteristic
61#41⋊C3—71no—
62#41⋊C3—141no—
63#41⋊C3—141no—
64C3×Q16—281no—
65C2×ΓL(1,𝔽8)—41no—
66H(5,𝔽2)—71no—
67H(5,𝔽2)AGL(1,𝔽7)11yescharacteristic
68AGL(1,ℤ/8)—211no—
69AGL(1,ℤ/8)—211no—
70#135—211no—
71C2×D8—211no—
72#131—71no—
73C22×C7—31no—
74C22×C7—61no—
75C22×C7—241no—
76C2×D7—61no—
77C2×D7—241no—
78C4×C7—21no—
79C4×C7—31no—
80C4×C7C22×A411yescharacteristic
81Dic7—21no—
82C3×Q8—281no—
83C3×Q8—281no—
84SL(2,𝔽3)—71no—
85C3×C8—281no—
86ΓL(1,𝔽8)—41noHall
87#41—21no—
88#41—31no—
89#41—141no—
90#41—211no—
91#41C2×AGL(1,𝔽7)11yescharacteristic
92C2×D4—31no—
93C2×D4—61no—
94C2×D4—211no—
95C2×D4—421no—
96Q16—71no—
97SD16—211no—
98SD16—211no—
99D8—211no—
100D8—212no—
101M16—211no—
102C2×C8—211no—
103C2×C7—61no—
104C2×C7—121no—
105C2×C7D4×A411yescharacteristic
106D7—121no—
107C3×C4—281no—
108C3×C4—561no—
109C3×C4—561no—
110C23—61no—
111C23—421no—
112Q8—71no—
113Q8(C22⋊D7)⋊C311yescharacteristic
114Q8D7⋊A411yescharacteristic
115D4—31no—
116D4—31no—
117D4—61no—
118D4—61no—
119D4—211no—
120D4—421no—
121D4—421no—
122C2×C4—31no—
123C2×C4—61no—
124C2×C4—421no—
125C8—71no—
126C8—211no—
127C7#576⋊C311yescharacteristic, complemented, Sylow
128C2×C3—281no—
129C22—31no—
130C22—61no—
131C22—241no—
132C22—421no—
133C22—1681no—
134C4—21no—
135C4—31no—
136C4—141no—
137C4C2×D7⋊A411yescharacteristic
138C3—281noSylow
139C2—61no—
140C2—121no—
141C2—841no—
142C2(C22⋊D7)⋊A411yescharacteristic
subgroupG/H
centerC2(C22⋊D7)⋊A4
commutator, cosocleC7×#41C22×C3
FrattiniC4C2×D7⋊A4
FittingC7×H(5,𝔽2)C2×C3
socleC2×C7D4×A4
hypercenterQ8D7⋊A4
hypocenterC7×Q8C3×D4
LCSG ▷ C7×#41 ▷ C7×Q8
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. #10621615C2semidirect
2. #10621539C2semidirect
3. #10620803C2semidirect
4. #90526C3semidirect
127. C7#576⋊C3semidirect

Maximal subgroups

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

subgroupindex
#106216152
#106215392
#106208032
#905263
#1144⋊C34
#576⋊C37