Finite Group Databasebeta

← #13062114

Subgroups — #13062114

Order 1344.

326 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 47 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
1C7×Q8×A4C211yesmaximal, characteristic, complemented
2C4⋊(C7⋊S4)C213yesmaximal, complemented
3C4×C7⋊S4C213yesmaximal, complemented
4#90508—31nomaximal, Hall
5C2×C7⋊S4C2213yes—
6#66732—41nomaximal
7C4×C7×A4C2213yes—
8C7⋊SL(2,ℤ/4)C2211yescharacteristic
9C22×C7×Q8D311yescharacteristic, complemented
10C2×#1161—31no—
11#6510—33no—
12#6509—33no—
13Q8×Dic7—31no—
14#6493—33no—
15C4×C22⋊D7—33no—
16#5919—71nomaximal, Hall
17C2×C7×A4C2311yescharacteristic
18C7⋊S4—23no—
19C3×C7×Q8—41no—
20D84—43no—
21C4×D21—43no—
22C2×C7×Q8—31no—
23C2×C7×Q8—33no—
24C22×C4×C7C2×D313yes—
25C2×C22⋊D7—33no—
26#1161—63no—
27#1161—61no—
28C2×D28—33no—
29C2×C4×D7—33no—
30C22⋊Dic7—31no—
31(C2×D7)⋊C4—33no—
32(C2×D7)⋊C4—36no—
33C4⋊Dic7—33no—
34Dic7⋊C4—33no—
35C4×Dic7—33no—
36Q8×A4D711yescharacteristic, complemented
37C4⋊S4—73no—
38C4×S4—73no—
39C2×D21—43no—
40C7×A4#4111yescharacteristic, complemented
41C3×C4×C7—43no—
42Dic21—41no—
43#548—211noSylow
44C23×C7C22×D311yescharacteristic
45C22×D7—33no—
46C7×Q8—33no—
47C7×Q8—61no—
48C7×Q8S411yescharacteristic, complemented
49C2×C4×C7—33no—
50C2×C4×C7—33no—
51C22⋊D7—63no—
52C22⋊D7—63no—
53C2×Dic7—31no—
54C2×Dic7—33no—
55C2×Dic7—31no—
56D28—63no—
57D28—63no—
58C4×D7—63no—
59C4×D7—63no—
60C2×S4—73no—
61#239—281no—
62C4×A4C2×D713yes—
63SL(2,ℤ/4)—71no—
64C2×C3×C7—41no—
65D21—83no—
66C2×#41—211no—
67C22×Q8D2111yescharacteristic, complemented
68#124—213no—
69C22⋊Q8—213no—
70#121—213no—
71C4×Q8—211no—
72C4×D4—213no—
73C22×C7—31no—
74C22×C7—31no—
75C22×C7#23911yescharacteristic, complemented
76C2×D7—63no—
77C2×D7—63no—
78C2×D7—63no—
79C4×C7—33no—
80C4×C7C2×S413yes—
81Dic7—61no—
82Dic7—63no—
83Dic7—61no—
84C2×A4C22×D711yescharacteristic
85S4—143no—
86C3×Q8—41no—
87D12—283no—
88C4×D3—283no—
89C3×C7—41noHall
90#41—423no—
91#41—421no—
92C2×Q8—31no—
93C2×Q8—33no—
94C2×D4—213no—
95C2×D4—213no—
96C22×C4—213no—
97C22×C4C2×D2113yes—
98C4⋊C4—213no—
99C4⋊C4—213no—
100C22⋊C4—213no—
101C22⋊C4—216no—
102C22⋊C4—211no—
103C42—213no—
104C2×C7—31no—
105C2×C7—31no—
106C2×C7C22×S411yescharacteristic
107D7—123no—
108C2×D3—283no—
109A4#116111yescharacteristic, complemented
110C3×C4—43no—
111Dic3—281no—
112C23—213no—
113C23C22×D2111yescharacteristic
114Q8—33no—
115Q8—61no—
116Q8C7⋊S411yescharacteristic, complemented
117D4—423no—
118D4—423no—
119D4—423no—
120D4—423no—
121C2×C4—33no—
122C2×C4—33no—
123C2×C4—211no—
124C2×C4—213no—
125C2×C4—211no—
126C2×C4—423no—
127C2×C4—423no—
128C7#591911yescharacteristic, complemented, Sylow
129C2×C3—41no—
130D3—563no—
131C22—31no—
132C22—31no—
133C22—423no—
134C22—423no—
135C22—423no—
136C22#6673211yescharacteristic, complemented
137C4—33no—
138C4—423no—
139C4—421no—
140C4—421no—
141C4C2×C7⋊S413yes—
142C3—41noSylow
143C2—31no—
144C2—31no—
145C2—843no—
146C2C22×C7⋊S411yescharacteristic
subgroupG/H
center, FrattiniC2C22×C7⋊S4
commutator, cosocleC2×C7×A4C23
FittingC22×C7×Q8D3
socleC23×C7C22×D3
hypercenterQ8C7⋊S4
hypocenterC7×A4#41
LCSG ▷ C2×C7×A4 ▷ C7×A4
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C7×Q8×A4C2semidirect
2. C4⋊(C7⋊S4)C2semidirect
3. C4×C7⋊S4C2semidirect
9. C22×C7×Q8D3semidirect
36. Q8×A4D7semidirect
40. C7×A4#41semidirect
48. C7×Q8S4semidirect
67. C22×Q8D21semidirect
75. C22×C7#239semidirect
109. A4#1161semidirect
116. Q8C7⋊S4semidirect
128. C7#5919semidirect
136. C22#66732semidirect

Maximal subgroups

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

subgroupindex
C7×Q8×A42
C4⋊(C7⋊S4)2
C4×C7⋊S42
#905083
#667324
#59197