Finite Group Databasebeta

← #13057184

Subgroups — #13057184

Order 1344.

140 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 27 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
2#10621496C211yesmaximal, characteristic
3C7⋊(A4⋊C8)C211yesmaximal, characteristic
4#90000—31nomaximal, Hall
5C4×C7×A4—21no—
6C4×C7×A4C2211yescharacteristic
7C7⋊SL(2,ℤ/4)—21no—
8#66635—41nomaximal
9C22×C7×Q8D311yescharacteristic
10C2×#1144—31no—
11C2×#1144—31no—
12#6487—31no—
13#6409—31no—
14#6409—31no—
15C22⋊(C7⋊C8)—31no—
16#5416—71nomaximal, Hall
17C2×C7×A4D411yescharacteristic
18C3×C7×Q8—41no—
19Dic42—41no—
20C3⋊(C7⋊C8)—41no—
21C2×C7×Q8—31no—
22C2×C7×Q8—31no—
23C2×C7×Q8—61no—
24C22×C4×C7—21no—
25C22×C4×C7C2×D311yescharacteristic
26C2×Dic14—31no—
27C22⋊Dic7—61no—
28#1144—62no—
29#1144—61no—
30#1144—61no—
31C4⋊Dic7—31no—
32Dic7⋊C4—61no—
33C2×C7⋊C8—31no—
34C2×C7⋊C8—31no—
35Q8×A4D711yescharacteristic
36A4⋊Q8—71no—
37A4⋊C8—71no—
38C7×A4Q1611yescharacteristic, complemented
39C3×C4×C7—41no—
40C3×C4×C7—81no—
41Dic21—81no—
42#451—211noSylow
43C23×C7#6711yescharacteristic
44C7×Q8—32no—
45C7×Q8—31no—
46C7×Q8—61no—
47C7×Q8S411yescharacteristic
48C2×C4×C7—31no—
49C2×C4×C7—31no—
50C2×C4×C7—61no—
51C2×C4×C7—61no—
52C2×Dic7—61no—
53C2×Dic7—61no—
54Dic14—61no—
55Dic14—61no—
56C7⋊C8—61no—
57C7⋊C8—61no—
58C4×A4—21no—
59C4×A4C2×D711yescharacteristic
60SL(2,ℤ/4)—141no—
61#216—281no—
62C2×C3×C7—41no—
63C22×Q8D2111yescharacteristic
64C2×Q16—211no—
65C2×Q16—211no—
66C22⋊Q8—211no—
67#103—211no—
68#103—211no—
69C22⋊C8—211no—
70C22×C7—31no—
71C22×C7—31no—
72C22×C7#21611yescharacteristic, complemented
73C4×C7—21no—
74C4×C7—31no—
75C4×C7—61no—
76C4×C7C2×S411yescharacteristic
77Dic7—121no—
78Dic7—121no—
79C2×A4C22⋊D711yescharacteristic
80C3×Q8—41no—
81Dic6—281no—
82C3⋊C8—281no—
83C3×C7—41noHall
84C2×Q8—31no—
85C2×Q8—31no—
86C2×Q8—61no—
87C2×Q8—211no—
88C22×C4—21no—
89C22×C4C2×D2111yescharacteristic
90Q16—422no—
91Q16—421no—
92Q16—421no—
93C2×C8—211no—
94C2×C8—211no—
95C4⋊C4—211no—
96C4⋊C4—421no—
97C22⋊C4—421no—
98C2×C7—31no—
99C2×C7—31no—
100C2×C7GL(2,ℤ/4)11yescharacteristic
101A4#114411yescharacteristic, complemented
102C3×C4—41no—
103C3×C4—81no—
104Dic3—561no—
105C23#426811yescharacteristic
106Q8—32no—
107Q8—31no—
108Q8—61no—
109Q8—421no—
110Q8—421no—
111Q8C7⋊S411yescharacteristic
112C2×C4—31no—
113C2×C4—31no—
114C2×C4—61no—
115C2×C4—61no—
116C2×C4—421no—
117C2×C4—421no—
118C8—421no—
119C8—421no—
120C7#541611yescharacteristic, complemented, Sylow
121C2×C3—41no—
122C22—31no—
123C22—31no—
124C22#6663511yescharacteristic, complemented
125C4—21no—
126C4—31no—
127C4—61no—
128C4—841no—
129C4—841no—
130C4C2×C7⋊S411yescharacteristic
131C3—41noSylow
132C2—31no—
133C2—31no—
134C2#1062150611yescharacteristic
subgroupG/H
centerC2#10621506
commutator, cosocleC4×C7×A4C22
FrattiniC4C2×C7⋊S4
FittingC22×C7×Q8D3
socleC23×C7#67
prime radicalC2×C7×A4D4
hypercenterQ8C7⋊S4
hypocenterC7×A4Q16
LCSG ▷ C4×C7×A4 ▷ C2×C7×A4 ▷ C7×A4
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
38. C7×A4Q16semidirect
72. C22×C7#216semidirect
101. A4#1144semidirect
120. C7#5416semidirect
124. C22#66635semidirect

Maximal subgroups

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

subgroupindex
C7×Q8×A42
#106214962
C7⋊(A4⋊C8)2
#900003
#666354
#54167