Finite Group Databasebeta

← #13051037

Subgroups — #13051037

Order 1344.

140 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 38 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
1C3×#6499C211yesmaximal, characteristic
2#10620673C211yesmaximal, characteristic
3C7×#834C211yesmaximal, characteristic
4#89268—31nomaximal, Hall
5C2×C3×C7×D4C2211yescharacteristic
6C2×C3×Dic14C411yescharacteristic, complemented
7#66636—21no—
8C7×#217—21no—
9C3×C22⋊Dic7—21no—
10C3×C22⋊Dic7—21no—
11C3×C4×Dic7C411yescharacteristic, complemented
12#6499D311yescharacteristic
13C7×#99—31no—
14#6408—31no—
15#4541—71nomaximal, Hall
16C23×C3×C7D411yescharacteristic
17C23×C3×C7D411yescharacteristic
18C3×C7×D4—41no—
19C2×C3×C4×C7C2×C411yescharacteristic
20C2×Dic21—41no—
21C2×C7×Dic3—41no—
22C2×C3×Dic7—21no—
23C2×C3×Dic7—21no—
24C3×Dic14—41no—
25C2×C7×D4C2×D311yescharacteristic
26C2×Dic14Dic311yescharacteristic, complemented
27C7×C22⋊C4—61no—
28C22⋊Dic7—21no—
29C22⋊Dic7—21no—
30C22⋊Dic7—61no—
31C4×Dic7Dic311yescharacteristic, complemented
32C3×#124—71no—
33#834—71no—
34#834D711yescharacteristic
35C22×C3×C7—21no—
36C22×C3×C7—21no—
37C22×C3×C7—41no—
38C22×C3×C7—41no—
39C22×C3×C7C22⋊C411yescharacteristic
40C3×C4×C7—21no—
41Dic21—81no—
42C3×Dic7—41no—
43C3×Dic7—41no—
44C7×Dic3—81no—
45#354—211noSylow
46C23×C7#6711yescharacteristic
47C23×C7#6711yescharacteristic
48C7×D4—41no—
49C2×C4×C7—121no—
50C2×C4×C7C2×Dic311yescharacteristic
51C2×Dic7—21no—
52C2×Dic7—21no—
53C2×Dic7—121no—
54Dic14—41no—
55C2×C3×Q8—71no—
56C2×C3×D4C2×D711yescharacteristic
57C3×C22⋊C4—141no—
58C3×C22⋊C4—141no—
59C3×C42—71no—
60#217—21no—
61#217—141no—
62C2×C3×C7—21no—
63C2×C3×C7—41no—
64C2×C3×C7—41no—
65C2×C3×C7#9911yescharacteristic
66#124—71no—
67#99—31no—
68#99—211no—
69C22×C7—21no—
70C22×C7—21no—
71C22×C7—41no—
72C22×C7—41no—
73C22×C7#21711yescharacteristic
74C4×C7—21no—
75C4×C7—241no—
76Dic7—41no—
77Dic7—41no—
78Dic7—241no—
79C23×C3D2811yescharacteristic
80C23×C3C22⋊D711yescharacteristic
81C3×Q8—281no—
82C3×D4—41no—
83C2×C3×C4—141no—
84C2×C3×C4—141no—
85C2×C3×C4C4×D711yescharacteristic
86C2×Dic3—41no—
87C2×Dic3—281no—
88C3×C7#35411yescharacteristic, complemented, Hall
89C2×Q8—71no—
90C2×D4D3×D711yescharacteristic
91C22⋊C4—61no—
92C22⋊C4—141no—
93C22⋊C4—141no—
94C22⋊C4—421no—
95C42—71no—
96C2×C7—21no—
97C2×C7—41no—
98C2×C7—41no—
99C2×C7#83411yescharacteristic
100C22×C3—21no—
101C22×C3—21no—
102C22×C3—41no—
103C22×C3—41no—
104C22×C3(C2×D7)⋊C411yescharacteristic
105C3×C4—21no—
106C3×C4—281no—
107C3×C4—281no—
108Dic3—81no—
109Dic3—561no—
110C23#424511yescharacteristic
111C23#424611yescharacteristic
112Q8—281no—
113D4—41no—
114C2×C4—121no—
115C2×C4—141no—
116C2×C4—141no—
117C2×C4—841no—
118C2×C4Dic3×D711yescharacteristic
119C7#454111yescharacteristic, complemented, Sylow
120C2×C3—21no—
121C2×C3—41no—
122C2×C3—41no—
123C2×C3#638011yescharacteristic
124C22—21no—
125C22—21no—
126C22—41no—
127C22—41no—
128C22(C2×D7)⋊Dic311yescharacteristic
129C4—21no—
130C4—241no—
131C4—281no—
132C4—281no—
133C4—1681no—
134C3#8926811yescharacteristic, complemented, Sylow
135C2—21no—
136C2—41no—
137C2—41no—
138C2#1062053011yescharacteristic
subgroupG/H
centerC2#10620530
commutatorC2×C3×C4×C7C2×C4
Frattini, hypercenterC2×D4D3×D7
Fitting, prime radical, cosocleC2×C3×C7×D4C22
socleC2×C3×C7#99
hypocenterC3×C7#354
LCSG ▷ C2×C3×C4×C7 ▷ C22×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
6. C2×C3×Dic14C4semidirect
11. C3×C4×Dic7C4semidirect
26. C2×Dic14Dic3semidirect
31. C4×Dic7Dic3semidirect
88. C3×C7#354semidirect
119. C7#4541semidirect
134. C3#89268semidirect

Maximal subgroups

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

subgroupindex
C3×#64992
#106206732
C7×#8342
#892683
#45417