Finite Group Databasebeta

← #13051376

Subgroups — #13051376

Order 1344.

136 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 37 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#10621418C211yesmaximal, characteristic
2C3×C7×#99C211yesmaximal, characteristic
3#10620672C211yesmaximal, characteristic
4#89267—31nomaximal, Hall
5C2×C3×C7×D4C2211yescharacteristic
6C22×Dic21C411yescharacteristic, complemented
7C3×C7×C22⋊C4—21no—
8#66636—21no—
9#66636C411yescharacteristic, complemented
10#66629—21no—
11#66627—21no—
12#6498—31no—
13C7×#99D311yescharacteristic
14#6407—31no—
15#4472—71nomaximal, Hall
16C23×C3×C7C2×C411yescharacteristic
17C23×C3×C7D411yescharacteristic
18C3×C7×D4—41no—
19C2×C3×C4×C7—41no—
20C2×C3×C4×C7D411yescharacteristic
21C2×Dic21—21no—
22C2×Dic21—21no—
23C2×Dic21—41no—
24C3⋊(C7⋊C8)—41no—
25C2×C7×D4C2×D311yescharacteristic
26C22×Dic7—31no—
27C7×C22⋊C4—21no—
28C22⋊Dic7—31no—
29C22⋊Dic7—61no—
30Dic7⋊C4—61no—
31#1136—61no—
32#935—71no—
33C3×#99D711yescharacteristic
34#833—71no—
35C22×C3×C7—21no—
36C22×C3×C7—21no—
37C22×C3×C7—41no—
38C22×C3×C7—41no—
39C22×C3×C7C22⋊C411yescharacteristic
40C3×C4×C7—21no—
41C3×C4×C7—81no—
42Dic21—41no—
43Dic21—41no—
44#352—211noSylow
45C23×C7C4×D311yescharacteristic
46C23×C7#6711yescharacteristic
47C7×D4—41no—
48C2×C4×C7—41no—
49C2×C4×C7D1211yescharacteristic
50C2×Dic7—61no—
51C2×Dic7—61no—
52C2×Dic7—121no—
53C7⋊C8—121no—
54C2×C3×D4C2×D711yescharacteristic
55C22×Dic3—71no—
56C3×C22⋊C4—21no—
57#217—71no—
58#217—141no—
59Dic3⋊C4—141no—
60#208—141no—
61C2×C3×C7—21no—
62C2×C3×C7—41no—
63C2×C3×C7—41no—
64C2×C3×C7#9911yescharacteristic
65#123—211no—
66#100—211no—
67#99D2111yescharacteristic
68C22×C7—21no—
69C22×C7—21no—
70C22×C7—41no—
71C22×C7—41no—
72C22×C7(C2×D3)⋊C411yescharacteristic
73C4×C7—21no—
74C4×C7—81no—
75Dic7—121no—
76Dic7—121no—
77C23×C3C4×D711yescharacteristic
78C23×C3C22⋊D711yescharacteristic
79C3×D4—41no—
80C2×C3×C4—41no—
81C2×C3×C4D2811yescharacteristic
82C2×Dic3—141no—
83C2×Dic3—141no—
84C2×Dic3—281no—
85C3⋊C8—281no—
86C3×C7#35211yescharacteristic, complemented, Hall
87C2×D4C2×D2111yescharacteristic
88C22×C4—211no—
89M16—421no—
90C4⋊C4—421no—
91C22⋊C4—21no—
92C22⋊C4—211no—
93C22⋊C4—421no—
94C2×C7—21no—
95C2×C7—41no—
96C2×C7—41no—
97C2×C7#80611yescharacteristic
98C22×C3—21no—
99C22×C3—21no—
100C22×C3—41no—
101C22×C3—41no—
102C22×C3(C2×D7)⋊C411yescharacteristic
103C3×C4—21no—
104C3×C4—81no—
105Dic3—281no—
106Dic3—281no—
107C23C4×D2111yescharacteristic
108C23#426811yescharacteristic
109D4—41no—
110C2×C4—41no—
111C2×C4—421no—
112C2×C4—421no—
113C2×C4—841no—
114C2×C4D8411yescharacteristic
115C8—841no—
116C7#447211yescharacteristic, complemented, Sylow
117C2×C3—21no—
118C2×C3—41no—
119C2×C3—41no—
120C2×C3#638011yescharacteristic
121C22—21no—
122C22—21no—
123C22—41no—
124C22—41no—
125C22#6663111yescharacteristic
126C4—21no—
127C4—81no—
128C4—841no—
129C4—841no—
130C3#8926711yescharacteristic, complemented, Sylow
131C2—21no—
132C2—41no—
133C2—41no—
134C2#1062064511yescharacteristic
subgroupG/H
centerC2#10620645
commutatorC23×C3×C7C2×C4
FrattiniC2×D4C2×D21
FittingC3×C7×#99C2
socleC2×C3×C7#99
prime radical, cosocleC2×C3×C7×D4C22
hypercenter#99D21
hypocenterC3×C7#352
LCSG ▷ C23×C3×C7 ▷ C22×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #99

Complements

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

NH ≅ G/Ntype
6. C22×Dic21C4semidirect
9. #66636C4semidirect
86. C3×C7#352semidirect
116. C7#4472semidirect
130. C3#89267semidirect

Maximal subgroups

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

subgroupindex
#106214182
C3×C7×#992
#106206722
#892673
#44727