Finite Group Databasebeta

← #20276387

Subgroups — #20276387

Order 6784.

270 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 73 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#16810597C212yesmaximal, complemented
2#16810416C214yesmaximal, complemented
3C53×#535C211yesmaximal, characteristic
4#13133176—22no—
5#13133132—22no—
6#13133132C2214yescomplemented
7C2×#41×C53C2211yescharacteristic
8C22×D4×C53C411yescharacteristic, complemented
9#127×C53C2212yes—
10#124×C53C412yescomplemented
11#121×C53—22no—
12#120×C53—22no—
13#117×C53C411yescharacteristic, complemented
14C24×C53—21no—
15#41×C53—42no—
16C2×Q8×C53C2×C411yescharacteristic
17C2×D4×C53—24no—
18C2×D4×C53—21no—
19C2×D4×C53—42no—
20C2×D4×C53C2×C412yes—
21C2×D4×C53D412yes—
22C2×D4×C53C2311yescharacteristic
23C22×C4×C53C2×C411yescharacteristic
24C22×C4×C53D412yes—
25C4⋊C4×C53—21no—
26C22⋊C4×C53—21no—
27C22⋊C4×C53—24no—
28C42×C53C2×C412yes—
29C22⋊Dic53—24no—
30C4⋊Dic53—42no—
31C4×Dic53—42no—
32C23×C53—22no—
33C23×C53—42no—
34C23×C53—82no—
35C23×C53C22⋊C411yescharacteristic
36C23×C53C2×D412yes—
37Q8×C53—41no—
38D4×C53—42no—
39D4×C53—41no—
40D4×C53—42no—
41D4×C53—42no—
42D4×C53—84no—
43C2×C4×C53—22no—
44C2×C4×C53—22no—
45C2×C4×C53—22no—
46C2×C4×C53—22no—
47C2×C4×C53C22⋊C412yes—
48C2×C4×C53C22⋊C411yescharacteristic
49C2×C4×C53C22×C411yescharacteristic
50C2×Dic53—44no—
51C22×C53—22no—
52C22×C53—21no—
53C22×C53—42no—
54C22×C53—41no—
55C22×C53—44no—
56C22×C53—42no—
57C22×C53—84no—
58C22×C53C2×C22⋊C411yescharacteristic
59C4×C53—22no—
60C4×C53—21no—
61C4×C53—42no—
62C4×C53#9912yes—
63Dic53—84no—
64#2128—531nomaximal, Sylow
65C2×C53—21no—
66C2×C53—42no—
67C2×C53—41no—
68C2×C53—82no—
69C2×C53C2×#9911yescharacteristic
70#535D5311yescharacteristic
71#410—532no—
72#353—534no—
73C53#212811yescharacteristic, complemented, Sylow
74C2×#41C2×D5311yescharacteristic
75C22×D4Dic5311yescharacteristic, complemented
76#127C2×D5312yes—
77#124Dic5312yescomplemented
78#121—22no—
79#120—22no—
80#117—1062no—
81#117Dic5311yescharacteristic, complemented
82#99—534no—
83#99—1062no—
84C24—21no—
85#41—42no—
86C2×Q8C2×Dic5311yescharacteristic
87C2×D4—24no—
88C2×D4—21no—
89C2×D4—42no—
90C2×D4C2×Dic5312yes—
91C2×D4C22⋊D5312yes—
92C2×D4C22×D5311yescharacteristic
93C22×C4C2×Dic5311yescharacteristic
94C22×C4C22⋊D5312yes—
95C4⋊C4—21no—
96C4⋊C4—2122no—
97C22⋊C4—21no—
98C22⋊C4—24no—
99C22⋊C4—1064no—
100C42—2122no—
101C42C2×Dic5312yes—
102C23—22no—
103C23—42no—
104C23—82no—
105C23C22⋊Dic5311yescharacteristic
106C23C2×C22⋊D5312yes—
107Q8—41no—
108D4—42no—
109D4—41no—
110D4—42no—
111D4—42no—
112D4—84no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—2124no—
118C2×C4C22⋊Dic5312yes—
119C2×C4C22⋊Dic5311yescharacteristic
120C2×C4C22×Dic5311yescharacteristic
121C22—22no—
122C22—21no—
123C22—42no—
124C22—41no—
125C22—44no—
126C22—42no—
127C22—84no—
128C22C2×C22⋊Dic5311yescharacteristic
129C4—21no—
130C4—22no—
131C4—42no—
132C4—4244no—
133C4#1313313212yes—
134C2—21no—
135C2—42no—
136C2—41no—
137C2—82no—
138C2C2×#1313313211yescharacteristic
subgroupG/H
centerC2C2×#13133132
commutatorC2×C4×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#535C2
socleC2×C53C2×#99
hypercenter#535D53
hypocenterC53#2128
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #16810597C2semidirect
2. #16810416C2semidirect
6. #13133132C22semidirect
8. C22×D4×C53C4semidirect
10. #124×C53C4semidirect
13. #117×C53C4semidirect
73. C53#2128semidirect
75. C22×D4Dic53semidirect
77. #124Dic53semidirect
81. #117Dic53semidirect