Finite Group Databasebeta

← #20280592

Subgroups — #20280592

Order 6784.

416 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×C4×Dic106C211yesmaximal, characteristic
2#16810562C212yesmaximal
3#16810561C212yesmaximal
4#16810554C212yesmaximal
5#16810543C212yesmaximal
6#16810504C214yesmaximal
7#16810491C211yesmaximal, characteristic
8C2×#125×C53C211yesmaximal, characteristic
9C22×Dic106C2211yescharacteristic
10C4×Dic106—24no—
11C2×C4⋊Dic53C2211yescharacteristic
12C2×#11719919C2214yes—
13C2×#11719919C2212yes—
14C2×C4×Dic53C2212yes—
15#13133124C2212yes—
16#13133124C2214yes—
17#13133124C2214yes—
18#13133124C2214yes—
19#13133124C2212yes—
20#13133124C2212yes—
21#125×C53—24no—
22C2×C4⋊C4×C53C2212yes—
23C2×C4⋊C4×C53C2212yes—
24C2×C4⋊C4×C53C2211yescharacteristic
25C2×C4⋊C4×C53C2211yescharacteristic
26C2×C42×C53C2211yescharacteristic
27C22×C4×C53C2312yes—
28C22×C4×C53C2312yes—
29C22×C4×C53C2311yescharacteristic
30C22×C4×C53C2311yescharacteristic
31C22×C4×C53C2311yescharacteristic
32C22×Dic53C2314yes—
33C22×Dic53C2312yes—
34C22×Dic53C2312yes—
35C2×Dic106—22no—
36C2×Dic106—24no—
37C4⋊C4×C53—24no—
38C4⋊C4×C53—24no—
39C4⋊C4×C53—22no—
40C4⋊C4×C53—22no—
41C42×C53D414yes—
42C4⋊Dic53—22no—
43#11719919—24no—
44#11719919—28no—
45C4×Dic53—24no—
46C23×C53C2411yescharacteristic
47C2×C4×C53—22no—
48C2×C4×C53—22no—
49C2×C4×C53—22no—
50C2×C4×C53—22no—
51C2×C4×C53—22no—
52C2×C4×C53—24no—
53C2×C4×C53—24no—
54C2×C4×C53C2×D412yes—
55C2×C4×C53C2×D412yes—
56C2×C4×C53C2×D412yes—
57C2×Dic53—22no—
58C2×Dic53—22no—
59C2×Dic53—22no—
60C2×Dic53—24no—
61C2×Dic53—24no—
62C2×Dic53—24no—
63C2×Dic53—24no—
64C2×Dic53#4114yes—
65Dic106—28no—
66C22×C53C22×D411yescharacteristic
67C22×C53C2×#4112yes—
68C22×C532−1+412yes—
69C22×C532−1+412yes—
70C4×C53—22no—
71C4×C53—22no—
72C4×C53—22no—
73C4×C53—42no—
74C4×C53—42no—
75Dic53—24no—
76Dic53—42no—
77Dic53—44no—
78#2634—531nomaximal, Sylow
79C2×C53#52511yescharacteristic
80C2×C53#53611yescharacteristic
81C2×C53#53611yescharacteristic
82C2×C53#54114yes—
83C2×#125D5311yescharacteristic
84C2×C4×Q8—531no—
85#400—532no—
86#400—532no—
87#398—532no—
88#395—532no—
89#387—534no—
90#383—531no—
91C53#263411yescharacteristic, complemented, Sylow
92C22×Q8—531no—
93#125—24no—
94C4×Q8—1064no—
95C2×C4⋊C4—532no—
96C2×C4⋊C4—531no—
97C2×C4⋊C4—534no—
98C2×C4⋊C4C2×D5312yes—
99C2×C4⋊C4C2×D5312yes—
100C2×C4⋊C4C2×D5311yescharacteristic
101C2×C4⋊C4C2×D5311yescharacteristic
102C2×C42—532no—
103C2×C42C2×D5311yescharacteristic
104#95—532no—
105#95—532no—
106#95—532no—
107#95—534no—
108#95—534no—
109#95—534no—
110C2×Q8—1064no—
111C2×Q8—1062no—
112C22×C4—532no—
113C22×C4—534no—
114C22×C4—532no—
115C22×C4C22×D5312yes—
116C22×C4C22×D5312yes—
117C22×C4C22×D5311yescharacteristic
118C22×C4C22×D5311yescharacteristic
119C22×C4C22×D5311yescharacteristic
120C4⋊C4—24no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—24no—
124C4⋊C4—1068no—
125C4⋊C4—1064no—
126C4⋊C4—1062no—
127C42—1064no—
128C42C22⋊D5314yes—
129C23C23×D5311yescharacteristic
130Q8—1068no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—24no—
137C2×C4—24no—
138C2×C4—534no—
139C2×C4—1062no—
140C2×C4—1062no—
141C2×C4—1062no—
142C2×C4—1064no—
143C2×C4—1064no—
144C2×C4—1064no—
145C2×C4—1064no—
146C2×C4C2×C22⋊D5312yes—
147C2×C4C2×C22⋊D5312yes—
148C2×C4C2×C22⋊D5312yes—
149C22C22×C22⋊D5311yescharacteristic
150C22#41×D5312yes—
151C22#1313326412yes—
152C22#1313326512yes—
153C4—22no—
154C4—22no—
155C4—22no—
156C4—42no—
157C4—42no—
158C4—1064no—
159C4—2122no—
160C4—2124no—
161C2#1681113011yescharacteristic
162C2#1681119411yescharacteristic
163C2#1681119711yescharacteristic
164C2#1681123214yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#125×C53C2
hypercenterC2×#125D53
hypocenterC53#2634
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
91. C53#2634semidirect