Finite Group Databasebeta

← #20282782

Subgroups — #20282782

Order 6784.

380 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133209C211yesmaximal, characteristic
2#16810562C212yesmaximal
3#16810561C212yesmaximal
4#16810556C214yesmaximal
5#16810554C212yesmaximal
6#16810553C212yesmaximal
7#16810489C211yesmaximal, characteristic
8C2×#125×C53C211yesmaximal, characteristic
9#13133209—24no—
10C2×C4⋊Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12C2×C4⋊Dic53C2212yes—
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×#11719919C2214yes—
16#13133124C2214yes—
17#13133124C2212yes—
18#13133124C2214yes—
19#13133124C2212yes—
20#13133124C2214yes—
21#125×C53—24no—
22C2×C4⋊C4×C53C2212yes—
23C2×C4⋊C4×C53C2211yescharacteristic
24C2×C4⋊C4×C53C2212yes—
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—
35C4⋊C4×C53—22no—
36C4⋊C4×C53—24no—
37C4⋊C4×C53—22no—
38C4⋊C4×C53—24no—
39C42×C53D414yes—
40C4⋊Dic53—22no—
41C4⋊Dic53—22no—
42C4⋊Dic53—42no—
43#11719919—24no—
44#11719919—24no—
45#11719919—44no—
46C23×C53C2411yescharacteristic
47C2×C4×C53—21no—
48C2×C4×C53—22no—
49C2×C4×C53—21no—
50C2×C4×C53—22no—
51C2×C4×C53—22no—
52C2×C4×C53—22no—
53C2×C4×C53—22no—
54C2×C4×C53—24no—
55C2×C4×C53—22no—
56C2×C4×C53C2×D412yes—
57C2×C4×C53C2×D412yes—
58C2×C4×C53C2×D412yes—
59C2×Dic53—22no—
60C2×Dic53—22no—
61C2×Dic53—22no—
62C2×Dic53—24no—
63C2×Dic53—22no—
64C2×Dic53—24no—
65C2×Dic53—24no—
66C2×Dic53—24no—
67C22×C53C22×D411yescharacteristic
68C22×C53H(5,𝔽2)11yescharacteristic
69C22×C532−1+412yes—
70C22×C532−1+411yescharacteristic
71C22×C532−1+412yes—
72C4×C53—22no—
73C4×C53—22no—
74C4×C53—22no—
75C4×C53—42no—
76C4×C53—42no—
77Dic53—42no—
78Dic53—42no—
79Dic53—44no—
80#2822—531nomaximal, Sylow
81C2×C53#53611yescharacteristic
82C2×C53#53611yescharacteristic
83C2×C53#53711yescharacteristic
84C2×C53#56312yes—
85C2×C53#56412yes—
86C2×#125—531no—
87C2×#125D5311yescharacteristic
88#400—532no—
89#400—532no—
90#398—532no—
91#398—532no—
92#398—534no—
93#382—531no—
94C53#282211yescharacteristic, complemented, Sylow
95#125—24no—
96#125—1064no—
97C2×C4⋊C4—532no—
98C2×C4⋊C4—532no—
99C2×C4⋊C4—531no—
100C2×C4⋊C4—531no—
101C2×C4⋊C4—534no—
102C2×C4⋊C4—532no—
103C2×C4⋊C4C2×D5312yes—
104C2×C4⋊C4C2×D5311yescharacteristic
105C2×C4⋊C4C2×D5312yes—
106C2×C4⋊C4C2×D5311yescharacteristic
107C2×C42C2×D5311yescharacteristic
108#95—532no—
109#95—532no—
110#95—534no—
111#95—534no—
112#95—534no—
113C22×C4—534no—
114C22×C4—532no—
115C22×C4—532no—
116C22×C4C22×D5312yes—
117C22×C4C22×D5312yes—
118C22×C4C22×D5311yescharacteristic
119C22×C4C22×D5311yescharacteristic
120C22×C4C22×D5311yescharacteristic
121C4⋊C4—24no—
122C4⋊C4—22no—
123C4⋊C4—24no—
124C4⋊C4—22no—
125C4⋊C4—1064no—
126C4⋊C4—1064no—
127C4⋊C4—1062no—
128C4⋊C4—1062no—
129C4⋊C4—2124no—
130C4⋊C4—2122no—
131C42C22⋊D5314yes—
132C23C23×D5311yescharacteristic
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—24no—
141C2×C4—22no—
142C2×C4—1062no—
143C2×C4—1062no—
144C2×C4—1064no—
145C2×C4—1062no—
146C2×C4—1064no—
147C2×C4—1062no—
148C2×C4—1064no—
149C2×C4—1064no—
150C2×C4C2×C22⋊D5312yes—
151C2×C4C2×C22⋊D5312yes—
152C2×C4C2×C22⋊D5312yes—
153C22C22×C22⋊D5311yescharacteristic
154C22#1313326311yescharacteristic
155C22#1313326412yes—
156C22#1313326411yescharacteristic
157C22#1313326512yes—
158C4—22no—
159C4—22no—
160C4—22no—
161C4—42no—
162C4—42no—
163C4—2124no—
164C4—2122no—
165C4—2122no—
166C2#1681119411yescharacteristic
167C2#1681119711yescharacteristic
168C2#1681120111yescharacteristic
169C2#1681137712yes—
170C2C53⋊#56412yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#125×C53C2
hypercenterC2×#125D53
hypocenterC53#2822
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
94. C53#2822semidirect