Finite Group Databasebeta

← #20282706

Subgroups — #20282706

Order 6784.

344 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 125 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#16810563C212yesmaximal
2#16810562C212yesmaximal
3#16810561C212yesmaximal
4#16810521C211yesmaximal, characteristic
5#16810521C211yesmaximal, characteristic
6#16810510C212yesmaximal
7#16810509C212yesmaximal
8#16810504C212yesmaximal
9C53×#389C211yesmaximal, characteristic
10C2×C4⋊Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12C2×C4⋊Dic53C2211yescharacteristic
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×C4×Dic53C2212yes—
17#13133124C2212yes—
18#13133124C2212yes—
19#13133124C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26C2×C4⋊C4×C53C2212yes—
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C42×C53C2211yescharacteristic
30#95×C53C2212yes—
31C22×C4×C53C2312yes—
32C22×C4×C53C2312yes—
33C22×C4×C53C2311yescharacteristic
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×Dic53C2312yes—
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C4⋊C4×C53—22no—
41C4⋊C4×C53—24no—
42C4⋊C4×C53—41no—
43C42×C53—41no—
44C4⋊Dic53—22no—
45C4⋊Dic53—22no—
46C4⋊Dic53—41no—
47C4⋊Dic53—41no—
48#11719919—24no—
49#11719919—24no—
50C4×Dic53—42no—
51C23×C53C2411yescharacteristic
52C2×C4×C53—21no—
53C2×C4×C53—21no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53#4112yes—
65C2×C4×C53#4112yes—
66C2×C4×C53#4112yes—
67C2×Dic53—22no—
68C2×Dic53—22no—
69C2×Dic53—22no—
70C2×Dic53—22no—
71C2×Dic53—22no—
72C2×Dic53—22no—
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C22×C53C2×#4111yescharacteristic
80C22×C53C2×#4111yescharacteristic
81C22×C53C2×#4111yescharacteristic
82C22×C53H(5,𝔽2)11yescharacteristic
83C22×C532−1+411yescharacteristic
84C22×C532−1+411yescharacteristic
85C22×C532−1+411yescharacteristic
86C4×C53—22no—
87C4×C53—22no—
88C4×C53—22no—
89C4×C53—42no—
90C4×C53—42no—
91Dic53—42no—
92Dic53—42no—
93Dic53—42no—
94Dic53—42no—
95#2810—531nomaximal, Sylow
96C2×C53#53911yescharacteristic
97C2×C53#53911yescharacteristic
98C2×C53#53911yescharacteristic
99C2×C53#55611yescharacteristic
100C2×C53#55611yescharacteristic
101C2×C53#55611yescharacteristic
102C2×C53#56411yescharacteristic
103#400—532no—
104#400—532no—
105#400—532no—
106#389—531no—
107#389—531no—
108#389D5311yescharacteristic
109#387—532no—
110#387—532no—
111#387—532no—
112C53#281011yescharacteristic, complemented, Sylow
113C2×C4⋊C4—532no—
114C2×C4⋊C4—532no—
115C2×C4⋊C4—531no—
116C2×C4⋊C4—531no—
117C2×C4⋊C4—531no—
118C2×C4⋊C4—531no—
119C2×C4⋊C4C2×D5312yes—
120C2×C4⋊C4C2×D5311yescharacteristic
121C2×C4⋊C4C2×D5311yescharacteristic
122C2×C42—532no—
123C2×C42C2×D5311yescharacteristic
124#95—532no—
125#95—532no—
126#95—532no—
127#95—532no—
128#95—532no—
129#95—532no—
130#95—532no—
131#95—532no—
132#95—532no—
133#95C2×D5312yes—
134C22×C4—532no—
135C22×C4—532no—
136C22×C4—532no—
137C22×C4—532no—
138C22×C4C22×D5312yes—
139C22×C4C22×D5312yes—
140C22×C4C22×D5311yescharacteristic
141C22×C4C22×D5311yescharacteristic
142C22×C4C22×D5311yescharacteristic
143C4⋊C4—22no—
144C4⋊C4—24no—
145C4⋊C4—41no—
146C4⋊C4—1064no—
147C4⋊C4—1062no—
148C4⋊C4—1064no—
149C4⋊C4—1062no—
150C4⋊C4—2121no—
151C4⋊C4—2121no—
152C42—41no—
153C42—2122no—
154C23C23×D5311yescharacteristic
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—1062no—
168C2×C4—1062no—
169C2×C4—1062no—
170C2×C4—1062no—
171C2×C4—1062no—
172C2×C4—1062no—
173C2×C4—1062no—
174C2×C4—1062no—
175C2×C4—1062no—
176C2×C4—1062no—
177C2×C4—1062no—
178C2×C4—1062no—
179C2×C4#1171994512yes—
180C2×C4#1171994712yes—
181C2×C4#1171994912yes—
182C22C2×#1171994711yescharacteristic
183C22C2×#1171994511yescharacteristic
184C22C2×#1171994911yescharacteristic
185C22#1313326211yescharacteristic
186C22#1313326411yescharacteristic
187C22#1313326411yescharacteristic
188C22#1313326511yescharacteristic
189C4—22no—
190C4—22no—
191C4—22no—
192C4—42no—
193C4—42no—
194C4—2122no—
195C4—2122no—
196C4—2122no—
197C4—2122no—
198C2#1681121811yescharacteristic
199C2#1681122011yescharacteristic
200C2#1681122211yescharacteristic
201C2#1681134311yescharacteristic
202C2#1681134611yescharacteristic
203C2#1681134711yescharacteristic
204C2C53⋊#56411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#389C2
hypercenter#389D53
hypocenterC53#2810
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
112. C53#2810semidirect