Finite Group Databasebeta

← #20282516

Subgroups — #20282516

Order 6784.

384 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 141 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#16810556C212yesmaximal
2#16810553C211yesmaximal, characteristic
3#16810542C212yesmaximal
4#16810520C212yesmaximal
5#16810519C212yesmaximal
6#16810508C212yesmaximal
7#16810506C212yesmaximal
8#16810492C211yesmaximal, characteristic
9C53×#395C211yesmaximal, characteristic
10C2×C4⋊Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12C2×C4⋊Dic53C2212yes—
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×C4×Dic53C2211yescharacteristic
19C2×C4×Dic53C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26C22×Q8×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2212yes—
28C2×C4⋊C4×C53C2211yescharacteristic
29#95×C53C2211yescharacteristic
30#95×C53C2212yes—
31C2×Q8×C53—41no—
32C2×Q8×C53—42no—
33C22×C4×C53C2311yescharacteristic
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C4⋊C4×C53—42no—
43C4⋊C4×C53—41no—
44C4⋊Dic53—41no—
45C4⋊Dic53—41no—
46C4⋊Dic53Q818yescomplemented
47#11719919—24no—
48#11719919—42no—
49#11719919—42no—
50#11719919—42no—
51#11719919—42no—
52C4×Dic53—22no—
53C4×Dic53—42no—
54C23×C53C2411yescharacteristic
55Q8×C53—44no—
56C2×C4×C53—22no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—21no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—21no—
67C2×C4×C53C2×Q814yes—
68C2×C4×C53#4112yes—
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—
79C2×Dic53C2×Q814yes—
80C2×Dic53C2×Q814yes—
81C22×C53C22×Q812yes—
82C22×C53C2×#4111yescharacteristic
83C22×C53H(5,𝔽2)11yescharacteristic
84C22×C53H(5,𝔽2)12yes—
85C22×C532−1+411yescharacteristic
86C4×C53—22no—
87C4×C53—24no—
88C4×C53—41no—
89C4×C53—42no—
90C4×C53—41no—
91Dic53—24no—
92Dic53—24no—
93Dic53—42no—
94Dic53—42no—
95#2792—531nomaximal, Sylow
96C2×C53#53911yescharacteristic
97C2×C53#54412yes—
98C2×C53#55412yes—
99C2×C53Q8211yescharacteristic
100C2×C53#56011yescharacteristic
101#398—531no—
102#398—532no—
103#395—532no—
104#395D5311yescharacteristic
105#389—532no—
106#389—532no—
107#387—532no—
108#387—532no—
109#384—531no—
110C53#279211yescharacteristic, complemented, Sylow
111C22×Q8C2×D5311yescharacteristic
112C2×C4⋊C4—532no—
113C2×C4⋊C4—532no—
114C2×C4⋊C4—532no—
115C2×C4⋊C4—532no—
116C2×C4⋊C4—532no—
117C2×C4⋊C4—532no—
118C2×C4⋊C4—531no—
119C2×C4⋊C4—531no—
120C2×C4⋊C4C2×D5311yescharacteristic
121C2×C4⋊C4C2×D5312yes—
122C2×C42—531no—
123C2×C42—532no—
124#95—532no—
125#95—532no—
126#95—532no—
127#95—532no—
128#95—532no—
129#95—531no—
130#95C2×D5311yescharacteristic
131#95C2×D5312yes—
132C2×Q8—42no—
133C2×Q8—41no—
134C22×C4—532no—
135C22×C4—532no—
136C22×C4—532no—
137C22×C4—532no—
138C22×C4C22×D5312yes—
139C22×C4C22×D5311yescharacteristic
140C22×C4C22×D5311yescharacteristic
141C22×C4C22×D5312yes—
142C22×C4C22×D5311yescharacteristic
143C4⋊C4—42no—
144C4⋊C4—41no—
145C4⋊C4—538no—
146C4⋊C4—1064no—
147C4⋊C4—2122no—
148C4⋊C4—2122no—
149C4⋊C4—2122no—
150C4⋊C4—2122no—
151C4⋊C4—2121no—
152C4⋊C4—2121no—
153C42—1062no—
154C42—2122no—
155C23C23×D5311yescharacteristic
156Q8—44no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—534no—
169C2×C4—534no—
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—1062no—
180C2×C4Q8×D5314yes—
181C2×C4#1171994912yes—
182C22C2×Q8×D5312yes—
183C22C2×#1171994911yescharacteristic
184C22#1313326212yes—
185C22#1313326211yescharacteristic
186C22#1313326411yescharacteristic
187C4—22no—
188C4—24no—
189C4—41no—
190C4—42no—
191C4—41no—
192C4—1064no—
193C4—1064no—
194C4—2122no—
195C4—2122no—
196C2#1681122211yescharacteristic
197C2#1681125212yes—
198C2#1681133412yes—
199C2#1681135311yescharacteristic
200C2#1681136211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2792
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
46. C4⋊Dic53Q8semidirect
110. C53#2792semidirect