Finite Group Databasebeta

← #20277900

Subgroups — #20277900

Order 6784.

408 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 173 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#16810520C212yesmaximal
3#16810518C212yesmaximal
4#16810510C212yesmaximal
5#16810508C212yesmaximal
6#16810506C212yesmaximal
7#16810504C212yesmaximal
8#16810489C211yesmaximal, characteristic
9C53×#382C211yesmaximal, characteristic
10#13133209C418yescomplemented
11C2×C4⋊Dic53C2211yescharacteristic
12C2×C4⋊Dic53C2211yescharacteristic
13C2×C4⋊Dic53C2211yescharacteristic
14C2×C4⋊Dic53C2211yescharacteristic
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×C4×Dic53C2212yes—
20C2×C4×Dic53C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C42×C53C2211yescharacteristic
30#95×C53C2212yes—
31#95×C53C2212yes—
32C22×C4×C53C2312yes—
33C22×C4×C53C2312yes—
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C4⋊C4×C53—41no—
42C4⋊C4×C53—41no—
43C42×C53C2×C414yes—
44C4⋊Dic53—41no—
45C4⋊Dic53—41no—
46C4⋊Dic53C2×C414yes—
47C4⋊Dic53C2×C414yes—
48#11719919—42no—
49#11719919—42no—
50#11719919C2×C418yes—
51#11719919C2×C418yes—
52C4×Dic53—42no—
53C4×Dic53—42no—
54C23×C53C2411yescharacteristic
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53C22×C412yes—
68C2×C4×C53C22×C412yes—
69C2×C4×C53C22×C412yes—
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×Dic53—22no—
80C2×Dic53C22×C414yes—
81C2×Dic53C22×C414yes—
82C22×C53C23×C411yescharacteristic
83C22×C53H(5,𝔽2)11yescharacteristic
84C22×C53H(5,𝔽2)11yescharacteristic
85C22×C532−1+411yescharacteristic
86C22×C532−1+411yescharacteristic
87C22×C532−1+411yescharacteristic
88C22×C532−1+411yescharacteristic
89C4×C53—22no—
90C4×C53—22no—
91C4×C53—22no—
92C4×C53—42no—
93C4×C53—42no—
94Dic53—24no—
95Dic53—24no—
96Dic53—42no—
97Dic53—42no—
98#2382—531nomaximal, Sylow
99C2×C53#51911yescharacteristic
100C2×C53#52011yescharacteristic
101C2×C53#52011yescharacteristic
102C2×C53#56211yescharacteristic
103C2×C53#56311yescharacteristic
104C2×C53#56311yescharacteristic
105C2×C53#56411yescharacteristic
106C2×#125—531no—
107#389—532no—
108#389—532no—
109#387—532no—
110#387—532no—
111#387—532no—
112#387—532no—
113#382—531no—
114#382D5311yescharacteristic
115C53#238211yescharacteristic, complemented, Sylow
116#125—538no—
117C2×C4⋊C4—532no—
118C2×C4⋊C4—532no—
119C2×C4⋊C4—531no—
120C2×C4⋊C4—531no—
121C2×C4⋊C4—532no—
122C2×C4⋊C4—532no—
123C2×C4⋊C4—531no—
124C2×C4⋊C4—531no—
125C2×C4⋊C4C2×D5311yescharacteristic
126C2×C4⋊C4C2×D5311yescharacteristic
127C2×C42—532no—
128C2×C42—532no—
129C2×C42C2×D5311yescharacteristic
130#95—532no—
131#95—532no—
132#95—532no—
133#95—532no—
134#95—532no—
135#95—532no—
136#95C2×D5312yes—
137#95C2×D5312yes—
138C22×C4—532no—
139C22×C4—532no—
140C22×C4—532no—
141C22×C4—532no—
142C22×C4C22×D5312yes—
143C22×C4C22×D5312yes—
144C22×C4C22×D5311yescharacteristic
145C22×C4C22×D5311yescharacteristic
146C22×C4C22×D5311yescharacteristic
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—538no—
150C4⋊C4—538no—
151C4⋊C4—534no—
152C4⋊C4—534no—
153C4⋊C4—2122no—
154C4⋊C4—2122no—
155C4⋊C4—2121no—
156C4⋊C4—2121no—
157C42—2122no—
158C42—2122no—
159C42C4×D5314yes—
160C23C23×D5311yescharacteristic
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—534no—
174C2×C4—534no—
175C2×C4—1062no—
176C2×C4—1062no—
177C2×C4—1062no—
178C2×C4—1062no—
179C2×C4—1062no—
180C2×C4—1062no—
181C2×C4—1062no—
182C2×C4—1062no—
183C2×C4—1062no—
184C2×C4—1062no—
185C2×C4C2×C4×D5312yes—
186C2×C4C2×C4×D5312yes—
187C2×C4C2×C4×D5312yes—
188C22C22×C4×D5311yescharacteristic
189C22#1313326211yescharacteristic
190C22#1313326311yescharacteristic
191C22#1313326411yescharacteristic
192C22#1313326411yescharacteristic
193C22#1313326411yescharacteristic
194C22#1313326511yescharacteristic
195C4—22no—
196C4—22no—
197C4—22no—
198C4—42no—
199C4—42no—
200C4—1064no—
201C4—1064no—
202C4—2122no—
203C4—2122no—
204C2#1681110011yescharacteristic
205C2#1681110311yescharacteristic
206C2#1681110411yescharacteristic
207C2#1681136711yescharacteristic
208C2#1681137311yescharacteristic
209C2#1681137711yescharacteristic
210C2C53⋊#56411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#382C2
socleC23×C53C24
hypercenter#382D53
hypocenterC53#2382
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
10. #13133209C4semidirect
115. C53#2382semidirect