Finite Group Databasebeta

← #20277521

Subgroups — #20277521

Order 6784.

428 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 217 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#16810522C212yesmaximal
2#16810509C212yesmaximal
3#16810509C212yesmaximal
4#16810509C212yesmaximal
5#16810509C212yesmaximal
6#16810491C211yesmaximal, characteristic
7C4⋊C4×Dic53C212yesmaximal
8C4×C4⋊Dic53C211yesmaximal, characteristic
9C2×#125×C53C211yesmaximal, characteristic
10C2×C4⋊Dic53C2212yes—
11C2×C4⋊Dic53C2212yes—
12C2×C4⋊Dic53C2212yes—
13C2×C4×Dic53C2212yes—
14C2×C4×Dic53C2211yescharacteristic
15C2×C4×Dic53C2212yes—
16C2×C4×Dic53C2211yescharacteristic
17#13133124C2212yes—
18#13133124C2212yes—
19#13133124C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2211yescharacteristic
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26#125×C53C418yescomplemented
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2212yes—
30C2×C4⋊C4×C53C2212yes—
31C2×C42×C53C2211yescharacteristic
32C22×C4×C53C2311yescharacteristic
33C22×C4×C53C2311yescharacteristic
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2312yes—
36C22×C4×C53C2312yes—
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C4⋊C4×C53C2×C414yes—
42C4⋊C4×C53C2×C414yes—
43C4⋊C4×C53C2×C418yes—
44C4⋊C4×C53C2×C418yes—
45C42×C53C2×C414yes—
46C4⋊Dic53—24no—
47C4⋊Dic53—24no—
48C4⋊Dic53—42no—
49C4×Dic53—24no—
50C4×Dic53—41no—
51C4×Dic53—42no—
52C4×Dic53—41no—
53C23×C53C2411yescharacteristic
54C2×C4×C53—22no—
55C2×C4×C53—22no—
56C2×C4×C53—22no—
57C2×C4×C53—21no—
58C2×C4×C53—22no—
59C2×C4×C53—21no—
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53C22×C412yes—
63C2×C4×C53C22×C412yes—
64C2×C4×C53C22×C412yes—
65C2×C4×C53C22×C414yes—
66C2×C4×C53C22×C414yes—
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#4114yes—
79C22×C53C23×C411yescharacteristic
80C22×C53C2×#4112yes—
81C22×C53H(5,𝔽2)11yescharacteristic
82C22×C532−1+411yescharacteristic
83C22×C532−1+412yes—
84C4×C53—24no—
85C4×C53—22no—
86C4×C53—22no—
87C4×C53—22no—
88C4×C53—24no—
89Dic53—24no—
90Dic53—42no—
91Dic53—42no—
92Dic53—42no—
93#2336—531nomaximal, Sylow
94C2×C53C4×#4111yescharacteristic
95C2×C53#51911yescharacteristic
96C2×C53#52011yescharacteristic
97C2×C53#53912yes—
98C2×C53#54112yes—
99C2×#125D5311yescharacteristic
100#389—532no—
101#387—532no—
102#387—532no—
103#387—532no—
104#387—532no—
105#383—531no—
106C4×C4⋊C4—532no—
107C4×C4⋊C4—531no—
108C53#233611yescharacteristic, complemented, Sylow
109#125Dic5318yescomplemented
110C2×C4⋊C4—532no—
111C2×C4⋊C4—532no—
112C2×C4⋊C4—532no—
113C2×C4⋊C4C2×D5312yes—
114C2×C4⋊C4C2×D5311yescharacteristic
115C2×C4⋊C4C2×D5311yescharacteristic
116C2×C4⋊C4C2×D5312yes—
117C2×C42—532no—
118C2×C42—531no—
119C2×C42—531no—
120C2×C42—532no—
121C2×C42C2×D5311yescharacteristic
122#95—532no—
123#95—532no—
124#95—532no—
125#95—531no—
126#95—532no—
127#95—531no—
128#95—532no—
129#95—532no—
130#95—532no—
131C22×C4—532no—
132C22×C4—532no—
133C22×C4—532no—
134C22×C4—532no—
135C22×C4C22×D5311yescharacteristic
136C22×C4C22×D5311yescharacteristic
137C22×C4C22×D5312yes—
138C22×C4C22×D5312yes—
139C22×C4C22×D5311yescharacteristic
140C4⋊C4—1064no—
141C4⋊C4—1064no—
142C4⋊C4—2122no—
143C4⋊C4C2×Dic5318yes—
144C4⋊C4C2×Dic5314yes—
145C4⋊C4C2×Dic5314yes—
146C4⋊C4C2×Dic5318yes—
147C42—1064no—
148C42—2121no—
149C42—2121no—
150C42—2122no—
151C42C2×Dic5314yes—
152C23C23×D5311yescharacteristic
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—534no—
162C2×C4—1062no—
163C2×C4—1062no—
164C2×C4—1062no—
165C2×C4—1062no—
166C2×C4—1062no—
167C2×C4—1062no—
168C2×C4—1062no—
169C2×C4—1062no—
170C2×C4—1062no—
171C2×C4—1062no—
172C2×C4—1062no—
173C2×C4C22×Dic5312yes—
174C2×C4C22×Dic5312yes—
175C2×C4C22×Dic5312yes—
176C2×C4C22×Dic5314yes—
177C2×C4C22×Dic5314yes—
178C22C23×Dic5311yescharacteristic
179C22#41×D5312yes—
180C22#1313326311yescharacteristic
181C22#1313326411yescharacteristic
182C22#1313326512yes—
183C4—24no—
184C4—22no—
185C4—22no—
186C4—22no—
187C4—24no—
188C4—1064no—
189C4—2122no—
190C4—2122no—
191C4—2122no—
192C2#41×Dic5311yescharacteristic
193C2#1681110111yescharacteristic
194C2#1681110611yescharacteristic
195C2#1681122112yes—
196C2#1681123212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC2×#125×C53C2
socleC23×C53C24
hypercenterC2×#125D53
hypocenterC53#2336
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
26. #125×C53C4semidirect
108. C53#2336semidirect
109. #125Dic53semidirect