Finite Group Databasebeta

← #20282765

Subgroups — #20282765

Order 6784.

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

normal subgroups only

This group has 464 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×#13133205C211yesmaximal, characteristic
2#16810558C212yesmaximal, complemented
3#16810549C212yesmaximal, complemented
4#16810545C214yesmaximal, complemented
5#16810543C212yesmaximal, complemented
6#16810540C212yesmaximal
7#16810489C211yesmaximal, characteristic, complemented
8C2×#125×C53C211yesmaximal, characteristic, complemented
9C22×Dic106C2211yescharacteristic
10C22×D212C2211yescharacteristic
11#13133205—24no—
12C2×C4⋊Dic53C2212yes—
13C2×#11719919C2214yes—
14C2×#11719921C2212yes—
15C2×#11719921C2212yes—
16C2×#11719921C2214yes—
17C2×#11719921C2214yes—
18#13133124C2212yes—
19#13133124C2214yes—
20#13133124C2212yes—
21#125×C53—24no—
22C2×C4⋊C4×C53C2212yes—
23C2×C4⋊C4×C53C2212yes—
24C2×C4⋊C4×C53C2211yescharacteristic
25C2×C4⋊C4×C53C2211yescharacteristic
26C2×C42×C53C2211yescharacteristic
27C23×D53C2312yes—
28C22×C4×C53C2312yes—
29C22×C4×C53C2312yes—
30C22×C4×C53C2311yescharacteristic
31C22×C4×C53C2311yescharacteristic
32C22×C4×C53C2311yescharacteristic
33C22×Dic53C2314yes—
34C22×Dic53C2312yes—
35C2×D212—22no—
36C2×D212—42no—
37C2×Dic106—22no—
38C2×Dic106—42no—
39C4⋊C4×C53—24no—
40C4⋊C4×C53—24no—
41C4⋊C4×C53—22no—
42C4⋊C4×C53—22no—
43C42×C53D414yes—
44#11719921—24no—
45#11719921—24no—
46#11719921—44no—
47#11719921—44no—
48C4⋊Dic53—42no—
49#11719919—44no—
50C23×C53C2411yescharacteristic
51C22×D53—22no—
52C22×D53—24no—
53C22×D53—22no—
54C22×D53—22no—
55C22×D53—24no—
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—24no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53C2×D412yes—
66C2×C4×C53C2×D412yes—
67C2×C4×C53C2×D412yes—
68C2×Dic53—22no—
69C2×Dic53—24no—
70C2×Dic53—24no—
71C2×Dic53—24no—
72C2×Dic53—24no—
73D212—44no—
74Dic106—44no—
75C22×C53C22×D411yescharacteristic
76C22×C53H(5,𝔽2)12yes—
77C22×C53H(5,𝔽2)11yescharacteristic
78C22×C532−1+411yescharacteristic
79C22×C532−1+412yes—
80C2×D53—42no—
81C2×D53—42no—
82C2×D53—42no—
83C2×D53—44no—
84C2×D53—44no—
85C4×C53—22no—
86C4×C53—22no—
87C4×C53—22no—
88C4×C53—42no—
89C4×C53—42no—
90Dic53—42no—
91Dic53—44no—
92#2820—531nomaximal, Sylow
93C2×C53#53511yescharacteristic
94C2×C53#53611yescharacteristic
95C2×C53#53711yescharacteristic
96C2×C53#56212yes—
97C2×C53#56312yes—
98D53—82no—
99C2×#125D5311yescharacteristic, complemented
100C2×#124—531no—
101#399—532no—
102#397—532no—
103#396—534no—
104#395—532no—
105#394—532no—
106#382—531no—
107C53#282011yescharacteristic, complemented, Sylow
108C22×Q8—531no—
109C22×D4—531no—
110#125—24no—
111#124—1064no—
112C2×C4⋊C4—534no—
113C2×C4⋊C4—532no—
114C2×C4⋊C4C2×D5312yes—
115C2×C4⋊C4C2×D5312yes—
116C2×C4⋊C4C2×D5311yescharacteristic
117C2×C4⋊C4C2×D5311yescharacteristic
118C2×C22⋊C4—532no—
119C2×C22⋊C4—532no—
120C2×C22⋊C4—534no—
121C2×C22⋊C4—534no—
122C2×C42C2×D5311yescharacteristic
123#95—532no—
124#95—532no—
125#95—534no—
126C24—532no—
127C2×Q8—1062no—
128C2×Q8—2122no—
129C2×D4—1062no—
130C2×D4—2122no—
131C22×C4—534no—
132C22×C4—532no—
133C22×C4C22×D5312yes—
134C22×C4C22×D5312yes—
135C22×C4C22×D5311yescharacteristic
136C22×C4C22×D5311yescharacteristic
137C22×C4C22×D5311yescharacteristic
138C4⋊C4—24no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—24no—
142C4⋊C4—2124no—
143C4⋊C4—2122no—
144C22⋊C4—1064no—
145C22⋊C4—1064no—
146C22⋊C4—2124no—
147C22⋊C4—2124no—
148C42C22⋊D5314yes—
149C23—1062no—
150C23—1062no—
151C23—1064no—
152C23—1062no—
153C23—1064no—
154C23C23×D5311yescharacteristic
155Q8—2124no—
156D4—2124no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—1062no—
167C2×C4—1064no—
168C2×C4—1064no—
169C2×C4—1064no—
170C2×C4—1064no—
171C2×C4C2×C22⋊D5312yes—
172C2×C4C2×C22⋊D5312yes—
173C2×C4C2×C22⋊D5312yes—
174C22—2124no—
175C22—2124no—
176C22—2122no—
177C22—2122no—
178C22—2122no—
179C22C22×C22⋊D5311yescharacteristic
180C22#1313326311yescharacteristic
181C22#1313326312yes—
182C22#1313326411yescharacteristic
183C22#1313326512yes—
184C4—22no—
185C4—22no—
186C4—22no—
187C4—42no—
188C4—42no—
189C4—2124no—
190C4—2122no—
191C2—4242no—
192C2#1681118711yescharacteristic
193C2#1681119711yescharacteristic
194C2#1681120111yescharacteristic
195C2#1681137112yes—
196C2#1681137712yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#125×C53C2
hypercenterC2×#125D53
hypocenterC53#2820
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #16810558C2semidirect
3. #16810549C2semidirect
4. #16810545C2semidirect
5. #16810543C2semidirect
7. #16810489C2semidirect
8. C2×#125×C53C2semidirect
99. C2×#125D53semidirect
107. C53#2820semidirect