Finite Group Databasebeta

← #20280584

Subgroups — #20280584

Order 6784.

512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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×C4×D212C211yesmaximal, characteristic
2#16810558C212yesmaximal, complemented
3#16810553C212yesmaximal, complemented
4#16810549C212yesmaximal, complemented
5#16810540C212yesmaximal
6#16810494C214yesmaximal, complemented
7#16810491C211yesmaximal, characteristic, complemented
8C2×#125×C53C211yesmaximal, characteristic, complemented
9C22×D212C2211yescharacteristic
10C22×C4×D53C2212yes—
11C4×D212—24no—
12C2×C4⋊Dic53C2211yescharacteristic
13C2×#11719919C2214yes—
14C2×#11719921C2214yes—
15C2×#11719921C2212yes—
16C2×#11719921C2214yes—
17#13133124C2212yes—
18#13133124C2214yes—
19#13133124C2212yes—
20#13133124C2212yes—
21#125×C53—24no—
22C2×C4⋊C4×C53C2212yes—
23C2×C4⋊C4×C53C2211yescharacteristic
24C2×C4⋊C4×C53C2211yescharacteristic
25C2×C4⋊C4×C53C2212yes—
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—24no—
36C2×D212—22no—
37C2×C4×D53—24no—
38C2×C4×D53—44no—
39C4⋊C4×C53—24no—
40C4⋊C4×C53—24no—
41C4⋊C4×C53—22no—
42C4⋊C4×C53—22no—
43C42×C53D414yes—
44#11719921—28no—
45#11719921—24no—
46#11719921—44no—
47C4⋊Dic53—22no—
48#11719919—44no—
49C23×C53C2411yescharacteristic
50C22×D53—22no—
51C22×D53—22no—
52C22×D53—24no—
53C22×D53—24no—
54C22×D53#4114yes—
55C2×C4×C53—22no—
56C2×C4×C53—22no—
57C2×C4×C53—22no—
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53—24no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53C2×D412yes—
64C2×C4×C53C2×D412yes—
65C2×C4×C53C2×D412yes—
66C2×Dic53—22no—
67C2×Dic53—24no—
68C2×Dic53—24no—
69C2×Dic53—24no—
70C2×Dic53—24no—
71D212—28no—
72C4×D53—48no—
73C22×C53C22×D411yescharacteristic
74C22×C53C2×#4112yes—
75C22×C53H(5,𝔽2)12yes—
76C22×C532−1+412yes—
77C2×D53—24no—
78C2×D53—28no—
79C2×D53—42no—
80C2×D53—42no—
81C2×D53—44no—
82C4×C53—22no—
83C4×C53—22no—
84C4×C53—22no—
85C4×C53—42no—
86C4×C53—42no—
87Dic53—42no—
88Dic53—44no—
89#2633—531nomaximal, Sylow
90C2×C53#52511yescharacteristic
91C2×C53#53511yescharacteristic
92C2×C53#53611yescharacteristic
93C2×C53#53914yes—
94D53—44no—
95C2×#125D5311yescharacteristic, complemented
96C2×C4×D4—531no—
97#399—532no—
98#398—532no—
99#397—532no—
100#394—532no—
101#385—534no—
102#383—531no—
103C53#263311yescharacteristic, complemented, Sylow
104C22×D4—531no—
105C23×C4—532no—
106#125—24no—
107C4×D4—1064no—
108C2×C4⋊C4—531no—
109C2×C4⋊C4—534no—
110C2×C4⋊C4C2×D5312yes—
111C2×C4⋊C4C2×D5312yes—
112C2×C4⋊C4C2×D5311yescharacteristic
113C2×C4⋊C4C2×D5311yescharacteristic
114C2×C22⋊C4—532no—
115C2×C22⋊C4—534no—
116C2×C22⋊C4—534no—
117C2×C42C2×D5311yescharacteristic
118#95—532no—
119#95—532no—
120#95—532no—
121#95—534no—
122C24—532no—
123C2×D4—1064no—
124C2×D4—1062no—
125C22×C4—534no—
126C22×C4—532no—
127C22×C4—1064no—
128C22×C4—2124no—
129C22×C4C22×D5312yes—
130C22×C4C22×D5312yes—
131C22×C4C22×D5311yescharacteristic
132C22×C4C22×D5311yescharacteristic
133C22×C4C22×D5311yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—24no—
137C4⋊C4—24no—
138C4⋊C4—1062no—
139C4⋊C4—2124no—
140C22⋊C4—1068no—
141C22⋊C4—1064no—
142C22⋊C4—2124no—
143C42C22⋊D5314yes—
144C23—534no—
145C23—1062no—
146C23—1062no—
147C23—1064no—
148C23—1064no—
149C23C23×D5311yescharacteristic
150D4—1068no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—1062no—
160C2×C4—1064no—
161C2×C4—1064no—
162C2×C4—1064no—
163C2×C4—1064no—
164C2×C4—2128no—
165C2×C4C2×C22⋊D5312yes—
166C2×C4C2×C22⋊D5312yes—
167C2×C4C2×C22⋊D5312yes—
168C22—1064no—
169C22—1068no—
170C22—2122no—
171C22—2122no—
172C22—2124no—
173C22C22×C22⋊D5311yescharacteristic
174C22#41×D5312yes—
175C22#1313326312yes—
176C22#1313326512yes—
177C4—22no—
178C4—22no—
179C4—22no—
180C4—42no—
181C4—42no—
182C4—2122no—
183C4—2124no—
184C2—2124no—
185C2#1681113011yescharacteristic
186C2#1681118711yescharacteristic
187C2#1681119711yescharacteristic
188C2#1681122114yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#125×C53C2
hypercenterC2×#125D53
hypocenterC53#2633
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. #16810553C2semidirect
4. #16810549C2semidirect
6. #16810494C2semidirect
7. #16810491C2semidirect
8. C2×#125×C53C2semidirect
95. C2×#125D53semidirect
103. C53#2633semidirect