Finite Group Databasebeta

← #20280195

Subgroups — #20280195

Order 6784.

596 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13133191C211yesmaximal, characteristic
2#16810562C212yesmaximal, complemented
3#16810540C212yesmaximal, complemented
4#16810513C214yesmaximal, complemented
5#16810502C212yesmaximal, complemented
6#16810502C212yesmaximal, complemented
7C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C53C211yesmaximal, characteristic, complemented
9C22×D212C2211yescharacteristic
10C22×C22⋊D53C2212yes—
11#13133191—24no—
12C2×C4⋊Dic53C2211yescharacteristic
13C2×C22⋊Dic53C2212yes—
14C2×#11719921C2214yes—
15C2×#11719921C2212yes—
16C2×#11719921C2214yes—
17C2×C4×Dic53C2212yes—
18#13133124C2214yes—
19#13133124C2214yes—
20#13133124C2212yes—
21C23×C4×C53C2211yescharacteristic
22C23×C4×C53C2211yescharacteristic
23C23×C4×C53C2211yescharacteristic
24C2×C4⋊C4×C53—24no—
25C2×C4⋊C4×C53C2212yes—
26C2×C4⋊C4×C53C2212yes—
27C24×C53C2311yescharacteristic
28C23×D53C2312yes—
29C22×C4×C53—22no—
30C22×C4×C53—24no—
31C22×C4×C53—22no—
32C22×C4×C53—42no—
33C22×C4×C53—42no—
34C22×C4×C53D414yes—
35C22×C4×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C2×C22⋊D53—24no—
40C2×C22⋊D53—28no—
41C22×Dic53C2312yes—
42C22×Dic53C2314yes—
43C2×D212—22no—
44C2×D212—42no—
45C4⋊C4×C53—24no—
46C4⋊C4×C53—24no—
47C22⋊Dic53—24no—
48#11719921—24no—
49#11719921—28no—
50#11719921—44no—
51C4⋊Dic53—22no—
52C4×Dic53—42no—
53C23×C53—22no—
54C23×C53—22no—
55C23×C53C2×D412yes—
56C23×C53#4114yes—
57C23×C53C2411yescharacteristic
58C22×D53—22no—
59C22×D53—24no—
60C22×D53—24no—
61C22×D53—24no—
62C2×C4×C53—28no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—24no—
69C2×C4×C53—44no—
70C2×C4×C53—44no—
71C2×C4×C53C2×D412yes—
72C2×C4×C53C2×D412yes—
73C2×C4×C53#4114yes—
74C22⋊D53—48no—
75C2×Dic53—22no—
76C2×Dic53—24no—
77C2×Dic53—24no—
78C2×Dic53—24no—
79C2×Dic53—24no—
80D212—44no—
81C22×C53—22no—
82C22×C53—22no—
83C22×C53—28no—
84C22×C53#12414yes—
85C22×C53C22×D411yescharacteristic
86C22×C53C2×#4112yes—
87C22×C53C2×#4112yes—
88C22×C53H(5,𝔽2)12yes—
89C2×D53—42no—
90C2×D53—42no—
91C2×D53—42no—
92C2×D53—48no—
93C4×C53—24no—
94C4×C53—22no—
95C4×C53—22no—
96C4×C53—42no—
97Dic53—42no—
98Dic53—44no—
99#2591—531nomaximal, Sylow
100C2×C53—24no—
101C2×C53#52511yescharacteristic
102C2×C53C2×#12411yescharacteristic
103C2×C53#53411yescharacteristic
104C2×C53#55314yes—
105D53—82no—
106C2×#121—531no—
107C22×C4⋊C4D5311yescharacteristic, complemented
108#400—532no—
109#394—532no—
110#388—534no—
111#386—532no—
112#386—532no—
113C4×C22⋊C4—531no—
114C53#259111yescharacteristic, complemented, Sylow
115C22×D4—532no—
116C22×D4—531no—
117C23×C4C2×D5311yescharacteristic
118C23×C4C2×D5311yescharacteristic
119C23×C4C2×D5311yescharacteristic
120#121—1064no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—531no—
123C2×C4⋊C4C2×D5312yes—
124C2×C4⋊C4C2×D5312yes—
125C2×C22⋊C4—532no—
126C2×C22⋊C4—532no—
127C2×C22⋊C4—534no—
128C2×C22⋊C4—534no—
129C2×C42—532no—
130#95—532no—
131#95—534no—
132#95—534no—
133C24—532no—
134C24C22×D5311yescharacteristic
135C2×D4—1068no—
136C2×D4—1064no—
137C2×D4—1062no—
138C2×D4—2122no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—532no—
145C22×C4—534no—
146C22×C4C22⋊D5314yes—
147C22×C4C22×D5312yes—
148C22×C4C22×D5312yes—
149C22×C4C22×D5311yescharacteristic
150C22×C4C22×D5311yescharacteristic
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—1062no—
154C22⋊C4—1064no—
155C22⋊C4—1068no—
156C22⋊C4—1064no—
157C22⋊C4—2124no—
158C42—2122no—
159C23—22no—
160C23—22no—
161C23—1062no—
162C23—1064no—
163C23—1064no—
164C23—1064no—
165C23#1171994914yes—
166C23C2×C22⋊D5312yes—
167C23C23×D5311yescharacteristic
168D4—2124no—
169D4—2128no—
170C2×C4—28no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—44no—
178C2×C4—44no—
179C2×C4—1062no—
180C2×C4—1064no—
181C2×C4—1064no—
182C2×C4—1064no—
183C2×C4—1064no—
184C2×C4#1171994514yes—
185C2×C4C2×C22⋊D5312yes—
186C2×C4C2×C22⋊D5312yes—
187C22—28no—
188C22—22no—
189C22—22no—
190C22—2128no—
191C22—2122no—
192C22—2122no—
193C22—2122no—
194C22#1313320614yes—
195C22C22×C22⋊D5311yescharacteristic
196C22C2×#1171994512yes—
197C22C2×#1171994912yes—
198C22#1313326212yes—
199C4—24no—
200C4—22no—
201C4—22no—
202C4—42no—
203C4—2124no—
204C4—2122no—
205C2—24no—
206C2—4242no—
207C2#1681113411yescharacteristic
208C2C2×#1313320611yescharacteristic
209C2#1681118211yescharacteristic
210C2#1681132114yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC22×C4⋊C4×C53C2
hypercenterC22×C4⋊C4D53
hypocenterC53#2591
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #16810562C2semidirect
3. #16810540C2semidirect
4. #16810513C2semidirect
5. #16810502C2semidirect
6. #16810502C2semidirect
7. C4×C22⋊Dic53C2semidirect
8. C22×C4⋊C4×C53C2semidirect
107. C22×C4⋊C4D53semidirect
114. C53#2591semidirect