Finite Group Databasebeta

← #20280459

Subgroups — #20280459

Order 6784.

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

normal subgroups only

This group has 632 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×#13133198C212yesmaximal, complemented
2C22×#11719919C211yesmaximal, characteristic, complemented
3#16810554C212yesmaximal, complemented
4#16810546C214yesmaximal, complemented
5#16810537C212yesmaximal, complemented
6#16810503C212yesmaximal, complemented
7#16810483C211yesmaximal, characteristic, complemented
8C2×#121×C53C211yesmaximal, characteristic
9C23×Dic53C2211yescharacteristic
10C23×Dic53C2211yescharacteristic
11#13133198—28no—
12C2×C4⋊Dic53C2212yes—
13C2×#11719919—24no—
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2214yes—
17C2×C22⋊Dic53C2212yes—
18C2×C22⋊Dic53C2214yes—
19C2×C22⋊Dic53C2214yes—
20#13133124C2214yes—
21#13133124C2212yes—
22C22×D4×C53C2212yes—
23C22×D4×C53C2211yescharacteristic
24C23×C4×C53C2211yescharacteristic
25#121×C53—24no—
26C2×C4⋊C4×C53C2211yescharacteristic
27C2×C22⋊C4×C53C2212yes—
28C24×C53C2312yes—
29C24×C53C2311yescharacteristic
30C2×D4×C53—24no—
31C2×D4×C53—24no—
32C2×D4×C53—24no—
33C2×D4×C53—22no—
34C2×D4×C53—42no—
35C22×C4×C53—42no—
36C22×C4×C53D414yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×Dic53—22no—
41C22×Dic53—22no—
42C22×Dic53—22no—
43C22×Dic53—42no—
44C22×Dic53D414yescomplemented
45C22×Dic53C2314yes—
46C22×Dic53C2312yes—
47C22×Dic53C2312yes—
48C4⋊C4×C53—22no—
49C22⋊C4×C53—24no—
50C22⋊Dic53—24no—
51C22⋊Dic53—28no—
52C22⋊Dic53—44no—
53C4⋊Dic53—42no—
54#11719919—24no—
55#11719919—24no—
56#11719919—28no—
57C23×C53—22no—
58C23×C53—22no—
59C23×C53—22no—
60C23×C53—22no—
61C23×C53—22no—
62C23×C53—24no—
63C23×C53—24no—
64C23×C53C2×D412yes—
65C23×C53C2×D412yes—
66C23×C53#4112yes—
67C23×C53C2411yescharacteristic
68D4×C53—44no—
69D4×C53—48no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—44no—
76C2×C4×C53C2×D412yes—
77C2×C4×C53C2×D412yes—
78C2×Dic53—28no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—24no—
83C2×Dic53—24no—
84C2×Dic53—24no—
85C2×Dic53—24no—
86C2×Dic53—44no—
87C2×Dic53C2×D414yes—
88C22×C53—22no—
89C22×C53—22no—
90C22×C53—24no—
91C22×C53—24no—
92C22×C53—42no—
93C22×C53—42no—
94C22×C53—42no—
95C22×C53—44no—
96C22×C53—44no—
97C22×C53#12114yes—
98C22×C53C22×D411yescharacteristic
99C22×C53C22×D411yescharacteristic
100C22×C53C2×#4111yescharacteristic
101C22×C53H(5,𝔽2)12yes—
102C22×C532−1+412yes—
103C4×C53—22no—
104C4×C53—22no—
105C4×C53—42no—
106Dic53—24no—
107Dic53—42no—
108Dic53—44no—
109#2618—531nomaximal, Sylow
110C2×C53—24no—
111C2×C53—82no—
112C2×C53C2×#12111yescharacteristic
113C2×C53#53411yescharacteristic
114C2×C53#53611yescharacteristic
115C2×C53#53712yes—
116C2×C53#53912yes—
117C2×#123—532no—
118C2×#121D5311yescharacteristic
119C22×C4⋊C4—531no—
120#398—532no—
121#396—534no—
122#394—532no—
123#386—532no—
124#380—531no—
125C53#261811yescharacteristic, complemented, Sylow
126C22×D4C2×D5312yes—
127C22×D4C2×D5311yescharacteristic
128C23×C4—531no—
129C23×C4—531no—
130C23×C4C2×D5311yescharacteristic
131#123—1068no—
132#121—24no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—532no—
135C2×C4⋊C4—534no—
136C2×C4⋊C4—532no—
137C2×C4⋊C4—1064no—
138C2×C4⋊C4C2×D5311yescharacteristic
139C2×C22⋊C4—532no—
140C2×C22⋊C4—534no—
141C2×C22⋊C4—534no—
142C2×C22⋊C4C2×D5312yes—
143#95—532no—
144#95—534no—
145C24C22×D5312yes—
146C24C22×D5311yescharacteristic
147C2×D4—24no—
148C2×D4—24no—
149C2×D4—24no—
150C2×D4—22no—
151C2×D4—42no—
152C22×C4—42no—
153C22×C4—534no—
154C22×C4—532no—
155C22×C4—534no—
156C22×C4—532no—
157C22×C4—1062no—
158C22×C4—1062no—
159C22×C4—1062no—
160C22×C4—2122no—
161C22×C4C22⋊D5314yes—
162C22×C4C22×D5312yes—
163C22×C4C22×D5311yescharacteristic
164C22×C4C22×D5311yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—1068no—
167C4⋊C4—1064no—
168C4⋊C4—1064no—
169C4⋊C4—2122no—
170C22⋊C4—24no—
171C22⋊C4—1068no—
172C22⋊C4—1064no—
173C22⋊C4—2124no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—24no—
181C23D4×D5312yes—
182C23#1171994712yes—
183C23C2×C22⋊D5312yes—
184C23C23×D5311yescharacteristic
185D4—44no—
186D4—48no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—44no—
193C2×C4—534no—
194C2×C4—1068no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1064no—
198C2×C4—1062no—
199C2×C4—1064no—
200C2×C4—1064no—
201C2×C4—1064no—
202C2×C4—2124no—
203C2×C4C2×C22⋊D5312yes—
204C2×C4C2×C22⋊D5312yes—
205C22—22no—
206C22—22no—
207C22—24no—
208C22—24no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—44no—
213C22—44no—
214C22#1313318814yes—
215C22C2×D4×D5311yescharacteristic
216C22C22×C22⋊D5311yescharacteristic
217C22C2×#1171994711yescharacteristic
218C22#1313326212yes—
219C22#1313326412yes—
220C4—22no—
221C4—22no—
222C4—42no—
223C4—1064no—
224C4—2122no—
225C4—2124no—
226C2—24no—
227C2—82no—
228C2C2×#1313318811yescharacteristic
229C2#1681118211yescharacteristic
230C2#1681119411yescharacteristic
231C2#1681120012yes—
232C2#1681121812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#121×C53C2
hypercenterC2×#121D53
hypocenterC53#2618
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#13133198C2semidirect
2. C22×#11719919C2semidirect
3. #16810554C2semidirect
4. #16810546C2semidirect
5. #16810537C2semidirect
6. #16810503C2semidirect
7. #16810483C2semidirect
44. C22×Dic53D4semidirect
125. C53#2618semidirect