Finite Group Databasebeta

← #20281058

Subgroups — #20281058

Order 6784.

568 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 141 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
2C2×#13133194C212yesmaximal, complemented
3#16810557C212yesmaximal, complemented
4#16810551C212yesmaximal, complemented
5#16810543C211yesmaximal, characteristic, complemented
6#16810511C212yesmaximal, complemented
7#16810498C212yesmaximal, complemented
8#16810492C211yesmaximal, characteristic, complemented
9C53×#394C211yesmaximal, characteristic
10C22×Dic106C2211yescharacteristic
11C23×Dic53C2212yes—
12#13133198—28no—
13#13133194—28no—
14C2×C4⋊Dic53C2212yes—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×#11719919C2212yes—
20C2×C22⋊Dic53C2212yes—
21C2×C22⋊Dic53C2212yes—
22C2×C22⋊Dic53C2212yes—
23C2×C4×Dic53C2211yescharacteristic
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2211yescharacteristic
28C22×D4×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2211yescharacteristic
30C2×C22⋊C4×C53C2212yes—
31C2×C22⋊C4×C53C2212yes—
32#95×C53C2211yescharacteristic
33C24×C53C2312yes—
34C2×D4×C53—24no—
35C2×D4×C53—41no—
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×Dic53—24no—
41C22×Dic53—24no—
42C22×Dic53—42no—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C22×Dic53C2312yes—
47C2×Dic106—24no—
48C2×Dic106—41no—
49C4⋊C4×C53—41no—
50C22⋊C4×C53—24no—
51C22⋊C4×C53—24no—
52C22⋊Dic53—24no—
53C22⋊Dic53—24no—
54C22⋊Dic53D418yes—
55C4⋊Dic53—24no—
56C4⋊Dic53—41no—
57#11719919—24no—
58#11719919—24no—
59#11719919—24no—
60#11719919—24no—
61C4×Dic53—22no—
62C23×C53—22no—
63C23×C53—22no—
64C23×C53—22no—
65C23×C53—22no—
66C23×C53—22no—
67C23×C53—22no—
68C23×C53C2×D414yes—
69C23×C53C2411yescharacteristic
70D4×C53—44no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—22no—
80C2×C4×C53#4112yes—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—48no—
92C2×Dic53C2×D414yes—
93C2×Dic53C2×D414yes—
94Dic106—44no—
95C22×C53—24no—
96C22×C53—24no—
97C22×C53—24no—
98C22×C53—42no—
99C22×C53—42no—
100C22×C53—42no—
101C22×C53—42no—
102C22×C53C22×D412yes—
103C22×C53C2×#4111yescharacteristic
104C22×C53H(5,𝔽2)11yescharacteristic
105C22×C532−1+411yescharacteristic
106C22×C532−1+412yes—
107C4×C53—22no—
108C4×C53—41no—
109C4×C53—41no—
110C4×C53—42no—
111Dic53—24no—
112Dic53—24no—
113Dic53—42no—
114Dic53—42no—
115#2680—531nomaximal, Sylow
116C2×C53—44no—
117C2×C53#53612yes—
118C2×C53#53911yescharacteristic
119C2×C53D4211yescharacteristic
120C2×C53#54712yes—
121C2×C53#56311yescharacteristic
122C2×#123—532no—
123C2×C22⋊Q8—532no—
124#398—532no—
125#397—532no—
126#395—531no—
127#394D5311yescharacteristic
128#388—532no—
129#385—532no—
130#384—531no—
131C53#268011yescharacteristic, complemented, Sylow
132C22×Q8—531no—
133C22×D4C2×D5311yescharacteristic
134C23×C4—532no—
135#123—1068no—
136C22⋊Q8—1068no—
137C2×C4⋊C4—532no—
138C2×C4⋊C4—532no—
139C2×C4⋊C4—532no—
140C2×C4⋊C4—532no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—531no—
143C2×C4⋊C4C2×D5311yescharacteristic
144C2×C22⋊C4—532no—
145C2×C22⋊C4—532no—
146C2×C22⋊C4—532no—
147C2×C22⋊C4C2×D5312yes—
148C2×C22⋊C4C2×D5312yes—
149C2×C42—531no—
150#95—532no—
151#95—532no—
152#95—532no—
153#95—531no—
154#95C2×D5311yescharacteristic
155C24C22×D5312yes—
156C2×Q8—1064no—
157C2×Q8—2121no—
158C2×D4—24no—
159C2×D4—41no—
160C22×C4—532no—
161C22×C4—532no—
162C22×C4—532no—
163C22×C4—532no—
164C22×C4—1064no—
165C22×C4—1064no—
166C22×C4—2122no—
167C22×C4C22×D5312yes—
168C22×C4C22×D5311yescharacteristic
169C22×C4C22×D5311yescharacteristic
170C22×C4C22×D5311yescharacteristic
171C4⋊C4—41no—
172C4⋊C4—1064no—
173C4⋊C4—1064no—
174C4⋊C4—1064no—
175C4⋊C4—1064no—
176C4⋊C4—1064no—
177C4⋊C4—2121no—
178C22⋊C4—24no—
179C22⋊C4—24no—
180C22⋊C4—538no—
181C22⋊C4—1064no—
182C22⋊C4—1064no—
183C42—1062no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23D4×D5314yes—
191C23C23×D5311yescharacteristic
192Q8—2124no—
193D4—44no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—534no—
204C2×C4—534no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—1062no—
209C2×C4—1062no—
210C2×C4—1062no—
211C2×C4—1062no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1062no—
215C2×C4—2128no—
216C2×C4#1171994712yes—
217C22—24no—
218C22—24no—
219C22—24no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22C2×D4×D5312yes—
225C22C2×#1171994711yescharacteristic
226C22#1313326211yescharacteristic
227C22#1313326412yes—
228C22#1313326411yescharacteristic
229C4—22no—
230C4—42no—
231C4—41no—
232C4—41no—
233C4—1064no—
234C4—1064no—
235C4—2122no—
236C4—2122no—
237C2—44no—
238C2#1681119512yes—
239C2#1681121811yescharacteristic
240C2#1681125511yescharacteristic
241C2#1681127312yes—
242C2#1681137311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2680
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#13133194C2semidirect
3. #16810557C2semidirect
4. #16810551C2semidirect
5. #16810543C2semidirect
6. #16810511C2semidirect
7. #16810498C2semidirect
8. #16810492C2semidirect
131. C53#2680semidirect