Finite Group Databasebeta

← #20279204

Subgroups — #20279204

Order 6784.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133192C211yesmaximal, characteristic
2C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
3#16810552C214yesmaximal, complemented
4#16810525C211yesmaximal, characteristic, complemented
5#16810524C211yesmaximal, characteristic, complemented
6#16810501C214yesmaximal, complemented
7#16810488C211yesmaximal, characteristic, complemented
8C4×#11719921C211yesmaximal, characteristic, complemented
9C53×#382C211yesmaximal, characteristic, complemented
10C22×D212C2212yes—
11C22×D212C2211yescharacteristic
12C22×C4×D53C2211yescharacteristic
13C22×C4×D53C2212yes—
14#13133192—24no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×#11719919C2212yes—
17C4⋊C4×D53—24no—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2214yes—
21C2×#11719921C2214yes—
22C2×C4×Dic53C2211yescharacteristic
23#13133124C2214yes—
24#13133124C2212yes—
25C2×C4⋊C4×C53C2211yescharacteristic
26C2×C4⋊C4×C53C2211yescharacteristic
27C2×C42×C53C2211yescharacteristic
28#95×C53C2214yes—
29C23×D53C2312yes—
30C23×D53C2311yescharacteristic
31C22×C4×C53C2312yes—
32C22×C4×C53C2312yes—
33C22×C4×C53C2312yes—
34C22×C4×C53C2311yescharacteristic
35C22×Dic53C2312yes—
36C22×Dic53C2312yes—
37C22×Dic53C2311yescharacteristic
38C2×D212—24no—
39C2×D212—28no—
40C2×D212—22no—
41C2×D212—22no—
42C2×D212—22no—
43C2×C4×D53—22no—
44C2×C4×D53—22no—
45C2×C4×D53—24no—
46C2×C4×D53—44no—
47C2×C4×D53D414yes—
48C4⋊C4×C53—22no—
49C4⋊C4×C53—22no—
50C42×C53—22no—
51#11719921—24no—
52#11719921—24no—
53#11719921—44no—
54#11719921—44no—
55C4⋊Dic53—22no—
56#11719919—24no—
57C4×Dic53—22no—
58C23×C53C2411yescharacteristic
59C22×D53—21no—
60C22×D53—21no—
61C22×D53—22no—
62C22×D53—22no—
63C22×D53—24no—
64C22×D53—22no—
65C22×D53—22no—
66C22×D53—24no—
67C22×D53C2×D412yes—
68C22×D53#4114yes—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—24no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—24no—
75C2×C4×C53—22no—
76C2×C4×C53C2×D412yes—
77C2×C4×C53#4112yes—
78C2×C4×C53#4112yes—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—24no—
85C2×Dic53—22no—
86C2×Dic53C2×D412yes—
87D212—48no—
88D212—44no—
89C4×D53—28no—
90C4×D53—48no—
91C22×C53C22×D411yescharacteristic
92C22×C53C2×#4112yes—
93C22×C53C2×#4111yescharacteristic
94C22×C53C2×#4111yescharacteristic
95C22×C53H(5,𝔽2)12yes—
96C2×D53—22no—
97C2×D53—22no—
98C2×D53—24no—
99C2×D53—24no—
100C2×D53—42no—
101C2×D53—42no—
102C2×D53—42no—
103C2×D53—44no—
104C2×D53—44no—
105C2×D53#12414yes—
106C4×C53—42no—
107C4×C53—42no—
108C4×C53—42no—
109C4×C53#12314yes—
110Dic53—22no—
111Dic53—42no—
112Dic53—42no—
113#2504—531nomaximal, Sylow
114C2×C53C2×#12311yescharacteristic
115C2×C53C2×#12411yescharacteristic
116C2×C53#53511yescharacteristic
117C2×C53#55312yes—
118C2×C53#55312yes—
119D53—24no—
120D53—82no—
121C2×#121—531no—
122C22×C4⋊C4—531no—
123#397—534no—
124#390—531no—
125#390—531no—
126#386—534no—
127#382—531no—
128#382D5311yescharacteristic, complemented
129C4×C22⋊C4—531no—
130C53#250411yescharacteristic, complemented, Sylow
131C22×D4—532no—
132C22×D4—531no—
133C23×C4—531no—
134C23×C4—532no—
135#121—1064no—
136C2×C4⋊C4—532no—
137C2×C4⋊C4—531no—
138C2×C4⋊C4—1064no—
139C2×C4⋊C4C2×D5311yescharacteristic
140C2×C4⋊C4C2×D5311yescharacteristic
141C2×C22⋊C4—532no—
142C2×C22⋊C4—532no—
143C2×C22⋊C4—534no—
144C2×C22⋊C4—534no—
145C2×C42—531no—
146C2×C42C2×D5311yescharacteristic
147#95—532no—
148#95—534no—
149#95C2×D5314yes—
150C24—531no—
151C24—532no—
152C2×D4—1064no—
153C2×D4—1068no—
154C2×D4—1062no—
155C2×D4—1062no—
156C2×D4—1062no—
157C22×C4—531no—
158C22×C4—534no—
159C22×C4—532no—
160C22×C4—532no—
161C22×C4—1064no—
162C22×C4—1062no—
163C22×C4—1062no—
164C22×C4—2124no—
165C22×C4C22×D5312yes—
166C22×C4C22×D5312yes—
167C22×C4C22×D5312yes—
168C22×C4C22×D5311yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—1064no—
172C4⋊C4—1062no—
173C22⋊C4—1064no—
174C22⋊C4—1064no—
175C22⋊C4—2124no—
176C22⋊C4—2124no—
177C42—22no—
178C42—1062no—
179C23—532no—
180C23—534no—
181C23—1061no—
182C23—1061no—
183C23—1062no—
184C23—1062no—
185C23—1064no—
186C23—1062no—
187C23—1062no—
188C23—1064no—
189C23C23×D5311yescharacteristic
190D4—2128no—
191D4—2124no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—532no—
200C2×C4—1061no—
201C2×C4—1061no—
202C2×C4—1068no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—1064no—
208C2×C4—2128no—
209C2×C4#1171994512yes—
210C2×C4D4×D5312yes—
211C2×C4#1171994912yes—
212C22—534no—
213C22—1064no—
214C22—1064no—
215C22—1062no—
216C22—1062no—
217C22—2124no—
218C22—2124no—
219C22—2122no—
220C22—2122no—
221C22—2122no—
222C22C2×D4×D5311yescharacteristic
223C22#41×D5312yes—
224C22C2×#1171994511yescharacteristic
225C22C2×#1171994911yescharacteristic
226C22#1313326312yes—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—1062no—
231C4—2122no—
232C4—2122no—
233C4#1313320214yes—
234C2—1064no—
235C2—4242no—
236C2C2×#1313320211yescharacteristic
237C2#124×D5311yescharacteristic
238C2#1681119111yescharacteristic
239C2#1681131712yes—
240C2#1681132612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#382C2
hypercenter#382D53
hypocenterC53#2504
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D53C2semidirect
3. #16810552C2semidirect
4. #16810525C2semidirect
5. #16810524C2semidirect
6. #16810501C2semidirect
7. #16810488C2semidirect
8. C4×#11719921C2semidirect
9. C53×#382C2semidirect
128. #382D53semidirect
130. C53#2504semidirect