Finite Group Databasebeta

← #20280302

Subgroups — #20280302

Order 6784.

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

normal subgroups only

This group has 604 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×C4×C22⋊D53C212yesmaximal, complemented
2#16810548C211yesmaximal, characteristic, complemented
3#16810539C212yesmaximal, complemented
4#16810532C211yesmaximal, characteristic
5#16810515C212yesmaximal, complemented
6#16810508C212yesmaximal, complemented
7#16810501C212yesmaximal, complemented
8#16810497C212yesmaximal, complemented
9C53×#381C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C22×C4×D53C2212yes—
13C4×C22⋊D53—28no—
14C2×C4⋊Dic53C2212yes—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×#11719919C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×C4×Dic53C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26#13133124C2211yescharacteristic
27C23×C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2211yescharacteristic
30#95×C53C2212yes—
31#95×C53C2212yes—
32C24×C53C2311yescharacteristic
33C23×D53C2312yes—
34C22×C4×C53—41no—
35C22×C4×C53—41no—
36C22×C4×C53D414yescomplemented
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C2×C22⋊D53—24no—
42C2×C22⋊D53—24no—
43C2×C22⋊D53—24no—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C22×Dic53C2312yes—
47C2×D212—24no—
48C2×D212—41no—
49C2×C4×D53—24no—
50C2×C4×D53—42no—
51C2×C4×D53—42no—
52C22⋊C4×C53—41no—
53C22⋊C4×C53—41no—
54C22⋊Dic53—24no—
55#11719921—24no—
56#11719921—24no—
57#11719921—24no—
58#11719921—42no—
59C4⋊Dic53—24no—
60C4⋊Dic53—41no—
61#11719919—24no—
62C4×Dic53—24no—
63C23×C53—22no—
64C23×C53—22no—
65C23×C53—21no—
66C23×C53—21no—
67C23×C53C2×D412yes—
68C23×C53C2411yescharacteristic
69C22×D53—22no—
70C22×D53—22no—
71C22×D53—22no—
72C22×D53—22no—
73C22×D53—22no—
74C22×D53—22no—
75C22×D53#4114yes—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—44no—
86C2×C4×C53C2×D414yes—
87C22⋊D53—216no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53#4114yes—
97D212—44no—
98C4×D53—48no—
99C22×C53—24no—
100C22×C53—22no—
101C22×C53—41no—
102C22×C53—42no—
103C22×C53—41no—
104C22×C53C22×D411yescharacteristic
105C22×C53C2×#4112yes—
106C22×C53C2×#4112yes—
107C22×C53H(5,𝔽2)11yescharacteristic
108C22×C53H(5,𝔽2)11yescharacteristic
109C2×D53—24no—
110C2×D53—24no—
111C2×D53—24no—
112C2×D53—42no—
113C2×D53—42no—
114C2×D53—42no—
115C2×D53—42no—
116C4×C53—24no—
117C4×C53—41no—
118C4×C53—42no—
119C4×C53—41no—
120Dic53—24no—
121Dic53—42no—
122Dic53—42no—
123#2600—531nomaximal, Sylow
124C2×C53—42no—
125C2×C53#52512yes—
126C2×C53#53411yescharacteristic
127C2×C53#55111yescharacteristic
128C2×C53#55312yes—
129C2×C53#55911yescharacteristic
130D53—44no—
131C2×C4×D4—532no—
132#397—531no—
133#394—532no—
134#392—531no—
135#388—532no—
136#387—532no—
137#386—532no—
138#385—532no—
139#381D5311yescharacteristic, complemented
140C53#260011yescharacteristic, complemented, Sylow
141C22×D4—532no—
142C22×D4—531no—
143C23×C4—532no—
144C23×C4C2×D5311yescharacteristic
145C4×D4—1068no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4—531no—
149C2×C22⋊C4—532no—
150C2×C22⋊C4—532no—
151C2×C22⋊C4—532no—
152C2×C22⋊C4—532no—
153C2×C22⋊C4—532no—
154C2×C22⋊C4C2×D5311yescharacteristic
155C2×C22⋊C4C2×D5311yescharacteristic
156C2×C42—532no—
157#95—532no—
158#95—532no—
159#95—531no—
160#95—531no—
161#95C2×D5312yes—
162#95C2×D5312yes—
163C24—532no—
164C24C22×D5311yescharacteristic
165C2×D4—1064no—
166C2×D4—1064no—
167C2×D4—1064no—
168C2×D4—1064no—
169C2×D4—2121no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—532no—
173C22×C4—532no—
174C22×C4—532no—
175C22×C4—1064no—
176C22×C4—2122no—
177C22×C4—2122no—
178C22×C4D21214yescomplemented
179C22×C4C22×D5311yescharacteristic
180C22×C4C22×D5311yescharacteristic
181C22×C4C22×D5312yes—
182C22×C4C22×D5312yes—
183C4⋊C4—1064no—
184C4⋊C4—1064no—
185C4⋊C4—2121no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—1064no—
189C22⋊C4—1064no—
190C22⋊C4—1064no—
191C22⋊C4—1064no—
192C22⋊C4—2122no—
193C42—1064no—
194C23—22no—
195C23—22no—
196C23—21no—
197C23—21no—
198C23—534no—
199C23—1062no—
200C23—1062no—
201C23—1062no—
202C23—1062no—
203C23—1062no—
204C23—1062no—
205C23C2×D21212yes—
206C23C23×D5311yescharacteristic
207D4—10616no—
208D4—2124no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—44no—
219C2×C4—534no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—2128no—
229C2×C4C2×D21214yes—
230C22—24no—
231C22—22no—
232C22—41no—
233C22—42no—
234C22—41no—
235C22—1064no—
236C22—1064no—
237C22—1064no—
238C22—2122no—
239C22—2122no—
240C22—2122no—
241C22—2122no—
242C22C22×D21211yescharacteristic
243C22#41×D5312yes—
244C22#41×D5312yes—
245C22#1313326211yescharacteristic
246C22#1313326211yescharacteristic
247C4—24no—
248C4—41no—
249C4—42no—
250C4—41no—
251C4—1064no—
252C4—2122no—
253C4—2122no—
254C2—42no—
255C2—2124no—
256C2#1681113512yes—
257C2#1681118511yescharacteristic
258C2#1681130111yescharacteristic
259C2#1681131812yes—
260C2#1681135711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#381C2
hypercenter#381D53
hypocenterC53#2600
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C2×C4×C22⋊D53C2semidirect
2. #16810548C2semidirect
3. #16810539C2semidirect
5. #16810515C2semidirect
6. #16810508C2semidirect
7. #16810501C2semidirect
8. #16810497C2semidirect
9. C53×#381C2semidirect
36. C22×C4×C53D4semidirect
139. #381D53semidirect
140. C53#2600semidirect
178. C22×C4D212semidirect