Finite Group Databasebeta

← #20281219

Subgroups — #20281219

Order 6784.

568 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 129 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×#13133199C212yesmaximal, complemented
2#16810565C212yesmaximal, complemented
3#16810545C212yesmaximal, complemented
4#16810532C211yesmaximal, characteristic
5#16810501C212yesmaximal, complemented
6#16810500C212yesmaximal, complemented
7#16810495C211yesmaximal, characteristic, complemented
8#16810482C212yesmaximal, complemented
9C53×#397C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C23×Dic53C2211yescharacteristic
13C22×C4×D53C2212yes—
14#13133199—28no—
15C2×C4⋊Dic53C2212yes—
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×C22⋊Dic53C2211yescharacteristic
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2212yes—
29C2×C22⋊C4×C53C2211yescharacteristic
30#95×C53C2212yes—
31#95×C53C2211yescharacteristic
32C24×C53C2311yescharacteristic
33C23×D53C2312yes—
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2312yes—
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C2×C22⋊D53—24no—
39C2×C22⋊D53—24no—
40C2×C22⋊D53—42no—
41C22×Dic53—42no—
42C22×Dic53—41no—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C2×D212—24no—
47C2×D212—41no—
48C2×C4×D53—24no—
49C2×C4×D53—24no—
50C2×C4×D53—42no—
51C4⋊C4×C53—41no—
52C22⋊C4×C53—24no—
53C22⋊C4×C53—41no—
54C22⋊Dic53D414yescomplemented
55#11719921—24no—
56#11719921—42no—
57#11719921—42no—
58#11719921—42no—
59C4⋊Dic53—24no—
60#11719919—24no—
61#11719919—42no—
62C23×C53—22no—
63C23×C53—22no—
64C23×C53—21no—
65C23×C53—21no—
66C23×C53C2×D412yes—
67C23×C53C2411yescharacteristic
68C22×D53—22no—
69C22×D53—22no—
70C22×D53—22no—
71C22×D53—22no—
72C22×D53—22no—
73C22×D53—22no—
74C22×D53#4114yes—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C22⋊D53—48no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—44no—
95C2×Dic53C2×D414yes—
96D212—44no—
97C4×D53—48no—
98C22×C53—22no—
99C22×C53—24no—
100C22×C53—42no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53C22×D411yescharacteristic
104C22×C53C2×#4112yes—
105C22×C53H(5,𝔽2)11yescharacteristic
106C22×C53H(5,𝔽2)12yes—
107C22×C532−1+411yescharacteristic
108C2×D53—24no—
109C2×D53—24no—
110C2×D53—24no—
111C2×D53—42no—
112C2×D53—42no—
113C2×D53—42no—
114C2×D53—42no—
115C4×C53—42no—
116C4×C53—42no—
117C4×C53—41no—
118C4×C53—41no—
119Dic53—24no—
120Dic53—42no—
121Dic53—42no—
122#2693—531nomaximal, Sylow
123C2×C53—42no—
124C2×C53#53711yescharacteristic
125C2×C53#53912yes—
126C2×C53#54612yes—
127C2×C53#55111yescharacteristic
128C2×C53#56011yescharacteristic
129D53—44no—
130C2×#123—532no—
131#400—532no—
132#397D5311yescharacteristic, complemented
133#396—532no—
134#392—531no—
135#386—532no—
136#386—532no—
137#385—531no—
138#380—532no—
139C53#269311yescharacteristic, complemented, Sylow
140C22×D4—532no—
141C22×D4—531no—
142C23×C4—531no—
143C23×C4—532no—
144#123—1068no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4C2×D5311yescharacteristic
149C2×C22⋊C4—531no—
150C2×C22⋊C4—532no—
151C2×C22⋊C4—532no—
152C2×C22⋊C4—532no—
153C2×C22⋊C4—532no—
154C2×C22⋊C4C2×D5312yes—
155C2×C22⋊C4C2×D5311yescharacteristic
156#95—532no—
157#95—532no—
158#95—532no—
159#95—531no—
160#95C2×D5311yescharacteristic
161#95C2×D5312yes—
162C24—532no—
163C24C22×D5311yescharacteristic
164C2×D4—1064no—
165C2×D4—1064no—
166C2×D4—1064no—
167C2×D4—2121no—
168C2×D4—2122no—
169C22×C4—532no—
170C22×C4—532no—
171C22×C4—532no—
172C22×C4—1064no—
173C22×C4—1064no—
174C22×C4—2122no—
175C22×C4—2121no—
176C22×C4—2122no—
177C22×C4C22×D5311yescharacteristic
178C22×C4C22×D5312yes—
179C22×C4C22×D5312yes—
180C22×C4C22×D5311yescharacteristic
181C4⋊C4—41no—
182C4⋊C4—1064no—
183C4⋊C4—1064no—
184C4⋊C4—2122no—
185C22⋊C4—24no—
186C22⋊C4—41no—
187C22⋊C4—534no—
188C22⋊C4—1064no—
189C22⋊C4—2122no—
190C22⋊C4—2122no—
191C22⋊C4—2122no—
192C23—22no—
193C23—21no—
194C23—22no—
195C23—21no—
196C23—534no—
197C23—1062no—
198C23—1062no—
199C23—1062no—
200C23—1062no—
201C23—1062no—
202C23—1062no—
203C23D4×D5312yes—
204C23C23×D5311yescharacteristic
205D4—2124no—
206D4—2128no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—534no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—2128no—
227C2×C4—2124no—
228C22—24no—
229C22—22no—
230C22—41no—
231C22—41no—
232C22—42no—
233C22—1064no—
234C22—1064no—
235C22—1064no—
236C22—2122no—
237C22—2122no—
238C22—2122no—
239C22—2122no—
240C22C2×D4×D5311yescharacteristic
241C22#41×D5312yes—
242C22#1313326211yescharacteristic
243C22#1313326312yes—
244C22#1313326411yescharacteristic
245C4—42no—
246C4—41no—
247C4—41no—
248C4—42no—
249C4—1064no—
250C4—2122no—
251C4—2122no—
252C2—42no—
253C2—2124no—
254C2#1681120011yescharacteristic
255C2#1681121412yes—
256C2#1681126512yes—
257C2#1681130311yescharacteristic
258C2#1681136311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2693
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#13133199C2semidirect
2. #16810565C2semidirect
3. #16810545C2semidirect
5. #16810501C2semidirect
6. #16810500C2semidirect
7. #16810495C2semidirect
8. #16810482C2semidirect
9. C53×#397C2semidirect
54. C22⋊Dic53D4semidirect
132. #397D53semidirect
139. C53#2693semidirect