Finite Group Databasebeta

← #20281928

Subgroups — #20281928

Order 6784.

464 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 137 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
1#16810559C212yesmaximal, complemented
2#16810556C212yesmaximal, complemented
3#16810546C212yesmaximal, complemented
4#16810507C212yesmaximal, complemented
5#16810503C211yesmaximal, characteristic
6#16810503C211yesmaximal, characteristic
7#16810495C212yesmaximal, complemented
8#16810481C212yesmaximal, complemented
9C53×#390C211yesmaximal, characteristic
10C23×Dic53C2211yescharacteristic
11C23×Dic53C2211yescharacteristic
12C2×C4⋊Dic53C2212yes—
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×C22⋊Dic53C2211yescharacteristic
19C2×C22⋊Dic53C2212yes—
20C2×C22⋊Dic53C2211yescharacteristic
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26C22×D4×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2212yes—
29C2×C22⋊C4×C53C2212yes—
30C2×C42×C53C2211yescharacteristic
31C24×C53C2311yescharacteristic
32C24×C53C2311yescharacteristic
33C2×D4×C53—41no—
34C2×D4×C53—41no—
35C2×D4×C53Q814yes—
36C22×C4×C53C2312yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×Dic53—24no—
40C22×Dic53—41no—
41C22×Dic53—41no—
42C22×Dic53—42no—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C22×Dic53C2312yes—
47C4⋊C4×C53—41no—
48C22⋊C4×C53—42no—
49C22⋊C4×C53—42no—
50C42×C53—41no—
51C22⋊Dic53—24no—
52C22⋊Dic53—42no—
53C22⋊Dic53—41no—
54C22⋊Dic53—41no—
55C4⋊Dic53—42no—
56#11719919—24no—
57#11719919—42no—
58#11719919—42no—
59#11719919—42no—
60C23×C53—22no—
61C23×C53—22no—
62C23×C53—21no—
63C23×C53—21no—
64C23×C53—22no—
65C23×C53—21no—
66C23×C53—21no—
67C23×C53—22no—
68C23×C53C2×Q812yes—
69C23×C53C2×Q812yes—
70C23×C53C2411yescharacteristic
71D4×C53—44no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53C2×Q812yes—
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—22no—
92C2×Dic53—44no—
93C2×Dic53—44no—
94C2×Dic53#4114yes—
95C22×C53—22no—
96C22×C53—22no—
97C22×C53—24no—
98C22×C53—24no—
99C22×C53—42no—
100C22×C53—42no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53—41no—
105C22×C53C22×Q811yescharacteristic
106C22×C53C2×#4112yes—
107C22×C53H(5,𝔽2)11yescharacteristic
108C22×C53H(5,𝔽2)12yes—
109C22×C53H(5,𝔽2)11yescharacteristic
110C4×C53—22no—
111C4×C53—42no—
112C4×C53—42no—
113Dic53—24no—
114Dic53—42no—
115Dic53—42no—
116Dic53—42no—
117#2748—531nomaximal, Sylow
118C2×C53—42no—
119C2×C53—42no—
120C2×C53#54012yes—
121C2×C53#54311yescharacteristic
122C2×C53#55111yescharacteristic
123C2×C53#55412yes—
124C2×C53#56011yescharacteristic
125#399—532no—
126#398—532no—
127#396—532no—
128#390D5311yescharacteristic
129#387—532no—
130#386—531no—
131#386—531no—
132#385—532no—
133#380—532no—
134C53#274811yescharacteristic, complemented, Sylow
135C22×D4C2×D5311yescharacteristic
136C23×C4—531no—
137C23×C4—531no—
138C2×C4⋊C4—532no—
139C2×C4⋊C4—532no—
140C2×C4⋊C4—532no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4C2×D5311yescharacteristic
144C2×C22⋊C4—532no—
145C2×C22⋊C4—532no—
146C2×C22⋊C4—531no—
147C2×C22⋊C4—531no—
148C2×C22⋊C4C2×D5312yes—
149C2×C22⋊C4C2×D5312yes—
150C2×C42C2×D5311yescharacteristic
151#95—532no—
152#95—532no—
153#95—532no—
154#95—532no—
155#95—532no—
156C24C22×D5311yescharacteristic
157C24C22×D5311yescharacteristic
158C2×D4—41no—
159C2×D4—41no—
160C2×D4Dic10614yes—
161C22×C4—532no—
162C22×C4—532no—
163C22×C4—532no—
164C22×C4—532no—
165C22×C4—1064no—
166C22×C4—2121no—
167C22×C4—2122no—
168C22×C4—2121no—
169C22×C4C22×D5312yes—
170C22×C4C22×D5312yes—
171C22×C4C22×D5311yescharacteristic
172C4⋊C4—41no—
173C4⋊C4—1064no—
174C4⋊C4—2122no—
175C4⋊C4—2122no—
176C4⋊C4—2122no—
177C4⋊C4—2122no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—1064no—
181C22⋊C4—2121no—
182C22⋊C4—2121no—
183C22⋊C4—2122no—
184C42—41no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23C2×Dic10612yes—
194C23C2×Dic10612yes—
195C23C23×D5311yescharacteristic
196D4—44no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—534no—
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—1062no—
216C2×C4—1062no—
217C2×C4—2124no—
218C2×C4—2124no—
219C2×C4C2×Dic10612yes—
220C22—24no—
221C22—24no—
222C22—22no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—42no—
229C22—42no—
230C22C22×Dic10611yescharacteristic
231C22#41×D5312yes—
232C22#1313326211yescharacteristic
233C22#1313326212yes—
234C22#1313326211yescharacteristic
235C4—22no—
236C4—42no—
237C4—42no—
238C4—1064no—
239C4—2122no—
240C4—2122no—
241C4—2122no—
242C2—42no—
243C2—42no—
244C2#1681122312yes—
245C2#1681124911yescharacteristic
246C2#1681130311yescharacteristic
247C2#1681133212yes—
248C2#1681136011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#390C2
hypercenter#390D53
hypocenterC53#2748
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #16810559C2semidirect
2. #16810556C2semidirect
3. #16810546C2semidirect
4. #16810507C2semidirect
7. #16810495C2semidirect
8. #16810481C2semidirect
134. C53#2748semidirect