Finite Group Databasebeta

← #20282658

Subgroups — #20282658

Order 6784.

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

normal subgroups only

This group has 480 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#16810565C212yesmaximal, complemented
2#16810535C212yesmaximal, complemented
3#16810535C212yesmaximal, complemented
4#16810527C212yesmaximal, complemented
5#16810526C211yesmaximal, characteristic
6#16810503C211yesmaximal, characteristic
7#16810496C212yesmaximal, complemented
8#16810495C212yesmaximal, complemented
9C53×#386C211yesmaximal, characteristic
10C22×Dic106C2212yes—
11C23×Dic53C2211yescharacteristic
12C2×C4⋊Dic53C2211yescharacteristic
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×C22⋊Dic53C2211yescharacteristic
19C2×C4×Dic53C2211yescharacteristic
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26C22×D4×C53C2211yescharacteristic
27C23×C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2212yes—
30#95×C53C2212yes—
31C24×C53C2311yescharacteristic
32C24×C53C2311yescharacteristic
33C2×D4×C53—41no—
34C2×D4×C53—41no—
35C2×D4×C53Q814yescomplemented
36C22×C4×C53—41no—
37C22×C4×C53—42no—
38C22×C4×C53C2312yes—
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×Dic53—41no—
42C22×Dic53—42no—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C22×Dic53C2312yes—
47C2×Dic106—24no—
48C2×Dic106—42no—
49C2×Dic106—42no—
50C22⋊C4×C53—41no—
51C22⋊C4×C53—42no—
52C22⋊Dic53—24no—
53C22⋊Dic53—24no—
54C22⋊Dic53—41no—
55C22⋊Dic53—42no—
56C4⋊Dic53—41no—
57#11719919—42no—
58#11719919—42no—
59C4×Dic53—41no—
60C23×C53—22no—
61C23×C53—22no—
62C23×C53—21no—
63C23×C53—22no—
64C23×C53—21no—
65C23×C53—22no—
66C23×C53—21no—
67C23×C53—21no—
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×C53—44no—
81C2×C4×C53C2×Q812yes—
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—22no—
93C2×Dic53—44no—
94C2×Dic53#4114yes—
95Dic106—48no—
96C22×C53—22no—
97C22×C53—22no—
98C22×C53—24no—
99C22×C53—24no—
100C22×C53—42no—
101C22×C53—42no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53—41no—
105C22×C53—41no—
106C22×C53C22×Q811yescharacteristic
107C22×C53C2×#4112yes—
108C22×C53H(5,𝔽2)11yescharacteristic
109C22×C53H(5,𝔽2)12yes—
110C22×C53H(5,𝔽2)11yescharacteristic
111C4×C53—22no—
112C4×C53—42no—
113C4×C53—42no—
114Dic53—24no—
115Dic53—42no—
116Dic53—42no—
117Dic53—42no—
118#2804—531nomaximal, Sylow
119C2×C53—42no—
120C2×C53—42no—
121C2×C53#53812yes—
122C2×C53#54311yescharacteristic
123C2×C53#55412yes—
124C2×C53#55911yescharacteristic
125C2×C53H(3,𝔽4)11yescharacteristic
126#400—532no—
127#393—532no—
128#393—532no—
129#391—532no—
130#390—531no—
131#386—531no—
132#386D5311yescharacteristic
133#385—532no—
134#385—532no—
135C53#280411yescharacteristic, complemented, Sylow
136C22×Q8—532no—
137C22×D4C2×D5311yescharacteristic
138C23×C4—531no—
139C23×C4C2×D5311yescharacteristic
140C2×C4⋊C4—532no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—531no—
143C2×C22⋊C4—532no—
144C2×C22⋊C4—532no—
145C2×C22⋊C4—532no—
146C2×C22⋊C4—531no—
147C2×C22⋊C4C2×D5312yes—
148C2×C22⋊C4C2×D5311yescharacteristic
149C2×C42—531no—
150#95—532no—
151#95—532no—
152#95—532no—
153#95—532no—
154#95—532no—
155#95—532no—
156#95C2×D5312yes—
157C24C22×D5311yescharacteristic
158C24C22×D5311yescharacteristic
159C2×Q8—1064no—
160C2×Q8—2122no—
161C2×Q8—2122no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4Dic10614yescomplemented
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—532no—
168C22×C4—532no—
169C22×C4—532no—
170C22×C4—532no—
171C22×C4—2122no—
172C22×C4—2121no—
173C22×C4C22×D5312yes—
174C22×C4C22×D5312yes—
175C22×C4C22×D5311yescharacteristic
176C4⋊C4—2122no—
177C4⋊C4—2122no—
178C4⋊C4—2121no—
179C22⋊C4—42no—
180C22⋊C4—41no—
181C22⋊C4—1064no—
182C22⋊C4—1064no—
183C22⋊C4—2122no—
184C22⋊C4—2121no—
185C42—2121no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—22no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23C2×Dic10612yes—
195C23C2×Dic10612yes—
196C23C23×D5311yescharacteristic
197Q8—2128no—
198D4—44no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—44no—
208C2×C4—534no—
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—1062no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—2124no—
221C2×C4C2×Dic10612yes—
222C22—24no—
223C22—24no—
224C22—22no—
225C22—22no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22C22×Dic10611yescharacteristic
233C22#41×D5312yes—
234C22#1313326212yes—
235C22#1313326211yescharacteristic
236C22#1313326211yescharacteristic
237C4—22no—
238C4—42no—
239C4—42no—
240C4—1064no—
241C4—2122no—
242C4—2122no—
243C4—2122no—
244C2—42no—
245C2—42no—
246C2#1681120512yes—
247C2#1681124911yescharacteristic
248C2#1681133212yes—
249C2#1681135711yescharacteristic
250C2#1681136611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#386C2
hypercenter#386D53
hypocenterC53#2804
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. #16810565C2semidirect
2. #16810535C2semidirect
3. #16810535C2semidirect
4. #16810527C2semidirect
7. #16810496C2semidirect
8. #16810495C2semidirect
35. C2×D4×C53Q8semidirect
135. C53#2804semidirect
164. C2×D4Dic106semidirect