Finite Group Databasebeta

← #20282759

Subgroups — #20282759

Order 6784.

580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13133203C212yesmaximal, complemented
2#16810540C211yesmaximal, characteristic, complemented
3#16810539C212yesmaximal, complemented
4#16810536C212yesmaximal, complemented
5#16810535C212yesmaximal, complemented
6#16810532C212yesmaximal, complemented
7#16810531C212yesmaximal, complemented
8#16810488C211yesmaximal, characteristic, complemented
9C53×#394C211yesmaximal, characteristic, complemented
10C22×Dic106C2212yes—
11C22×D212C2211yescharacteristic
12C22×C22⋊D53C2212yes—
13C22×C22⋊D53C2212yes—
14#13133203—28no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×C4×Dic53C2211yescharacteristic
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26#13133124C2212yes—
27C22×D4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2212yes—
30C2×C22⋊C4×C53C2212yes—
31#95×C53C2211yescharacteristic
32C24×C53C2312yes—
33C23×D53C2312yes—
34C2×D4×C53—42no—
35C2×D4×C53—41no—
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C2×C22⋊D53—24no—
41C2×C22⋊D53—42no—
42C2×C22⋊D53—42no—
43C2×C22⋊D53—42no—
44C2×C22⋊D53—42no—
45C2×C22⋊D53—42no—
46C22×Dic53C2312yes—
47C22×Dic53C2312yes—
48C22×Dic53C2312yes—
49C2×D212—42no—
50C2×D212—41no—
51C2×Dic106—24no—
52C2×Dic106—42no—
53C2×Dic106—42no—
54C4⋊C4×C53—41no—
55C22⋊C4×C53—24no—
56C22⋊C4×C53—42no—
57C22⋊Dic53—24no—
58C22⋊Dic53—42no—
59#11719921—24no—
60#11719921—24no—
61#11719921—42no—
62#11719921—42no—
63C4⋊Dic53—41no—
64C4×Dic53D414yescomplemented
65C23×C53—22no—
66C23×C53—22no—
67C23×C53—22no—
68C23×C53—22no—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53C2411yescharacteristic
73C22×D53—22no—
74C22×D53—22no—
75C22×D53—22no—
76C22×D53—22no—
77C22×D53—22no—
78C22×D53—22no—
79C22×D53—22no—
80D4×C53—44no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53C2×D412yes—
91C22⋊D53—48no—
92C22⋊D53—48no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53C2×D414yes—
102D212—44no—
103Dic106—48no—
104C22×C53—42no—
105C22×C53—42no—
106C22×C53—42no—
107C22×C53—42no—
108C22×C53—42no—
109C22×C53—42no—
110C22×C53—42no—
111C22×C53C22×D411yescharacteristic
112C22×C53H(5,𝔽2)12yes—
113C22×C53H(5,𝔽2)11yescharacteristic
114C22×C53H(5,𝔽2)12yes—
115C22×C532−1+411yescharacteristic
116C2×D53—42no—
117C2×D53—42no—
118C2×D53—42no—
119C2×D53—42no—
120C2×D53—42no—
121C2×D53—42no—
122C2×D53—42no—
123C4×C53—22no—
124C4×C53—42no—
125C4×C53—41no—
126C4×C53—41no—
127Dic53—24no—
128Dic53—42no—
129Dic53—42no—
130#2819—531nomaximal, Sylow
131C2×C53—82no—
132C2×C53#53512yes—
133C2×C53#53711yescharacteristic
134C2×C53H(3,𝔽4)11yescharacteristic
135C2×C53H(3,𝔽4)11yescharacteristic
136C2×C53#56212yes—
137D53—82no—
138C2×#124—532no—
139#394—531no—
140#394—532no—
141#394D5311yescharacteristic, complemented
142#393—532no—
143#393—532no—
144#392—532no—
145#392—532no—
146#382—531no—
147C53#281911yescharacteristic, complemented, Sylow
148C22×Q8—532no—
149C22×D4—532no—
150C22×D4—532no—
151C22×D4—531no—
152C22×D4C2×D5311yescharacteristic
153#124—1068no—
154C2×C4⋊C4—531no—
155C2×C4⋊C4C2×D5311yescharacteristic
156C2×C22⋊C4—532no—
157C2×C22⋊C4—532no—
158C2×C22⋊C4—532no—
159C2×C22⋊C4—532no—
160C2×C22⋊C4—532no—
161C2×C22⋊C4—532no—
162C2×C22⋊C4C2×D5312yes—
163C2×C22⋊C4C2×D5312yes—
164C2×C42—531no—
165#95—532no—
166#95—532no—
167#95—532no—
168#95—531no—
169#95C2×D5311yescharacteristic
170C24—532no—
171C24C22×D5312yes—
172C2×Q8—1064no—
173C2×Q8—2122no—
174C2×Q8—2122no—
175C2×D4—42no—
176C2×D4—41no—
177C2×D4—1064no—
178C2×D4—2122no—
179C2×D4—2121no—
180C2×D4—2122no—
181C2×D4—2122no—
182C2×D4—2122no—
183C2×D4—2122no—
184C2×D4—2122no—
185C22×C4—532no—
186C22×C4—532no—
187C22×C4—532no—
188C22×C4C22×D5311yescharacteristic
189C22×C4C22×D5311yescharacteristic
190C22×C4C22×D5312yes—
191C22×C4C22×D5311yescharacteristic
192C4⋊C4—41no—
193C4⋊C4—2121no—
194C22⋊C4—24no—
195C22⋊C4—42no—
196C22⋊C4—1064no—
197C22⋊C4—1064no—
198C22⋊C4—1064no—
199C22⋊C4—2122no—
200C22⋊C4—2122no—
201C22⋊C4—2122no—
202C42—534no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—1062no—
211C23—1062no—
212C23—1062no—
213C23—1062no—
214C23—1062no—
215C23—1062no—
216C23—1062no—
217C23C23×D5311yescharacteristic
218Q8—2128no—
219D4—44no—
220D4—2124no—
221D4—2128no—
222D4—2128no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—534no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4D4×D5312yes—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—2122no—
250C22—2122no—
251C22—2122no—
252C22—2122no—
253C22—2122no—
254C22—2122no—
255C22—2122no—
256C22C2×D4×D5311yescharacteristic
257C22#1313326211yescharacteristic
258C22#1313326212yes—
259C22#1313326312yes—
260C22#1313326411yescharacteristic
261C4—22no—
262C4—41no—
263C4—41no—
264C4—42no—
265C4—1064no—
266C4—2122no—
267C4—2122no—
268C2—82no—
269C2—4242no—
270C2#1681118912yes—
271C2#1681120011yescharacteristic
272C2#1681136511yescharacteristic
273C2#1681136611yescharacteristic
274C2#1681137012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2819
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#13133203C2semidirect
2. #16810540C2semidirect
3. #16810539C2semidirect
4. #16810536C2semidirect
5. #16810535C2semidirect
6. #16810532C2semidirect
7. #16810531C2semidirect
8. #16810488C2semidirect
9. C53×#394C2semidirect
64. C4×Dic53D4semidirect
141. #394D53semidirect
147. C53#2819semidirect