Finite Group Databasebeta

← #20280678

Subgroups — #20280678

Order 6784.

420 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 133 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#16810552C212yesmaximal, complemented
2#16810550C212yesmaximal, complemented
3#16810549C212yesmaximal, complemented
4#16810545C212yesmaximal, complemented
5#16810525C211yesmaximal, characteristic
6#16810524C211yesmaximal, characteristic
7#16810524C211yesmaximal, characteristic
8#16810522C211yesmaximal, characteristic, complemented
9#16810519C211yesmaximal, characteristic, complemented
10#16810469C211yesmaximal, characteristic, complemented
11C53×#389C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C2×C4⋊Dic53C2211yescharacteristic
14C2×C4⋊Dic53C2212yes—
15C2×#11719919C2212yes—
16C2×#11719921C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×C4×Dic53C2211yescharacteristic
23C2×C4×Dic53C2211yescharacteristic
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2211yescharacteristic
27#13133124C2211yescharacteristic
28#13133124C2211yescharacteristic
29#13133124C2211yescharacteristic
30C2×C4⋊C4×C53C2211yescharacteristic
31C2×C4⋊C4×C53C2211yescharacteristic
32C2×C4⋊C4×C53C2212yes—
33C2×C42×C53C2211yescharacteristic
34#95×C53C2212yes—
35C23×D53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2312yes—
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×Dic53C2311yescharacteristic
42C22×Dic53C2311yescharacteristic
43C22×Dic53C2311yescharacteristic
44C22×Dic53C2311yescharacteristic
45C22×Dic53C2312yes—
46C2×D212—22no—
47C2×D212—22no—
48C2×D212—22no—
49C4⋊C4×C53—22no—
50C4⋊C4×C53—22no—
51C4⋊C4×C53—42no—
52C42×C53—22no—
53#11719921—42no—
54#11719921—42no—
55#11719921—42no—
56#11719921—42no—
57#11719921—42no—
58#11719921—42no—
59C4⋊Dic53—22no—
60C4⋊Dic53—42no—
61#11719919—42no—
62C4×Dic53—22no—
63C4×Dic53—22no—
64C23×C53C2411yescharacteristic
65C22×D53—22no—
66C22×D53—22no—
67C22×D53—22no—
68C22×D53—22no—
69C22×D53—22no—
70C22×D53—22no—
71C22×D53—22no—
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—21no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53#4112yes—
85C2×C4×C53#4112yes—
86C2×C4×C53#4112yes—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—21no—
91C2×Dic53—22no—
92C2×Dic53—21no—
93C2×Dic53—22no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—21no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102D212—44no—
103C22×C53C2×#4111yescharacteristic
104C22×C53C2×#4111yescharacteristic
105C22×C53C2×#4111yescharacteristic
106C22×C53H(5,𝔽2)11yescharacteristic
107C22×C53H(5,𝔽2)11yescharacteristic
108C22×C53H(5,𝔽2)11yescharacteristic
109C22×C532−1+411yescharacteristic
110C2×D53—42no—
111C2×D53—42no—
112C2×D53—42no—
113C2×D53—42no—
114C2×D53—42no—
115C2×D53—42no—
116C2×D53—42no—
117C4×C53—42no—
118C4×C53—41no—
119C4×C53—42no—
120C4×C53—41no—
121C4×C53#12614yes—
122Dic53—41no—
123Dic53—41no—
124Dic53—42no—
125Dic53—41no—
126Dic53—41no—
127#2644—531nomaximal, Sylow
128C2×C53C2×#12611yescharacteristic
129C2×C53#54011yescharacteristic
130C2×C53#54011yescharacteristic
131C2×C53#54011yescharacteristic
132C2×C53#54211yescharacteristic
133C2×C53#54211yescharacteristic
134C2×C53#54211yescharacteristic
135D53—82no—
136#397—532no—
137#397—532no—
138#397—532no—
139#396—532no—
140#390—531no—
141#390—531no—
142#390—531no—
143#389—531no—
144#389—531no—
145#389D5311yescharacteristic, complemented
146#376—531no—
147C53#264411yescharacteristic, complemented, Sylow
148C22×D4—531no—
149C2×C4⋊C4—531no—
150C2×C4⋊C4—532no—
151C2×C4⋊C4—532no—
152C2×C4⋊C4C2×D5311yescharacteristic
153C2×C4⋊C4C2×D5311yescharacteristic
154C2×C4⋊C4C2×D5312yes—
155C2×C22⋊C4—532no—
156C2×C22⋊C4—532no—
157C2×C22⋊C4—532no—
158C2×C22⋊C4—532no—
159C2×C22⋊C4—532no—
160C2×C22⋊C4—532no—
161C2×C42—531no—
162C2×C42—531no—
163C2×C42C2×D5311yescharacteristic
164#95—531no—
165#95—531no—
166#95—531no—
167#95—531no—
168#95—532no—
169#95—532no—
170#95C2×D5312yes—
171C24—532no—
172C2×D4—1062no—
173C2×D4—1062no—
174C2×D4—1062no—
175C22×C4—531no—
176C22×C4—531no—
177C22×C4—531no—
178C22×C4—531no—
179C22×C4—532no—
180C22×C4C22×D5311yescharacteristic
181C22×C4C22×D5311yescharacteristic
182C22×C4C22×D5312yes—
183C22×C4C22×D5312yes—
184C22×C4C22×D5311yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—42no—
188C4⋊C4—1062no—
189C4⋊C4—2122no—
190C4⋊C4—2122no—
191C22⋊C4—2122no—
192C22⋊C4—2122no—
193C22⋊C4—2122no—
194C22⋊C4—2122no—
195C22⋊C4—2122no—
196C22⋊C4—2122no—
197C42—22no—
198C42—1062no—
199C42—1062no—
200C23—1062no—
201C23—1062no—
202C23—1062no—
203C23—1062no—
204C23—1062no—
205C23—1062no—
206C23—1062no—
207C23C23×D5311yescharacteristic
208D4—2124no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—1061no—
222C2×C4—1061no—
223C2×C4—1061no—
224C2×C4—1061no—
225C2×C4—1062no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—1061no—
229C2×C4—1061no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1061no—
233C2×C4—1061no—
234C2×C4—1061no—
235C2×C4—1061no—
236C2×C4#1171994512yes—
237C2×C4#1171994712yes—
238C2×C4#1171994912yes—
239C22—2122no—
240C22—2122no—
241C22—2122no—
242C22—2122no—
243C22—2122no—
244C22—2122no—
245C22—2122no—
246C22C2×#1171994711yescharacteristic
247C22C2×#1171994511yescharacteristic
248C22C2×#1171994911yescharacteristic
249C22#1313326211yescharacteristic
250C22#1313326311yescharacteristic
251C22#1313326311yescharacteristic
252C22#1313326511yescharacteristic
253C4—42no—
254C4—41no—
255C4—42no—
256C4—41no—
257C4—2121no—
258C4—2121no—
259C4—2122no—
260C4—2121no—
261C4—2121no—
262C4#1313321214yes—
263C2—4242no—
264C2C2×#1313321211yescharacteristic
265C2#1681122411yescharacteristic
266C2#1681122611yescharacteristic
267C2#1681122911yescharacteristic
268C2#1681124511yescharacteristic
269C2#1681124611yescharacteristic
270C2#1681124711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#389C2
hypercenter#389D53
hypocenterC53#2644
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #16810552C2semidirect
2. #16810550C2semidirect
3. #16810549C2semidirect
4. #16810545C2semidirect
8. #16810522C2semidirect
9. #16810519C2semidirect
10. #16810469C2semidirect
11. C53×#389C2semidirect
145. #389D53semidirect
147. C53#2644semidirect