Finite Group Databasebeta

← #20279890

Subgroups — #20279890

Order 6784.

584 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 151 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×#13133197C211yesmaximal, characteristic, complemented
2C2×#13133177C211yesmaximal, characteristic
3#16810561C211yesmaximal, characteristic, complemented
4#16810544C211yesmaximal, characteristic, complemented
5#16810535C211yesmaximal, characteristic, complemented
6#16810509C212yesmaximal, complemented
7#16810499C212yesmaximal
8#16810498C212yesmaximal, complemented
9#16810480C212yesmaximal, complemented
10#95×D53C211yesmaximal, characteristic, complemented
11C53×#399C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×C22⋊D53C2211yescharacteristic
14C23×Dic53C2211yescharacteristic
15C22×C4×D53C2212yes—
16C22×C4×D53C2211yescharacteristic
17#13133197—24no—
18#13133177—24no—
19C2×C4⋊Dic53C2212yes—
20C2×C4⋊Dic53C2212yes—
21C2×#11719919C2211yescharacteristic
22C2×C22⋊Dic53C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2211yescharacteristic
25C2×C4×Dic53C2211yescharacteristic
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C2×C4⋊C4×C53C2212yes—
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2212yes—
35C2×C22⋊C4×C53C2211yescharacteristic
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C2×C22⋊D53—22no—
44C2×C22⋊D53—22no—
45C2×C22⋊D53Q814yescomplemented
46C22×Dic53—22no—
47C22×Dic53—42no—
48C22×Dic53C2312yes—
49C22×Dic53C2312yes—
50C22×Dic53C2311yescharacteristic
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2311yescharacteristic
53C2×C4×D53—24no—
54C2×C4×D53—24no—
55C2×C4×D53—42no—
56C2×C4×D53—42no—
57C2×C4×D53D414yes—
58C2×Dic106—22no—
59C2×Dic106—42no—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—24no—
62C22⋊C4×C53—22no—
63C22⋊C4×C53—24no—
64C22⋊Dic53—42no—
65#11719921—22no—
66#11719921—24no—
67C4⋊Dic53—24no—
68C4⋊Dic53—42no—
69#11719919—22no—
70C4×Dic53—22no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53C2×Q812yes—
76C23×C53C2411yescharacteristic
77C22×D53—22no—
78C22×D53—22no—
79C22×D53C2×D412yes—
80C22×D53C2×Q812yes—
81C22×D53#4112yes—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53C2×D412yes—
91C2×C4×C53#4114yes—
92C22⋊D53—28no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—21no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—44no—
105C2×Dic53C2×D412yes—
106C2×Dic53C2×Q812yes—
107C4×D53—44no—
108C4×D53—48no—
109Dic106—44no—
110C22×C53—22no—
111C22×C53—24no—
112C22×C53—41no—
113C22×C53—41no—
114C22×C53—42no—
115C22×C53C22×D411yescharacteristic
116C22×C53C22×Q811yescharacteristic
117C22×C53C2×#4111yescharacteristic
118C22×C53C2×#4112yes—
119C22×C53H(5,𝔽2)12yes—
120C2×D53—22no—
121C2×D53—22no—
122C2×D53—24no—
123C2×D53—24no—
124C2×D53C22⋊Q814yes—
125C4×C53—22no—
126C4×C53—24no—
127C4×C53—41no—
128C4×C53—42no—
129Dic53—22no—
130Dic53—22no—
131Dic53—42no—
132Dic53—41no—
133Dic53—42no—
134#2564—531nomaximal, Sylow
135C2×C53—42no—
136C2×C53C2×C22⋊Q811yescharacteristic
137C2×C53#53211yescharacteristic
138C2×C53#53411yescharacteristic
139C2×C53#55112yes—
140C2×C53#55412yes—
141D53—24no—
142C2×C22⋊Q8—531no—
143C2×C4×D4—531no—
144#400—531no—
145#399D5311yescharacteristic, complemented
146#395—531no—
147#393—531no—
148#387—532no—
149#386—532no—
150#385—532no—
151#380—532no—
152C2×#95—531no—
153C53#256411yescharacteristic, complemented, Sylow
154C22×Q8—531no—
155C22×D4—531no—
156C23×C4—531no—
157C23×C4—531no—
158C23×C4—532no—
159C22⋊Q8—1064no—
160C4×D4—1064no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—532no—
163C2×C4⋊C4—532no—
164C2×C4⋊C4C2×D5311yescharacteristic
165C2×C4⋊C4C2×D5312yes—
166C2×C22⋊C4—531no—
167C2×C22⋊C4—532no—
168C2×C22⋊C4—532no—
169C2×C22⋊C4C2×D5311yescharacteristic
170C2×C22⋊C4C2×D5312yes—
171C2×C42—531no—
172#95—532no—
173#95—531no—
174#95—532no—
175#95—532no—
176#95—532no—
177#95—532no—
178#95C2×D5311yescharacteristic
179C24—531no—
180C24C22×D5311yescharacteristic
181C2×Q8—1062no—
182C2×Q8—2122no—
183C2×D4—534no—
184C2×D4—1062no—
185C2×D4—1062no—
186C22×C4—531no—
187C22×C4—531no—
188C22×C4—534no—
189C22×C4—532no—
190C22×C4—532no—
191C22×C4—531no—
192C22×C4—1064no—
193C22×C4—1062no—
194C22×C4—1064no—
195C22×C4—2122no—
196C22×C4—2122no—
197C22×C4—2122no—
198C22×C4C22×D5311yescharacteristic
199C22×C4C22×D5312yes—
200C22×C4C22×D5311yescharacteristic
201C22×C4C22×D5312yes—
202C4⋊C4—22no—
203C4⋊C4—24no—
204C4⋊C4—1064no—
205C4⋊C4—1062no—
206C4⋊C4—2122no—
207C22⋊C4—22no—
208C22⋊C4—24no—
209C22⋊C4—1062no—
210C22⋊C4—1064no—
211C22⋊C4—2122no—
212C42—1062no—
213C23—21no—
214C23—21no—
215C23—22no—
216C23—22no—
217C23—532no—
218C23—532no—
219C23—532no—
220C23—1062no—
221C23—1062no—
222C23Q8×D5312yes—
223C23C23×D5311yescharacteristic
224Q8—2124no—
225D4—1068no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—532no—
235C2×C4—532no—
236C2×C4—1061no—
237C2×C4—1061no—
238C2×C4—1062no—
239C2×C4—1061no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—2124no—
248C2×C4—2128no—
249C2×C4—2124no—
250C2×C4D4×D5312yes—
251C2×C4#1171994714yes—
252C22—24no—
253C22—22no—
254C22—42no—
255C22—41no—
256C22—41no—
257C22—534no—
258C22—1064no—
259C22—1064no—
260C22—1062no—
261C22—1062no—
262C22C2×D4×D5311yescharacteristic
263C22C2×Q8×D5311yescharacteristic
264C22#41×D5311yescharacteristic
265C22C2×#1171994712yes—
266C22#1313326312yes—
267C4—24no—
268C4—22no—
269C4—41no—
270C4—42no—
271C4—1062no—
272C4—1062no—
273C4—2122no—
274C4—2121no—
275C4—2122no—
276C2—42no—
277C2—1064no—
278C2C22⋊Q8×D5311yescharacteristic
279C2#1681117111yescharacteristic
280C2#1681118411yescharacteristic
281C2#1681129612yes—
282C2#1681133112yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2564
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#13133197C2semidirect
3. #16810561C2semidirect
4. #16810544C2semidirect
5. #16810535C2semidirect
6. #16810509C2semidirect
8. #16810498C2semidirect
9. #16810480C2semidirect
10. #95×D53C2semidirect
11. C53×#399C2semidirect
45. C2×C22⋊D53Q8semidirect
145. #399D53semidirect
153. C53#2564semidirect