Finite Group Databasebeta

← #20280501

Subgroups — #20280501

Order 6784.

526 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133201C211yesmaximal, characteristic
2C22×#11719919C211yesmaximal, characteristic, complemented
3#16810565C212yesmaximal, complemented
4#16810558C212yesmaximal, complemented
5#16810556C212yesmaximal, complemented
6#16810552C212yesmaximal, complemented
7#16810502C211yesmaximal, characteristic
8#16810500C211yesmaximal, characteristic
9#16810495C211yesmaximal, characteristic, complemented
10#16810484C211yesmaximal, characteristic, complemented
11C53×#385C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2211yescharacteristic
13C23×Dic53C2211yescharacteristic
14C23×Dic53C2211yescharacteristic
15#13133201—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×C4⋊Dic53C2211yescharacteristic
18C2×#11719919—24no—
19C2×#11719919C2212yes—
20C2×#11719919C2212yes—
21C2×#11719919C2212yes—
22C2×C22⋊Dic53C2211yescharacteristic
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32C23×C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2212yes—
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2211yescharacteristic
36#95×C53C2212yes—
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C22×C4×C53—22no—
40C22×C4×C53—42no—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C2×C22⋊D53—22no—
46C2×C22⋊D53—22no—
47C2×C22⋊D53—22no—
48C22×Dic53—22no—
49C22×Dic53—22no—
50C22×Dic53—22no—
51C22×Dic53—42no—
52C22×Dic53D414yes—
53C22×Dic53C2312yes—
54C22×Dic53C2311yescharacteristic
55C22×Dic53C2312yes—
56C22×Dic53C2311yescharacteristic
57C22×Dic53C2311yescharacteristic
58C4⋊C4×C53—42no—
59C22⋊C4×C53—22no—
60C22⋊C4×C53—22no—
61C22⋊Dic53—22no—
62#11719921—22no—
63#11719921—22no—
64#11719921—42no—
65#11719921—42no—
66C4⋊Dic53—22no—
67C4⋊Dic53—22no—
68#11719919—24no—
69#11719919—24no—
70#11719919—42no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53C2×D412yes—
74C23×C53#4112yes—
75C23×C53#4112yes—
76C23×C53C2411yescharacteristic
77C22×D53—21no—
78C22×D53—21no—
79C22×D53—21no—
80C22×D53—22no—
81C22×D53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—44no—
93C22⋊D53—44no—
94C2×Dic53—28no—
95C2×Dic53—21no—
96C2×Dic53—21no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—22no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—44no—
109C2×Dic53C2×D412yes—
110C2×Dic53C2×D412yes—
111C22×C53—22no—
112C22×C53—22no—
113C22×C53—24no—
114C22×C53—24no—
115C22×C53#12314yes—
116C22×C53C22×D411yescharacteristic
117C22×C53C2×#4111yescharacteristic
118C22×C53C2×#4111yescharacteristic
119C22×C53H(5,𝔽2)12yes—
120C22×C532−1+412yes—
121C2×D53—41no—
122C2×D53—41no—
123C2×D53—41no—
124C2×D53—42no—
125C2×D53—42no—
126C4×C53—41no—
127C4×C53—42no—
128C4×C53—41no—
129C4×C53—42no—
130Dic53—22no—
131Dic53—22no—
132Dic53—41no—
133Dic53—42no—
134Dic53—42no—
135#2623—531nomaximal, Sylow
136C2×C53—24no—
137C2×C53C2×#12311yescharacteristic
138C2×C53#53511yescharacteristic
139C2×C53#53611yescharacteristic
140C2×C53#53912yes—
141C2×C53#53912yes—
142D53—81no—
143C2×#123—531no—
144C22×C4⋊C4—531no—
145#400—532no—
146#399—532no—
147#398—532no—
148#397—532no—
149#386—531no—
150#386—531no—
151#385—531no—
152#385D5311yescharacteristic, complemented
153#381—531no—
154C53#262311yescharacteristic, complemented, Sylow
155C22×D4—531no—
156C23×C4—531no—
157C23×C4—531no—
158C23×C4C2×D5311yescharacteristic
159#123—1064no—
160C2×C4⋊C4—532no—
161C2×C4⋊C4—532no—
162C2×C4⋊C4—531no—
163C2×C4⋊C4—531no—
164C2×C4⋊C4—532no—
165C2×C4⋊C4—1064no—
166C2×C4⋊C4C2×D5312yes—
167C2×C22⋊C4—531no—
168C2×C22⋊C4—531no—
169C2×C22⋊C4—531no—
170C2×C22⋊C4—532no—
171C2×C22⋊C4—532no—
172C2×C22⋊C4C2×D5311yescharacteristic
173C2×C22⋊C4C2×D5311yescharacteristic
174#95—532no—
175#95—532no—
176#95—532no—
177#95—532no—
178#95—532no—
179#95C2×D5312yes—
180C24—531no—
181C24C22×D5311yescharacteristic
182C2×D4—1062no—
183C2×D4—1062no—
184C2×D4—1062no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—534no—
188C22×C4—531no—
189C22×C4—531no—
190C22×C4—532no—
191C22×C4—531no—
192C22×C4—532no—
193C22×C4—1062no—
194C22×C4—1062no—
195C22×C4—1062no—
196C22×C4—2122no—
197C22×C4C22×D5312yes—
198C22×C4C22×D5311yescharacteristic
199C22×C4C22×D5312yes—
200C22×C4C22×D5311yescharacteristic
201C4⋊C4—42no—
202C4⋊C4—1064no—
203C4⋊C4—1064no—
204C4⋊C4—1062no—
205C4⋊C4—1062no—
206C4⋊C4—2122no—
207C22⋊C4—22no—
208C22⋊C4—22no—
209C22⋊C4—1062no—
210C22⋊C4—1062no—
211C22⋊C4—1062no—
212C22⋊C4—2122no—
213C22⋊C4—2122no—
214C23—22no—
215C23—22no—
216C23—1061no—
217C23—1061no—
218C23—1061no—
219C23—1062no—
220C23—1062no—
221C23#1171994512yes—
222C23D4×D5312yes—
223C23#1171994912yes—
224C23C23×D5311yescharacteristic
225D4—2124no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—44no—
237C2×C4—532no—
238C2×C4—532no—
239C2×C4—1068no—
240C2×C4—1061no—
241C2×C4—1061no—
242C2×C4—1061no—
243C2×C4—1061no—
244C2×C4—1062no—
245C2×C4—1061no—
246C2×C4—1061no—
247C2×C4—1062no—
248C2×C4—1061no—
249C2×C4—1062no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—2124no—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—24no—
258C22—2121no—
259C22—2121no—
260C22—2121no—
261C22—2122no—
262C22—2122no—
263C22#1313320214yes—
264C22C2×D4×D5311yescharacteristic
265C22C2×#1171994511yescharacteristic
266C22C2×#1171994911yescharacteristic
267C22#1313326212yes—
268C22#1313326412yes—
269C4—42no—
270C4—41no—
271C4—41no—
272C4—42no—
273C4—1062no—
274C4—1062no—
275C4—2121no—
276C4—2122no—
277C4—2122no—
278C2—24no—
279C2—4241no—
280C2C2×#1313320211yescharacteristic
281C2#1681118811yescharacteristic
282C2#1681119511yescharacteristic
283C2#1681121512yes—
284C2#1681122212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#385C2
hypercenter#385D53
hypocenterC53#2623
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C22×#11719919C2semidirect
3. #16810565C2semidirect
4. #16810558C2semidirect
5. #16810556C2semidirect
6. #16810552C2semidirect
9. #16810495C2semidirect
10. #16810484C2semidirect
11. C53×#385C2semidirect
152. #385D53semidirect
154. C53#2623semidirect