Finite Group Databasebeta

← #20282740

Subgroups — #20282740

Order 6784.

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

normal subgroups only

This group has 644 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×#13133189C212yesmaximal, complemented
2#16810552C212yesmaximal, complemented
3#16810551C212yesmaximal, complemented
4#16810532C212yesmaximal, complemented
5#16810532C211yesmaximal, characteristic, complemented
6#16810531C212yesmaximal, complemented
7#16810531C212yesmaximal, complemented
8#16810484C211yesmaximal, characteristic, complemented
9C53×#392C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C22×C22⋊D53C2212yes—
13C22×C22⋊D53C2212yes—
14C23×Dic53C2211yescharacteristic
15#13133189—28no—
16C2×#11719919C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2211yescharacteristic
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26#13133124C2212yes—
27C22×D4×C53C2212yes—
28C22×D4×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2212yes—
30C2×C22⋊C4×C53C2211yescharacteristic
31#95×C53C2211yescharacteristic
32C24×C53C2312yes—
33C24×C53C2311yescharacteristic
34C23×D53C2312yes—
35C2×D4×C53—24no—
36C2×D4×C53—42no—
37C2×D4×C53—42no—
38C2×D4×C53—42no—
39C2×D4×C53—41no—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C2×C22⋊D53—24no—
44C2×C22⋊D53—24no—
45C2×C22⋊D53—42no—
46C2×C22⋊D53—42no—
47C2×C22⋊D53—42no—
48C2×C22⋊D53—42no—
49C2×C22⋊D53—42no—
50C2×C22⋊D53—42no—
51C2×C22⋊D53—42no—
52C22×Dic53—41no—
53C22×Dic53—41no—
54C22×Dic53D414yescomplemented
55C22×Dic53C2312yes—
56C22×Dic53C2312yes—
57C22×Dic53C2312yes—
58C2×D212—41no—
59C2×D212—42no—
60C22⋊C4×C53—41no—
61C22⋊C4×C53—42no—
62C22⋊Dic53—24no—
63C22⋊Dic53—41no—
64C22⋊Dic53—42no—
65#11719921—24no—
66#11719921—42no—
67#11719921—42no—
68#11719919—24no—
69C23×C53—21no—
70C23×C53—21no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53C2×D412yes—
81C23×C53C2411yescharacteristic
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89D4×C53—48no—
90D4×C53—44no—
91C2×C4×C53—22no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—21no—
98C22⋊D53—48no—
99C22⋊D53—48no—
100C22⋊D53—48no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—44no—
110C2×Dic53C2×D414yes—
111D212—44no—
112C22×C53—22no—
113C22×C53—24no—
114C22×C53—41no—
115C22×C53—42no—
116C22×C53—41no—
117C22×C53—42no—
118C22×C53—42no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53—42no—
123C22×C53—42no—
124C22×C53C22×D411yescharacteristic
125C22×C53H(5,𝔽2)11yescharacteristic
126C22×C53H(5,𝔽2)12yes—
127C22×C53H(5,𝔽2)12yes—
128C22×C53H(5,𝔽2)11yescharacteristic
129C2×D53—42no—
130C2×D53—42no—
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—42no—
135C2×D53—42no—
136C4×C53—42no—
137C4×C53—41no—
138C4×C53—41no—
139Dic53—24no—
140Dic53—42no—
141Dic53—42no—
142#2815—531nomaximal, Sylow
143C2×C53—42no—
144C2×C53—82no—
145C2×C53#53411yescharacteristic
146C2×C53#53512yes—
147C2×C53#56011yescharacteristic
148C2×C53#56012yes—
149C2×C53#56011yescharacteristic
150D53—82no—
151C2×#121—532no—
152#397—532no—
153#397—532no—
154#392—532no—
155#392—532no—
156#392—532no—
157#392—531no—
158#392D5311yescharacteristic, complemented
159#381—531no—
160C53#281511yescharacteristic, complemented, Sylow
161C22×D4—532no—
162C22×D4—532no—
163C22×D4—532no—
164C22×D4—531no—
165C22×D4C2×D5312yes—
166C22×D4C2×D5311yescharacteristic
167C23×C4—531no—
168#121—1068no—
169C2×C4⋊C4—532no—
170C2×C22⋊C4—532no—
171C2×C22⋊C4—532no—
172C2×C22⋊C4—531no—
173C2×C22⋊C4—532no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4—532no—
176C2×C22⋊C4C2×D5311yescharacteristic
177C2×C22⋊C4C2×D5312yes—
178#95—532no—
179#95—532no—
180#95—532no—
181#95—531no—
182#95C2×D5311yescharacteristic
183C24—532no—
184C24C22×D5312yes—
185C24C22×D5311yescharacteristic
186C2×D4—24no—
187C2×D4—42no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—1064no—
192C2×D4—1064no—
193C2×D4—2122no—
194C2×D4—2121no—
195C2×D4—2122no—
196C2×D4—2122no—
197C2×D4—2122no—
198C2×D4—2122no—
199C2×D4—2122no—
200C2×D4—2122no—
201C2×D4—2122no—
202C22×C4—532no—
203C22×C4—534no—
204C22×C4—532no—
205C22×C4—532no—
206C22×C4—2121no—
207C22×C4—2121no—
208C22×C4C22×D5311yescharacteristic
209C22×C4C22×D5311yescharacteristic
210C22×C4C22×D5312yes—
211C4⋊C4—1064no—
212C22⋊C4—41no—
213C22⋊C4—42no—
214C22⋊C4—1064no—
215C22⋊C4—1064no—
216C22⋊C4—2122no—
217C22⋊C4—2122no—
218C22⋊C4—2122no—
219C22⋊C4—2121no—
220C23—21no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—1062no—
232C23—1062no—
233C23—1062no—
234C23—1062no—
235C23—1062no—
236C23—1062no—
237C23—1062no—
238C23D4×D5312yes—
239C23C23×D5311yescharacteristic
240D4—48no—
241D4—44no—
242D4—2124no—
243D4—2128no—
244D4—2128no—
245D4—2128no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—534no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—2124no—
263C22—22no—
264C22—24no—
265C22—41no—
266C22—42no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—2122no—
276C22—2122no—
277C22—2122no—
278C22—2122no—
279C22—2122no—
280C22—2122no—
281C22—2122no—
282C22C2×D4×D5311yescharacteristic
283C22#1313326212yes—
284C22#1313326211yescharacteristic
285C22#1313326212yes—
286C22#1313326311yescharacteristic
287C4—41no—
288C4—41no—
289C4—42no—
290C4—1064no—
291C4—2122no—
292C4—2122no—
293C2—42no—
294C2—82no—
295C2—4242no—
296C2#1681118111yescharacteristic
297C2#1681118812yes—
298C2#1681136011yescharacteristic
299C2#1681136112yes—
300C2#1681136211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2815
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#13133189C2semidirect
2. #16810552C2semidirect
3. #16810551C2semidirect
4. #16810532C2semidirect
5. #16810532C2semidirect
6. #16810531C2semidirect
7. #16810531C2semidirect
8. #16810484C2semidirect
9. C53×#392C2semidirect
54. C22×Dic53D4semidirect
158. #392D53semidirect
160. C53#2815semidirect