Finite Group Databasebeta

← #20282771

Subgroups — #20282771

Order 6784.

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

normal subgroups only

This group has 464 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×#13133208C211yesmaximal, characteristic, complemented
2C2×#13133203C211yesmaximal, characteristic
3#16810560C211yesmaximal, characteristic, complemented
4#16810559C211yesmaximal, characteristic, complemented
5#16810549C211yesmaximal, characteristic, complemented
6#16810547C212yesmaximal, complemented
7#16810546C212yesmaximal, complemented
8#16810544C212yesmaximal, complemented
9#16810539C212yesmaximal
10#16810487C211yesmaximal, characteristic, complemented
11C53×#397C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×C22⋊D53C2211yescharacteristic
14#13133208—24no—
15#13133203—24no—
16C2×C4⋊Dic53C2212yes—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2212yes—
19C2×#11719919C2212yes—
20C2×#11719919C2212yes—
21C2×#11719919C2212yes—
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2212yes—
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×C4×Dic53C2211yescharacteristic
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C22⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C2×C22⋊D53—22no—
44C2×C22⋊D53—42no—
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2312yes—
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2311yescharacteristic
50C2×Dic106—22no—
51C2×Dic106—42no—
52C4⋊C4×C53—22no—
53C22⋊C4×C53—22no—
54C22⋊C4×C53—42no—
55C22⋊Dic53—22no—
56C22⋊Dic53—42no—
57#11719921—22no—
58#11719921—22no—
59#11719921—42no—
60#11719921—42no—
61C4⋊Dic53—22no—
62C4⋊Dic53—24no—
63C4⋊Dic53—42no—
64#11719919—24no—
65#11719919—42no—
66#11719919—42no—
67C4×Dic53D414yes—
68C23×C53—21no—
69C23×C53—22no—
70C23×C53—21no—
71C23×C53—22no—
72C23×C53—21no—
73C23×C53C2411yescharacteristic
74C22×D53—21no—
75C22×D53—22no—
76C22×D53—21no—
77C22×D53—22no—
78C22×D53—21no—
79C2×C4×C53—22no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53C2×D412yes—
89C22⋊D53—44no—
90C2×Dic53—22no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53C2×D412yes—
102C2×Dic53C2×D412yes—
103Dic106—44no—
104C22×C53—41no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—42no—
108C22×C53—42no—
109C22×C53C22×D411yescharacteristic
110C22×C53H(5,𝔽2)11yescharacteristic
111C22×C53H(5,𝔽2)12yes—
112C22×C532−1+412yes—
113C22×C532−1+411yescharacteristic
114C2×D53—41no—
115C2×D53—41no—
116C2×D53—41no—
117C2×D53—42no—
118C2×D53—42no—
119C4×C53—22no—
120C4×C53—41no—
121C4×C53—42no—
122C4×C53—42no—
123Dic53—22no—
124Dic53—22no—
125Dic53—41no—
126Dic53—42no—
127Dic53—42no—
128#2820—531nomaximal, Sylow
129C2×C53—81no—
130C2×C53#53511yescharacteristic
131C2×C53#53611yescharacteristic
132C2×C53#53711yescharacteristic
133C2×C53#56212yes—
134C2×C53#56312yes—
135D53—81no—
136C2×#125—531no—
137C2×#124—531no—
138#399—531no—
139#399—531no—
140#397—531no—
141#397D5311yescharacteristic, complemented
142#396—532no—
143#396—532no—
144#395—532no—
145#394—532no—
146#382—531no—
147C53#282011yescharacteristic, complemented, Sylow
148C22×Q8—531no—
149C22×D4—531no—
150#125—1064no—
151#124—1064no—
152C2×C4⋊C4—532no—
153C2×C4⋊C4—532no—
154C2×C4⋊C4—531no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4C2×D5311yescharacteristic
159C2×C22⋊C4—531no—
160C2×C22⋊C4—531no—
161C2×C22⋊C4—531no—
162C2×C22⋊C4—532no—
163C2×C22⋊C4—532no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4C2×D5311yescharacteristic
166C2×C22⋊C4C2×D5312yes—
167C2×C42—531no—
168#95—532no—
169#95—531no—
170#95—532no—
171#95C2×D5311yescharacteristic
172#95C2×D5312yes—
173C24—531no—
174C24C22×D5311yescharacteristic
175C2×Q8—1062no—
176C2×Q8—2122no—
177C2×D4—1062no—
178C2×D4—2122no—
179C22×C4—531no—
180C22×C4—531no—
181C22×C4—532no—
182C22×C4—531no—
183C22×C4—532no—
184C22×C4C22×D5312yes—
185C22×C4C22×D5312yes—
186C22×C4C22×D5311yescharacteristic
187C22×C4C22×D5311yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—1064no—
190C4⋊C4—1064no—
191C4⋊C4—1062no—
192C4⋊C4—2122no—
193C4⋊C4—2122no—
194C4⋊C4—2122no—
195C22⋊C4—22no—
196C22⋊C4—42no—
197C22⋊C4—1062no—
198C22⋊C4—1062no—
199C22⋊C4—1062no—
200C22⋊C4—2122no—
201C22⋊C4—2122no—
202C22⋊C4—2122no—
203C42—534no—
204C23—21no—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—1061no—
210C23—1061no—
211C23—1062no—
212C23—1062no—
213C23—1061no—
214C23C23×D5311yescharacteristic
215Q8—2124no—
216D4—2124no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—532no—
227C2×C4—532no—
228C2×C4—1061no—
229C2×C4—1062no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4D4×D5312yes—
240C22—42no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—2122no—
246C22—2122no—
247C22—2121no—
248C22—2121no—
249C22—2121no—
250C22C2×D4×D5311yescharacteristic
251C22#1313326211yescharacteristic
252C22#1313326312yes—
253C22#1313326412yes—
254C22#1313326411yescharacteristic
255C4—22no—
256C4—42no—
257C4—41no—
258C4—42no—
259C4—1062no—
260C4—1062no—
261C4—2122no—
262C4—2122no—
263C4—2121no—
264C2—81no—
265C2—4241no—
266C2#1681119111yescharacteristic
267C2#1681119511yescharacteristic
268C2#1681120211yescharacteristic
269C2#1681137012yes—
270C2#1681137512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2820
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#13133208C2semidirect
3. #16810560C2semidirect
4. #16810559C2semidirect
5. #16810549C2semidirect
6. #16810547C2semidirect
7. #16810546C2semidirect
8. #16810544C2semidirect
10. #16810487C2semidirect
11. C53×#397C2semidirect
141. #397D53semidirect
147. C53#2820semidirect