Finite Group Databasebeta

← #20277972

Subgroups — #20277972

Order 6784.

560 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 157 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×C4×C22⋊D53C211yesmaximal, characteristic
2#16810561C211yesmaximal, characteristic, complemented
3#16810558C211yesmaximal, characteristic, complemented
4#16810554C211yesmaximal, characteristic, complemented
5#16810551C212yesmaximal, complemented
6#16810549C211yesmaximal, characteristic, complemented
7#16810538C212yesmaximal
8C4×#11719919C212yesmaximal, complemented
9C4×#11719921C212yesmaximal, complemented
10C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
11C2×#117×C53C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14C4×C22⋊D53—24no—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×#11719919C2211yescharacteristic
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2211yescharacteristic
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2211yescharacteristic
23C2×C4×Dic53C2211yescharacteristic
24C2×C4×Dic53C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2211yescharacteristic
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2211yescharacteristic
31C23×C4×C53C2211yescharacteristic
32#117×C53—24no—
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36C2×C42×C53C2212yes—
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C22×C4×C53—24no—
40C22×C4×C53D414yes—
41C22×C4×C53C2312yes—
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C2×C22⋊D53—22no—
46C2×C22⋊D53—22no—
47C2×C22⋊D53—22no—
48C22×Dic53C2312yes—
49C22×Dic53C2311yescharacteristic
50C22×Dic53C2311yescharacteristic
51C22×Dic53C2312yes—
52C22×Dic53C2311yescharacteristic
53C2×C4×D53—22no—
54C2×C4×D53—24no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—22no—
57C22⋊C4×C53—24no—
58C42×C53—24no—
59C22⋊Dic53—22no—
60C22⋊Dic53—24no—
61#11719921—24no—
62#11719921—22no—
63#11719921—24no—
64#11719919—22no—
65#11719919—24no—
66#11719919—24no—
67C4×Dic53—22no—
68C4×Dic53—24no—
69C23×C53—21no—
70C23×C53—21no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53C2×D412yes—
74C23×C53C2411yescharacteristic
75C22×D53—21no—
76C22×D53—22no—
77C22×D53—21no—
78C22×D53—22no—
79C22×D53#4112yes—
80C2×C4×C53—22no—
81C2×C4×C53—28no—
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—
89C2×C4×C53C2×D412yes—
90C2×C4×C53#4114yes—
91C22⋊D53—28no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—21no—
97C2×Dic53—21no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53#4112yes—
104C2×Dic53#4114yes—
105C4×D53—44no—
106C22×C53—22no—
107C22×C53—22no—
108C22×C53—22no—
109C22×C53—42no—
110C22×C53—42no—
111C22×C53C22×D411yescharacteristic
112C22×C53C2×#4111yescharacteristic
113C22×C53C2×#4111yescharacteristic
114C22×C53C2×#4112yes—
115C22×C53C2×#4112yes—
116C2×D53—22no—
117C2×D53—24no—
118C2×D53—41no—
119C2×D53—42no—
120C2×D53—41no—
121C4×C53—22no—
122C4×C53—42no—
123C4×C53—42no—
124C4×C53#12314yes—
125Dic53—22no—
126Dic53—24no—
127Dic53—41no—
128Dic53—42no—
129Dic53—41no—
130#2395—531nomaximal, Sylow
131C2×C53—42no—
132C2×C53C2×#12311yescharacteristic
133C2×C53#52511yescharacteristic
134C2×C53#52912yes—
135C2×C53#52912yes—
136C2×C53#53211yescharacteristic
137D53—42no—
138C2×C4×D4—531no—
139C2×#117D5311yescharacteristic, complemented
140#400—531no—
141#399—531no—
142#398—531no—
143#397—531no—
144#397—532no—
145#394—532no—
146C4×C4⋊C4—532no—
147C4×C22⋊C4—531no—
148C4×C22⋊C4—532no—
149C53#239511yescharacteristic, complemented, Sylow
150C22×D4—531no—
151C23×C4—531no—
152C23×C4C2×D5311yescharacteristic
153C4×D4—1064no—
154#117—24no—
155C2×C4⋊C4—531no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4C2×D5311yescharacteristic
159C2×C4⋊C4C2×D5311yescharacteristic
160C2×C22⋊C4—531no—
161C2×C22⋊C4—532no—
162C2×C22⋊C4—531no—
163C2×C22⋊C4—532no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4C2×D5312yes—
166C2×C42—531no—
167C2×C42—532no—
168C2×C42C2×D5312yes—
169#95—532no—
170#95—531no—
171#95—532no—
172#95—531no—
173#95—532no—
174#95—532no—
175C24—531no—
176C24C22×D5311yescharacteristic
177C2×D4—1062no—
178C2×D4—1062no—
179C2×D4—1062no—
180C22×C4—24no—
181C22×C4—532no—
182C22×C4—531no—
183C22×C4—532no—
184C22×C4—531no—
185C22×C4—531no—
186C22×C4—1064no—
187C22×C4—1062no—
188C22×C4C22⋊D5314yes—
189C22×C4C22×D5312yes—
190C22×C4C22×D5312yes—
191C22×C4C22×D5311yescharacteristic
192C22×C4C22×D5311yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—1064no—
196C4⋊C4—1064no—
197C4⋊C4—1062no—
198C22⋊C4—24no—
199C22⋊C4—1064no—
200C22⋊C4—1064no—
201C22⋊C4—1064no—
202C22⋊C4—1062no—
203C22⋊C4—1062no—
204C42—24no—
205C42—1064no—
206C42—1062no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—532no—
212C23—1061no—
213C23—1062no—
214C23—1061no—
215C23—1062no—
216C23C2×C22⋊D5312yes—
217C23C23×D5311yescharacteristic
218D4—1068no—
219C2×C4—22no—
220C2×C4—28no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—534no—
228C2×C4—532no—
229C2×C4—1061no—
230C2×C4—1062no—
231C2×C4—1061no—
232C2×C4—1062no—
233C2×C4—1061no—
234C2×C4—1061no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—2124no—
241C2×C4#1171994514yes—
242C2×C4C2×C22⋊D5312yes—
243C2×C4C2×C22⋊D5312yes—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—1062no—
250C22—1064no—
251C22—2121no—
252C22—2122no—
253C22—2121no—
254C22C22×C22⋊D5311yescharacteristic
255C22#41×D5311yescharacteristic
256C22#41×D5311yescharacteristic
257C22#41×D5312yes—
258C22C2×#1171994512yes—
259C4—22no—
260C4—42no—
261C4—42no—
262C4—1062no—
263C4—1064no—
264C4—2121no—
265C4—2122no—
266C4—2121no—
267C4#1313320014yes—
268C2—42no—
269C2—2122no—
270C2C2×#1313320011yescharacteristic
271C2#1681113011yescharacteristic
272C2#1681115112yes—
273C2#1681115112yes—
274C2#1681116911yescharacteristic
subgroupG/H
centerC22×C4C22×D53
commutator, socleC23×C53C24
FrattiniC23C23×D53
FittingC2×#117×C53C2
hypercenterC2×#117D53
hypocenterC53#2395
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #16810561C2semidirect
3. #16810558C2semidirect
4. #16810554C2semidirect
5. #16810551C2semidirect
6. #16810549C2semidirect
8. C4×#11719919C2semidirect
9. C4×#11719921C2semidirect
10. C4×C22⋊Dic53C2semidirect
11. C2×#117×C53C2semidirect
139. C2×#117D53semidirect
149. C53#2395semidirect