Finite Group Databasebeta

← #20280091

Subgroups — #20280091

Order 6784.

544 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 159 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×#13133193C211yesmaximal, characteristic
2C2×C4⋊C4×D53C211yesmaximal, characteristic
3#16810555C211yesmaximal, characteristic, complemented
4#16810553C211yesmaximal, characteristic, complemented
5#16810542C211yesmaximal, characteristic, complemented
6#16810520C212yesmaximal, complemented
7#16810519C212yesmaximal, complemented
8#16810497C212yesmaximal
9#16810494C212yesmaximal
10#16810470C211yesmaximal, characteristic
11C53×#395C211yesmaximal, characteristic, complemented
12C22×C4×D53C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15#13133193—24no—
16C2×C4⋊Dic53C2212yes—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2212yes—
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2212yes—
24C4⋊C4×D53—24no—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×C4×Dic53C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C22×Q8×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C23×D53C2311yescharacteristic
38C2×Q8×C53—22no—
39C2×Q8×C53—42no—
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C22×Dic53C2312yes—
49C22×Dic53C2311yescharacteristic
50C2×C4×D53—22no—
51C2×C4×D53—22no—
52C2×C4×D53—22no—
53C2×C4×D53—22no—
54C2×C4×D53—42no—
55C2×C4×D53—42no—
56C2×C4×D53D414yes—
57C4⋊C4×C53—22no—
58C4⋊C4×C53—42no—
59#11719921—24no—
60#11719921Q818yescomplemented
61C4⋊Dic53—22no—
62C4⋊Dic53—22no—
63C4⋊Dic53—24no—
64#11719919—22no—
65#11719919—24no—
66#11719919—22no—
67#11719919—24no—
68#11719919—42no—
69C4×Dic53—42no—
70C23×C53C2411yescharacteristic
71C22×D53—21no—
72C22×D53—21no—
73C22×D53—22no—
74C22×D53C2×D412yes—
75C22×D53C2×Q814yes—
76Q8×C53—44no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53C2×D412yes—
88C2×C4×C53C2×Q814yes—
89C2×C4×C53#4112yes—
90C2×Dic53—21no—
91C2×Dic53—21no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—21no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53C2×D412yes—
101C2×Dic53C2×Q814yes—
102C2×Dic53#4112yes—
103C4×D53—28no—
104C4×D53—44no—
105C4×D53—44no—
106C22×C53C22×D411yescharacteristic
107C22×C53C22×Q812yes—
108C22×C53C2×#4111yescharacteristic
109C22×C53C2×#4111yescharacteristic
110C22×C53H(5,𝔽2)12yes—
111C2×D53—22no—
112C2×D53—22no—
113C2×D53—24no—
114C2×D53—24no—
115C2×D53C4⋊Q814yes—
116C4×C53—22no—
117C4×C53—22no—
118C4×C53—24no—
119C4×C53—41no—
120C4×C53—42no—
121Dic53—22no—
122Dic53—22no—
123Dic53—24no—
124Dic53—42no—
125Dic53—41no—
126#2583—531nomaximal, Sylow
127C2×C53#52511yescharacteristic
128C2×C53C2×C4⋊Q811yescharacteristic
129C2×C53#53411yescharacteristic
130C2×C53#55412yes—
131C2×C53#55412yes—
132D53—24no—
133C2×C22⋊Q8—531no—
134C22×C4⋊C4—531no—
135#398—531no—
136#398—531no—
137#395—531no—
138#395D5311yescharacteristic, complemented
139#389—532no—
140#389—532no—
141#385—532no—
142#385—532no—
143C4×C22⋊C4—531no—
144C53#258311yescharacteristic, complemented, Sylow
145C22×Q8C2×D5311yescharacteristic
146C23×C4—531no—
147C23×C4—531no—
148C23×C4—531no—
149C22⋊Q8—1064no—
150C2×C4⋊C4—531no—
151C2×C4⋊C4—531no—
152C2×C4⋊C4—531no—
153C2×C4⋊C4—532no—
154C2×C4⋊C4—531no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4—1064no—
159C2×C4⋊C4C2×D5311yescharacteristic
160C2×C4⋊C4C2×D5312yes—
161C2×C22⋊C4—532no—
162C2×C22⋊C4—532no—
163C2×C42—532no—
164#95—531no—
165#95—532no—
166#95—532no—
167#95—532no—
168#95C2×D5311yescharacteristic
169#95C2×D5312yes—
170C24—531no—
171C2×Q8—22no—
172C2×Q8—42no—
173C22×C4—532no—
174C22×C4—531no—
175C22×C4—531no—
176C22×C4—534no—
177C22×C4—532no—
178C22×C4—531no—
179C22×C4—1062no—
180C22×C4—1062no—
181C22×C4—1062no—
182C22×C4—1062no—
183C22×C4—2122no—
184C22×C4—2122no—
185C22×C4C22×D5311yescharacteristic
186C22×C4C22×D5312yes—
187C22×C4C22×D5311yescharacteristic
188C22×C4C22×D5312yes—
189C22×C4C22×D5311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—1062no—
193C4⋊C4—1062no—
194C4⋊C4—1064no—
195C4⋊C4—1064no—
196C4⋊C4—1062no—
197C4⋊C4—1062no—
198C4⋊C4—1064no—
199C4⋊C4—2122no—
200C22⋊C4—538no—
201C22⋊C4—1064no—
202C42—2122no—
203C23—532no—
204C23—534no—
205C23—1061no—
206C23—1061no—
207C23—1062no—
208C23C23×D5311yescharacteristic
209Q8—44no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—532no—
221C2×C4—534no—
222C2×C4—532no—
223C2×C4—1068no—
224C2×C4—1061no—
225C2×C4—1062no—
226C2×C4—1061no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1061no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—2124no—
235C2×C4—2124no—
236C2×C4D4×D5312yes—
237C2×C4Q8×D5314yes—
238C2×C4#1171994912yes—
239C22—534no—
240C22—1064no—
241C22—1064no—
242C22—1062no—
243C22—1062no—
244C22C2×D4×D5311yescharacteristic
245C22C2×Q8×D5312yes—
246C22#41×D5311yescharacteristic
247C22C2×#1171994911yescharacteristic
248C22#1313326212yes—
249C4—24no—
250C4—22no—
251C4—22no—
252C4—41no—
253C4—42no—
254C4—1064no—
255C4—1062no—
256C4—1062no—
257C4—2122no—
258C4—2121no—
259C2—1064no—
260C2#1681113611yescharacteristic
261C2C4⋊Q8×D5311yescharacteristic
262C2#1681118311yescharacteristic
263C2#1681132712yes—
264C2#1681133412yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2583
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
3. #16810555C2semidirect
4. #16810553C2semidirect
5. #16810542C2semidirect
6. #16810520C2semidirect
7. #16810519C2semidirect
11. C53×#395C2semidirect
60. #11719921Q8semidirect
138. #395D53semidirect
144. C53#2583semidirect