Finite Group Databasebeta

← #20282726

Subgroups — #20282726

Order 6784.

478 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133200C211yesmaximal, characteristic
2#16810556C212yesmaximal, complemented
3#16810554C211yesmaximal, characteristic, complemented
4#16810551C212yesmaximal, complemented
5#16810549C211yesmaximal, characteristic, complemented
6#16810545C212yesmaximal, complemented
7#16810542C211yesmaximal, characteristic, complemented
8#16810538C212yesmaximal
9#16810534C211yesmaximal, characteristic, complemented
10#16810483C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2211yescharacteristic
13#13133200—24no—
14C2×C4⋊Dic53C2212yes—
15C2×#11719919C2211yescharacteristic
16C2×#11719919C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×C22⋊Dic53C2211yescharacteristic
20C2×C22⋊Dic53C2212yes—
21C2×#11719921C2211yescharacteristic
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25#13133124C2212yes—
26#13133124C2211yescharacteristic
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2211yescharacteristic
30#13133124C2212yes—
31C22×Q8×C53C2211yescharacteristic
32C23×C4×C53C2211yescharacteristic
33C22⋊Q8×C53—24no—
34C2×C4⋊C4×C53C2212yes—
35C2×C4⋊C4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2212yes—
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C2×Q8×C53—22no—
40C2×Q8×C53—42no—
41C22×C4×C53—42no—
42C22×C4×C53D414yes—
43C22×C4×C53C2312yes—
44C22×C4×C53C2312yes—
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—22no—
48C2×C22⋊D53—42no—
49C22×Dic53C2311yescharacteristic
50C22×Dic53C2312yes—
51C22×Dic53C2312yes—
52C22×Dic53C2311yescharacteristic
53C22×Dic53C2311yescharacteristic
54C4⋊C4×C53—24no—
55C4⋊C4×C53—22no—
56C22⋊C4×C53—24no—
57C22⋊Dic53—22no—
58C22⋊Dic53—42no—
59#11719921—22no—
60#11719921—22no—
61#11719921—42no—
62#11719921—42no—
63C4⋊Dic53—42no—
64#11719919—22no—
65#11719919—22no—
66#11719919—42no—
67#11719919—42no—
68C23×C53—22no—
69C23×C53—21no—
70C23×C53—21no—
71C23×C53—22no—
72C23×C53C2×D412yes—
73C23×C53C2411yescharacteristic
74C22×D53—21no—
75C22×D53—22no—
76C22×D53—21no—
77C22×D53—21no—
78C22×D53—22no—
79Q8×C53—44no—
80C2×C4×C53—22no—
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×C53—44no—
89C2×C4×C53C2×D412yes—
90C2×C4×C53C2×D412yes—
91C22⋊D53—44no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—21no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—21no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C22×C53—22no—
106C22×C53—22no—
107C22×C53—22no—
108C22×C53—42no—
109C22×C53—42no—
110C22×C53C22×D411yescharacteristic
111C22×C53H(5,𝔽2)12yes—
112C22×C53H(5,𝔽2)11yescharacteristic
113C22×C532−1+411yescharacteristic
114C22×C532−1+412yes—
115C2×D53—41no—
116C2×D53—41no—
117C2×D53—41no—
118C2×D53—42no—
119C2×D53—42no—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—42no—
123C4×C53—42no—
124Dic53—41no—
125Dic53—41no—
126Dic53—41no—
127Dic53—42no—
128Dic53—42no—
129#2814—531nomaximal, Sylow
130C2×C53—42no—
131C2×C53#53411yescharacteristic
132C2×C53#53611yescharacteristic
133C2×C53#53711yescharacteristic
134C2×C53#56212yes—
135C2×C53#56312yes—
136D53—81no—
137C2×#123—531no—
138C2×C22⋊Q8D5311yescharacteristic, complemented
139#398—531no—
140#398—532no—
141#397—531no—
142#397—532no—
143#396—532no—
144#395—531no—
145#394—532no—
146#393—531no—
147#380—531no—
148C53#281411yescharacteristic, complemented, Sylow
149C22×Q8C2×D5311yescharacteristic
150C22×D4—531no—
151C23×C4C2×D5311yescharacteristic
152#123—1064no—
153C22⋊Q8—24no—
154C2×C4⋊C4—531no—
155C2×C4⋊C4—531no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4—532no—
159C2×C4⋊C4C2×D5312yes—
160C2×C4⋊C4C2×D5311yescharacteristic
161C2×C22⋊C4—531no—
162C2×C22⋊C4—531no—
163C2×C22⋊C4—531no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4—532no—
166C2×C22⋊C4—532no—
167C2×C22⋊C4C2×D5312yes—
168#95—531no—
169#95—531no—
170#95—532no—
171#95—532no—
172#95—532no—
173#95—532no—
174C24—531no—
175C24C22×D5311yescharacteristic
176C2×Q8—22no—
177C2×Q8—42no—
178C2×D4—1062no—
179C2×D4—2122no—
180C22×C4—42no—
181C22×C4—531no—
182C22×C4—532no—
183C22×C4—532no—
184C22×C4—531no—
185C22×C4—531no—
186C22×C4C22⋊D5314yes—
187C22×C4C22×D5312yes—
188C22×C4C22×D5312yes—
189C22×C4C22×D5311yescharacteristic
190C22×C4C22×D5311yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—1062no—
194C4⋊C4—1062no—
195C4⋊C4—2122no—
196C4⋊C4—2122no—
197C4⋊C4—2122no—
198C22⋊C4—24no—
199C22⋊C4—1062no—
200C22⋊C4—1062no—
201C22⋊C4—1062no—
202C22⋊C4—2122no—
203C22⋊C4—2122no—
204C22⋊C4—2122no—
205C23—22no—
206C23—21no—
207C23—21no—
208C23—22no—
209C23—1061no—
210C23—1062no—
211C23—1061no—
212C23—1061no—
213C23—1062no—
214C23C2×C22⋊D5312yes—
215C23C23×D5311yescharacteristic
216Q8—44no—
217D4—2124no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—44no—
227C2×C4—1061no—
228C2×C4—1061no—
229C2×C4—1061no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—1061no—
236C2×C4—1061no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4C2×C22⋊D5312yes—
241C2×C4C2×C22⋊D5312yes—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—2122no—
248C22—2122no—
249C22—2121no—
250C22—2121no—
251C22—2121no—
252C22C22×C22⋊D5311yescharacteristic
253C22#1313326212yes—
254C22#1313326311yescharacteristic
255C22#1313326411yescharacteristic
256C22#1313326512yes—
257C4—22no—
258C4—22no—
259C4—42no—
260C4—42no—
261C4—2122no—
262C4—2122no—
263C4—2121no—
264C4—2121no—
265C4—2121no—
266C2—42no—
267C2—4241no—
268C2#1681118211yescharacteristic
269C2#1681119711yescharacteristic
270C2#1681120111yescharacteristic
271C2#1681136912yes—
272C2#1681137412yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C22⋊Q8×C53C2
hypercenterC2×C22⋊Q8D53
hypocenterC53#2814
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16810556C2semidirect
3. #16810554C2semidirect
4. #16810551C2semidirect
5. #16810549C2semidirect
6. #16810545C2semidirect
7. #16810542C2semidirect
9. #16810534C2semidirect
10. #16810483C2semidirect
11. C2×C22⋊Q8×C53C2semidirect
138. C2×C22⋊Q8D53semidirect
148. C53#2814semidirect