Finite Group Databasebeta

← #20280496

Subgroups — #20280496

Order 6784.

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

normal subgroups only

This group has 526 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×#13133202C211yesmaximal, characteristic
2C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
3#16810564C212yesmaximal, complemented
4#16810558C212yesmaximal, complemented
5#16810553C211yesmaximal, characteristic, complemented
6#16810550C211yesmaximal, characteristic, complemented
7#16810549C211yesmaximal, characteristic, complemented
8#16810501C212yesmaximal
9#16810494C212yesmaximal, complemented
10#16810485C211yesmaximal, characteristic, complemented
11C53×#398C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14C22×C4×D53C2212yes—
15#13133202—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2212yes—
20C4⋊C4×D53—24no—
21C2×#11719921C2212yes—
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27#13133124C2212yes—
28#13133124C2211yescharacteristic
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C23×D53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×Dic53C2311yescharacteristic
45C22×Dic53C2312yes—
46C22×Dic53C2312yes—
47C22×Dic53C2311yescharacteristic
48C2×D212—22no—
49C2×D212—24no—
50C2×C4×D53—24no—
51C2×C4×D53—24no—
52C2×C4×D53—42no—
53C2×C4×D53—42no—
54C2×C4×D53D414yes—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—22no—
57C4⋊C4×C53—42no—
58#11719921—22no—
59#11719921—24no—
60#11719921—22no—
61#11719921—22no—
62#11719921—42no—
63#11719921—42no—
64C4⋊Dic53—22no—
65#11719919—24no—
66#11719919—22no—
67#11719919—42no—
68C23×C53C2411yescharacteristic
69C22×D53—21no—
70C22×D53—21no—
71C22×D53—22no—
72C22×D53—21no—
73C22×D53—22no—
74C22×D53—21no—
75C22×D53—21no—
76C22×D53—22no—
77C22×D53C2×D412yes—
78C22×D53#4114yes—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53C2×D412yes—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53C2×D412yes—
102D212—44no—
103C4×D53—28no—
104C4×D53—48no—
105C22×C53C22×D411yescharacteristic
106C22×C53C2×#4112yes—
107C22×C53H(5,𝔽2)11yescharacteristic
108C22×C53H(5,𝔽2)11yescharacteristic
109C22×C532−1+412yes—
110C2×D53—22no—
111C2×D53—22no—
112C2×D53—24no—
113C2×D53—24no—
114C2×D53—41no—
115C2×D53—41no—
116C2×D53—41no—
117C2×D53—42no—
118C2×D53—42no—
119C2×D53#12314yes—
120C4×C53—22no—
121C4×C53—41no—
122C4×C53—42no—
123C4×C53—41no—
124C4×C53—42no—
125Dic53—22no—
126Dic53—42no—
127Dic53—41no—
128Dic53—42no—
129#2623—531nomaximal, Sylow
130C2×C53C2×#12311yescharacteristic
131C2×C53#53511yescharacteristic
132C2×C53#53611yescharacteristic
133C2×C53#53912yes—
134C2×C53#53912yes—
135D53—24no—
136D53—81no—
137C2×#123—531no—
138C22×C4⋊C4—531no—
139#400—532no—
140#399—532no—
141#398—531no—
142#398D5311yescharacteristic, complemented
143#397—531no—
144#397—531no—
145#386—532no—
146#385—532no—
147#381—531no—
148C53#262311yescharacteristic, complemented, Sylow
149C22×D4—531no—
150C23×C4—531no—
151C23×C4—532no—
152#123—1064no—
153C2×C4⋊C4—532no—
154C2×C4⋊C4—531no—
155C2×C4⋊C4—531no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—1064no—
158C2×C4⋊C4C2×D5311yescharacteristic
159C2×C4⋊C4C2×D5311yescharacteristic
160C2×C4⋊C4C2×D5312yes—
161C2×C22⋊C4—531no—
162C2×C22⋊C4—531no—
163C2×C22⋊C4—531no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4—532no—
166C2×C22⋊C4—532no—
167#95—531no—
168#95—532no—
169#95—532no—
170#95—532no—
171#95—532no—
172#95C2×D5311yescharacteristic
173#95C2×D5312yes—
174C24—531no—
175C24—531no—
176C2×D4—1062no—
177C2×D4—1064no—
178C22×C4—531no—
179C22×C4—534no—
180C22×C4—531no—
181C22×C4—532no—
182C22×C4—532no—
183C22×C4—1064no—
184C22×C4—1064no—
185C22×C4—2122no—
186C22×C4—2122no—
187C22×C4C22×D5312yes—
188C22×C4C22×D5311yescharacteristic
189C22×C4C22×D5312yes—
190C22×C4C22×D5311yescharacteristic
191C22×C4C22×D5311yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—42no—
195C4⋊C4—1064no—
196C4⋊C4—1062no—
197C4⋊C4—1062no—
198C4⋊C4—2122no—
199C22⋊C4—1062no—
200C22⋊C4—1062no—
201C22⋊C4—1064no—
202C22⋊C4—1062no—
203C22⋊C4—2122no—
204C22⋊C4—2122no—
205C23—532no—
206C23—534no—
207C23—1061no—
208C23—1061no—
209C23—1062no—
210C23—1061no—
211C23—1062no—
212C23—1061no—
213C23—1061no—
214C23—1062no—
215C23C23×D5311yescharacteristic
216D4—2124no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—532no—
230C2×C4—1068no—
231C2×C4—1062no—
232C2×C4—1061no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—2128no—
241C2×C4D4×D5312yes—
242C22—534no—
243C22—1062no—
244C22—1062no—
245C22—1064no—
246C22—1064no—
247C22—2121no—
248C22—2121no—
249C22—2121no—
250C22—2122no—
251C22—2122no—
252C22C2×D4×D5311yescharacteristic
253C22#41×D5312yes—
254C22#1313326211yescharacteristic
255C22#1313326311yescharacteristic
256C22#1313326512yes—
257C4—22no—
258C4—41no—
259C4—42no—
260C4—41no—
261C4—42no—
262C4—1062no—
263C4—2122no—
264C4—2121no—
265C4—2122no—
266C2—1064no—
267C2—4241no—
268C2#123×D5311yescharacteristic
269C2#1681118911yescharacteristic
270C2#1681119811yescharacteristic
271C2#1681121912yes—
272C2#1681122112yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2623
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D53C2semidirect
3. #16810564C2semidirect
4. #16810558C2semidirect
5. #16810553C2semidirect
6. #16810550C2semidirect
7. #16810549C2semidirect
9. #16810494C2semidirect
10. #16810485C2semidirect
11. C53×#398C2semidirect
142. #398D53semidirect
148. C53#2623semidirect