Finite Group Databasebeta

← #20280868

Subgroups — #20280868

Order 6784.

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

normal subgroups only

This group has 498 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×#13133199C211yesmaximal, characteristic
2C2×#13133197C211yesmaximal, characteristic, complemented
3#16810561C211yesmaximal, characteristic, complemented
4#16810549C212yesmaximal, complemented
5#16810544C211yesmaximal, characteristic, complemented
6#16810539C212yesmaximal
7#16810535C211yesmaximal, characteristic, complemented
8#16810522C212yesmaximal, complemented
9#16810516C212yesmaximal, complemented
10#16810470C211yesmaximal, characteristic, complemented
11C53×#399C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×C22⋊D53C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15#13133199—24no—
16#13133197—24no—
17C2×C4⋊Dic53C2212yes—
18C2×C4⋊Dic53C2211yescharacteristic
19C2×C4⋊Dic53C2212yes—
20C2×#11719919C2211yescharacteristic
21C2×C22⋊Dic53C2211yescharacteristic
22C2×C22⋊Dic53C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2211yescharacteristic
26C2×C4×Dic53C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C2×C4⋊C4×C53C2212yes—
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2212yes—
35C2×C22⋊C4×C53C2211yescharacteristic
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C2×C22⋊D53—22no—
44C2×C22⋊D53—42no—
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2312yes—
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2311yescharacteristic
50C2×C4×D53—42no—
51C2×C4×D53D414yes—
52C2×Dic106—22no—
53C2×Dic106—42no—
54C4⋊C4×C53—22no—
55C4⋊C4×C53—24no—
56C22⋊C4×C53—22no—
57C22⋊C4×C53—24no—
58C22⋊Dic53—22no—
59C22⋊Dic53—42no—
60#11719921—24no—
61#11719921—22no—
62#11719921—24no—
63C4⋊Dic53—24no—
64C4⋊Dic53—22no—
65C4⋊Dic53—42no—
66#11719919—22no—
67C4×Dic53—42no—
68C23×C53—21no—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53C2411yescharacteristic
74C22×D53—22no—
75C22×D53—22no—
76C22×D53—21no—
77C22×D53—21no—
78C22×D53C2×D412yes—
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×C53#4114yes—
89C22⋊D53—44no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—22no—
102C2×Dic53C2×D412yes—
103C4×D53—44no—
104Dic106—44no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—41no—
108C22×C53—42no—
109C22×C53—42no—
110C22×C53C22×D411yescharacteristic
111C22×C53C2×#4112yes—
112C22×C53H(5,𝔽2)12yes—
113C22×C532−1+411yescharacteristic
114C22×C532−1+411yescharacteristic
115C2×D53—22no—
116C2×D53—22no—
117C2×D53—22no—
118C2×D53—42no—
119C2×D53—42no—
120C4×C53—22no—
121C4×C53—24no—
122C4×C53—41no—
123C4×C53—42no—
124Dic53—22no—
125Dic53—42no—
126Dic53—41no—
127Dic53—41no—
128Dic53—42no—
129#2663—531nomaximal, Sylow
130C2×C53—81no—
131C2×C53#53211yescharacteristic
132C2×C53#53411yescharacteristic
133C2×C53#53611yescharacteristic
134C2×C53#53912yes—
135C2×C53#54212yes—
136D53—42no—
137C2×#123—531no—
138C2×C22⋊Q8—531no—
139#400—531no—
140#399D5311yescharacteristic, complemented
141#397—532no—
142#395—531no—
143#394—532no—
144#393—531no—
145#389—532no—
146#388—532no—
147C4×C22⋊C4—531no—
148C53#266311yescharacteristic, complemented, Sylow
149C22×Q8—531no—
150C22×D4—531no—
151C23×C4—531no—
152#123—1064no—
153C22⋊Q8—1064no—
154C2×C4⋊C4—532no—
155C2×C4⋊C4—531no—
156C2×C4⋊C4—531no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4C2×D5311yescharacteristic
159C2×C4⋊C4C2×D5312yes—
160C2×C22⋊C4—531no—
161C2×C22⋊C4—532no—
162C2×C22⋊C4—531no—
163C2×C22⋊C4—532no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4C2×D5311yescharacteristic
166C2×C22⋊C4C2×D5312yes—
167C2×C42—532no—
168#95—531no—
169#95—532no—
170#95—532no—
171#95—532no—
172#95—532no—
173#95C2×D5311yescharacteristic
174C24—531no—
175C24C22×D5311yescharacteristic
176C2×Q8—1062no—
177C2×Q8—2122no—
178C2×D4—1062no—
179C2×D4—2122no—
180C22×C4—531no—
181C22×C4—534no—
182C22×C4—532no—
183C22×C4—531no—
184C22×C4—532no—
185C22×C4—531no—
186C22×C4—2122no—
187C22×C4C22×D5311yescharacteristic
188C22×C4C22×D5312yes—
189C22×C4C22×D5311yescharacteristic
190C22×C4C22×D5312yes—
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—1064no—
194C4⋊C4—1062no—
195C4⋊C4—1062no—
196C4⋊C4—2122no—
197C22⋊C4—22no—
198C22⋊C4—24no—
199C22⋊C4—1064no—
200C22⋊C4—1062no—
201C22⋊C4—1062no—
202C22⋊C4—1064no—
203C22⋊C4—2122no—
204C42—2122no—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—532no—
211C23—1062no—
212C23—1062no—
213C23—1061no—
214C23—1061no—
215C23C23×D5311yescharacteristic
216Q8—2124no—
217D4—2124no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—532no—
227C2×C4—1062no—
228C2×C4—1061no—
229C2×C4—1062no—
230C2×C4—1061no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1061no—
237C2×C4—1061no—
238C2×C4—1062no—
239C2×C4—2124no—
240C2×C4D4×D5312yes—
241C2×C4#1171994714yes—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—1062no—
248C22—1062no—
249C22—1062no—
250C22—2122no—
251C22—2122no—
252C22C2×D4×D5311yescharacteristic
253C22C2×#1171994712yes—
254C22#1313326312yes—
255C22#1313326411yescharacteristic
256C22#1313326511yescharacteristic
257C4—22no—
258C4—24no—
259C4—41no—
260C4—42no—
261C4—1062no—
262C4—2121no—
263C4—2122no—
264C4—2121no—
265C4—2122no—
266C2—81no—
267C2—2122no—
268C2#1681117111yescharacteristic
269C2#1681118411yescharacteristic
270C2#1681119611yescharacteristic
271C2#1681121712yes—
272C2#1681124612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2663
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×#13133197C2semidirect
3. #16810561C2semidirect
4. #16810549C2semidirect
5. #16810544C2semidirect
7. #16810535C2semidirect
8. #16810522C2semidirect
9. #16810516C2semidirect
10. #16810470C2semidirect
11. C53×#399C2semidirect
140. #399D53semidirect
148. C53#2663semidirect