Finite Group Databasebeta

← #20280580

Subgroups — #20280580

Order 6784.

500 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 131 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×#13133204C211yesmaximal, characteristic
2C2×#13133182C211yesmaximal, characteristic, complemented
3#16810559C212yesmaximal, complemented
4#16810544C211yesmaximal, characteristic, complemented
5#16810543C211yesmaximal, characteristic, complemented
6#16810537C211yesmaximal, characteristic
7#16810535C212yesmaximal, complemented
8#16810517C212yesmaximal, complemented
9#16810516C212yesmaximal, complemented
10#16810490C211yesmaximal, characteristic, complemented
11C53×#394C211yesmaximal, characteristic
12C22×Dic106C2211yescharacteristic
13C22×Dic106C2211yescharacteristic
14#13133204—24no—
15#13133182—24no—
16C2×C4⋊Dic53C2212yes—
17C2×#11719919C2211yescharacteristic
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×C22⋊Dic53C2212yes—
21C2×C22⋊Dic53C2212yes—
22C2×C22⋊Dic53C2212yes—
23C2×C22⋊Dic53C2212yes—
24C2×C4×Dic53C2211yescharacteristic
25C2×C4×Dic53C2211yescharacteristic
26C2×C4×Dic53C2211yescharacteristic
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C22×D4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2212yes—
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2312yes—
38C2×D4×C53—22no—
39C2×D4×C53—42no—
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2311yescharacteristic
50C2×Dic106—22no—
51C2×Dic106—22no—
52C2×Dic106—22no—
53C2×Dic106—22no—
54C2×Dic106—42no—
55C4⋊C4×C53—22no—
56C22⋊C4×C53—24no—
57C22⋊C4×C53—42no—
58C22⋊Dic53—24no—
59C22⋊Dic53—24no—
60C22⋊Dic53—24no—
61C22⋊Dic53—42no—
62C4⋊Dic53—42no—
63#11719919—22no—
64#11719919—22no—
65#11719919—42no—
66C4×Dic53—22no—
67C4×Dic53—22no—
68C4×Dic53D414yes—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53C2411yescharacteristic
77D4×C53—44no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53C2×D412yes—
87C2×C4×C53#4112yes—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—21no—
91C2×Dic53—21no—
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—
102C2×Dic53C2×D412yes—
103C2×Dic53#4112yes—
104Dic106—28no—
105Dic106—44no—
106C22×C53—42no—
107C22×C53—42no—
108C22×C53—42no—
109C22×C53—42no—
110C22×C53—42no—
111C22×C53—42no—
112C22×C53—42no—
113C22×C53C22×D411yescharacteristic
114C22×C53C2×#4111yescharacteristic
115C22×C53C2×#4111yescharacteristic
116C22×C53H(5,𝔽2)12yes—
117C22×C532−1+412yes—
118C4×C53—22no—
119C4×C53—22no—
120C4×C53—41no—
121C4×C53—42no—
122Dic53—22no—
123Dic53—22no—
124Dic53—22no—
125Dic53—42no—
126Dic53—41no—
127Dic53—42no—
128#2632—531nomaximal, Sylow
129C2×C53—82no—
130C2×C53#52511yescharacteristic
131C2×C53#53511yescharacteristic
132C2×C53#53611yescharacteristic
133C2×C53#54212yes—
134C2×C53#54212yes—
135C2×#124—531no—
136C2×C4×Q8—531no—
137#399—532no—
138#395—531no—
139#395—531no—
140#394—531no—
141#394D5311yescharacteristic
142#393—532no—
143#388—532no—
144#388—532no—
145#383—531no—
146C53#263211yescharacteristic, complemented, Sylow
147C22×Q8—531no—
148C22×Q8—531no—
149C22×D4C2×D5311yescharacteristic
150#124—1064no—
151C4×Q8—1064no—
152C2×C4⋊C4—531no—
153C2×C4⋊C4—531no—
154C2×C4⋊C4—532no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4C2×D5311yescharacteristic
157C2×C22⋊C4—532no—
158C2×C22⋊C4—532no—
159C2×C22⋊C4—532no—
160C2×C22⋊C4—532no—
161C2×C22⋊C4C2×D5312yes—
162C2×C22⋊C4C2×D5312yes—
163C2×C42—531no—
164C2×C42—531no—
165C2×C42—531no—
166#95—532no—
167#95—532no—
168#95—531no—
169#95—532no—
170#95—532no—
171#95C2×D5311yescharacteristic
172C24C22×D5312yes—
173C2×Q8—1062no—
174C2×Q8—1062no—
175C2×Q8—1062no—
176C2×Q8—1062no—
177C2×Q8—2122no—
178C2×D4—22no—
179C2×D4—42no—
180C22×C4—531no—
181C22×C4—531no—
182C22×C4—531no—
183C22×C4—532no—
184C22×C4—532no—
185C22×C4—531no—
186C22×C4C22×D5312yes—
187C22×C4C22×D5311yescharacteristic
188C22×C4C22×D5311yescharacteristic
189C22×C4C22×D5311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—1062no—
192C4⋊C4—1062no—
193C4⋊C4—2122no—
194C4⋊C4—2122no—
195C22⋊C4—24no—
196C22⋊C4—42no—
197C22⋊C4—1064no—
198C22⋊C4—1064no—
199C22⋊C4—1064no—
200C22⋊C4—2122no—
201C42—534no—
202C42—1062no—
203C42—1062no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23C23×D5311yescharacteristic
212Q8—1068no—
213Q8—2124no—
214D4—44no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—532no—
224C2×C4—532no—
225C2×C4—532no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—1061no—
229C2×C4—1061no—
230C2×C4—1062no—
231C2×C4—1061no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4D4×D5312yes—
240C2×C4#1171994712yes—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×D4×D5311yescharacteristic
249C22#41×D5311yescharacteristic
250C22C2×#1171994711yescharacteristic
251C22#1313326212yes—
252C22#1313326412yes—
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—1062no—
258C4—1062no—
259C4—1062no—
260C4—2121no—
261C4—2122no—
262C4—2122no—
263C2—82no—
264C2#1681113311yescharacteristic
265C2#1681118811yescharacteristic
266C2#1681119311yescharacteristic
267C2#1681124012yes—
268C2#1681124112yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2632
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#13133182C2semidirect
3. #16810559C2semidirect
4. #16810544C2semidirect
5. #16810543C2semidirect
7. #16810535C2semidirect
8. #16810517C2semidirect
9. #16810516C2semidirect
10. #16810490C2semidirect
146. C53#2632semidirect