Finite Group Databasebeta

← #20280286

Subgroups — #20280286

Order 6784.

492 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 145 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×#13133174C211yesmaximal, characteristic
2#16810554C211yesmaximal, characteristic, complemented
3#16810549C211yesmaximal, characteristic
4#16810542C211yesmaximal, characteristic, complemented
5#16810534C211yesmaximal, characteristic
6#16810514C212yesmaximal
7#16810507C212yesmaximal, complemented
8#16810504C212yesmaximal, complemented
9#16810494C212yesmaximal
10#16810491C211yesmaximal, characteristic, complemented
11C2×C4×Q8×C53C211yesmaximal, characteristic, complemented
12C22×C4×D53C2211yescharacteristic
13#13133174—24no—
14C2×#11719919C2212yes—
15C2×#11719919C2211yescharacteristic
16C2×#11719919C2212yes—
17C2×#11719919C2211yescharacteristic
18C2×#11719921C2211yescharacteristic
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2211yescharacteristic
22C2×C4×Dic53C2211yescharacteristic
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2211yescharacteristic
30#13133124C2211yescharacteristic
31C22×Q8×C53C2211yescharacteristic
32C4×Q8×C53—24no—
33C2×C4⋊C4×C53C2212yes—
34C2×C4⋊C4×C53C2211yescharacteristic
35C2×C42×C53C2212yes—
36C2×C42×C53C2211yescharacteristic
37C23×D53C2311yescharacteristic
38C2×Q8×C53—22no—
39C2×Q8×C53—24no—
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2312yes—
45C22×Dic53C2312yes—
46C22×Dic53C2312yes—
47C22×Dic53C2311yescharacteristic
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2311yescharacteristic
50C2×C4×D53—22no—
51C2×C4×D53—22no—
52C2×C4×D53—22no—
53C4⋊C4×C53—24no—
54C4⋊C4×C53—22no—
55C42×C53—24no—
56C42×C53D414yes—
57#11719921—24no—
58#11719921—22no—
59#11719921—22no—
60#11719921—24no—
61#11719919—22no—
62#11719919—22no—
63#11719919—24no—
64#11719919—24no—
65C4×Dic53—22no—
66C23×C53C2411yescharacteristic
67C22×D53—21no—
68C22×D53—21no—
69C22×D53—22no—
70C22×D53—22no—
71C22×D53#4112yes—
72Q8×C53—28no—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53C2×D412yes—
83C2×C4×C53C2×D412yes—
84C2×C4×C53C2×D412yes—
85C2×C4×C53#4114yes—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53#4112yes—
99C4×D53—28no—
100C22×C53C22×D411yescharacteristic
101C22×C53C2×#4111yescharacteristic
102C22×C53C2×#4111yescharacteristic
103C22×C53C2×#4112yes—
104C22×C532−1+412yes—
105C2×D53—22no—
106C2×D53—24no—
107C2×D53—41no—
108C2×D53—41no—
109C2×D53—42no—
110C4×C53—22no—
111C4×C53—22no—
112C4×C53—24no—
113C4×C53—22no—
114C4×C53—42no—
115Dic53—22no—
116Dic53—41no—
117Dic53—41no—
118Dic53—42no—
119Dic53—42no—
120#2599—531nomaximal, Sylow
121C2×C53#52511yescharacteristic
122C2×C53#52511yescharacteristic
123C2×C53#53611yescharacteristic
124C2×C53#55212yes—
125C2×C53#55612yes—
126D53—42no—
127C2×C4×Q8D5311yescharacteristic, complemented
128C2×#117—531no—
129#398—531no—
130#397—531no—
131#395—531no—
132#393—531no—
133#388—532no—
134#387—532no—
135#387—532no—
136#385—532no—
137#383—531no—
138C53#259911yescharacteristic, complemented, Sylow
139C22×Q8C2×D5311yescharacteristic
140C23×C4—531no—
141C4×Q8—24no—
142#117—1064no—
143C2×C4⋊C4—531no—
144C2×C4⋊C4—531no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4C2×D5312yes—
148C2×C4⋊C4C2×D5311yescharacteristic
149C2×C22⋊C4—531no—
150C2×C22⋊C4—531no—
151C2×C22⋊C4—532no—
152C2×C22⋊C4—532no—
153C2×C42—531no—
154C2×C42C2×D5312yes—
155C2×C42C2×D5311yescharacteristic
156#95—532no—
157#95—532no—
158#95—531no—
159#95—531no—
160#95—532no—
161#95—532no—
162#95—532no—
163#95—532no—
164C24—531no—
165C2×Q8—24no—
166C2×Q8—22no—
167C22×C4—531no—
168C22×C4—532no—
169C22×C4—532no—
170C22×C4—531no—
171C22×C4—531no—
172C22×C4—1062no—
173C22×C4—1062no—
174C22×C4—1062no—
175C22×C4C22×D5312yes—
176C22×C4C22×D5312yes—
177C22×C4C22×D5311yescharacteristic
178C22×C4C22×D5311yescharacteristic
179C22×C4C22×D5311yescharacteristic
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—1064no—
183C4⋊C4—1064no—
184C4⋊C4—1062no—
185C4⋊C4—1062no—
186C22⋊C4—1064no—
187C22⋊C4—1064no—
188C22⋊C4—1062no—
189C22⋊C4—1062no—
190C42—24no—
191C42—1062no—
192C42C22⋊D5314yes—
193C23—532no—
194C23—1061no—
195C23—1061no—
196C23—1062no—
197C23—1062no—
198C23C23×D5311yescharacteristic
199Q8—28no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—532no—
210C2×C4—1068no—
211C2×C4—1061no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—1061no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4#1171994514yes—
224C2×C4C2×C22⋊D5312yes—
225C2×C4C2×C22⋊D5312yes—
226C2×C4C2×C22⋊D5312yes—
227C22—1064no—
228C22—1062no—
229C22—2122no—
230C22—2121no—
231C22—2121no—
232C22C22×C22⋊D5311yescharacteristic
233C22#41×D5311yescharacteristic
234C22#41×D5311yescharacteristic
235C22C2×#1171994512yes—
236C22#1313326512yes—
237C4—22no—
238C4—24no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—1062no—
243C4—2122no—
244C4—2122no—
245C4—2121no—
246C4—2121no—
247C2—2122no—
248C2#1681113011yescharacteristic
249C2#1681113411yescharacteristic
250C2#1681119711yescharacteristic
251C2#1681131412yes—
252C2#1681134512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C4×Q8×C53C2
hypercenterC2×C4×Q8D53
hypocenterC53#2599
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #16810554C2semidirect
4. #16810542C2semidirect
7. #16810507C2semidirect
8. #16810504C2semidirect
10. #16810491C2semidirect
11. C2×C4×Q8×C53C2semidirect
127. C2×C4×Q8D53semidirect
138. C53#2599semidirect