Finite Group Databasebeta

← #20281724

Subgroups — #20281724

Order 6784.

492 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 137 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×#13133215C211yesmaximal, characteristic, complemented
2C2×#13133206C211yesmaximal, characteristic
3#16810545C212yesmaximal, complemented
4#16810541C212yesmaximal
5#16810536C212yesmaximal, complemented
6#16810515C212yesmaximal, complemented
7#16810514C212yesmaximal, complemented
8#16810508C212yesmaximal, complemented
9C53×#391C211yesmaximal, characteristic, complemented
10C22×Dic106C2211yescharacteristic
11C22×D212C2211yescharacteristic
12#13133215—24no—
13#13133206—24no—
14C2×C4⋊Dic53C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×C4×Dic53C2211yescharacteristic
24C2×C4×Dic53C2211yescharacteristic
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28C22×Q8×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2211yescharacteristic
30C2×C42×C53C2211yescharacteristic
31#95×C53C2212yes—
32#95×C53C2212yes—
33C23×D53C2312yes—
34C2×Q8×C53—41no—
35C2×Q8×C53—41no—
36C2×Q8×C53D414yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C2×D212—22no—
46C2×D212—41no—
47C2×D212—41no—
48C2×Dic106—22no—
49C2×Dic106—41no—
50C2×Dic106—41no—
51C4⋊C4×C53—41no—
52C42×C53—41no—
53#11719921—24no—
54#11719921—24no—
55#11719921—24no—
56#11719921—24no—
57#11719921—42no—
58#11719921—42no—
59C4⋊Dic53—24no—
60#11719919—24no—
61#11719919—24no—
62C4×Dic53—22no—
63C4×Dic53—22no—
64C23×C53C2411yescharacteristic
65C22×D53—22no—
66C22×D53—22no—
67C22×D53—22no—
68C22×D53—22no—
69C22×D53—22no—
70C22×D53—22no—
71C22×D53—22no—
72Q8×C53—44no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53C2×D412yes—
86C2×C4×C53C2×D412yes—
87C2×C4×C53C2×D412yes—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53#4114yes—
97D212—44no—
98Dic106—44no—
99C22×C53C22×D411yescharacteristic
100C22×C53C2×#4112yes—
101C22×C53H(5,𝔽2)11yescharacteristic
102C22×C53H(5,𝔽2)11yescharacteristic
103C22×C532−1+412yes—
104C2×D53—42no—
105C2×D53—42no—
106C2×D53—42no—
107C2×D53—42no—
108C2×D53—42no—
109C2×D53—42no—
110C2×D53—42no—
111C4×C53—22no—
112C4×C53—22no—
113C4×C53—22no—
114C4×C53—42no—
115C4×C53—42no—
116Dic53—24no—
117Dic53—42no—
118Dic53—42no—
119#2731—531nomaximal, Sylow
120C2×C53#53411yescharacteristic
121C2×C53#54212yes—
122C2×C53#54812yes—
123C2×C53#55511yescharacteristic
124C2×C53#56311yescharacteristic
125D53—82no—
126C2×C4⋊Q8—531no—
127C2×#124—531no—
128#396—532no—
129#394—532no—
130#393—532no—
131#391D5311yescharacteristic, complemented
132#388—532no—
133#388—532no—
134#387—532no—
135C53#273111yescharacteristic, complemented, Sylow
136C22×Q8—531no—
137C22×Q8C2×D5311yescharacteristic
138C22×D4—531no—
139C4⋊Q8—1064no—
140#124—1064no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4—532no—
144C2×C4⋊C4C2×D5311yescharacteristic
145C2×C22⋊C4—532no—
146C2×C22⋊C4—532no—
147C2×C22⋊C4—532no—
148C2×C22⋊C4—532no—
149C2×C22⋊C4—532no—
150C2×C22⋊C4—532no—
151C2×C42—531no—
152C2×C42—531no—
153C2×C42C2×D5311yescharacteristic
154#95—532no—
155#95—532no—
156#95—532no—
157#95C2×D5312yes—
158#95C2×D5312yes—
159C24—532no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—1062no—
163C2×Q8—2121no—
164C2×Q8—2121no—
165C2×Q8D21214yes—
166C2×D4—1062no—
167C2×D4—2121no—
168C2×D4—2121no—
169C22×C4—532no—
170C22×C4—532no—
171C22×C4—532no—
172C22×C4C22×D5312yes—
173C22×C4C22×D5312yes—
174C22×C4C22×D5311yescharacteristic
175C22×C4C22×D5311yescharacteristic
176C22×C4C22×D5311yescharacteristic
177C4⋊C4—41no—
178C4⋊C4—1064no—
179C4⋊C4—1064no—
180C4⋊C4—1064no—
181C22⋊C4—1064no—
182C22⋊C4—1064no—
183C22⋊C4—1064no—
184C22⋊C4—1064no—
185C22⋊C4—2122no—
186C22⋊C4—2122no—
187C42—41no—
188C42—1062no—
189C42—1062no—
190C23—1062no—
191C23—1062no—
192C23—1062no—
193C23—1062no—
194C23—1062no—
195C23—1062no—
196C23—1062no—
197C23C23×D5311yescharacteristic
198Q8—44no—
199Q8—2124no—
200D4—2124no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—534no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4C2×D21212yes—
223C2×C4C2×D21212yes—
224C2×C4C2×D21212yes—
225C22—2122no—
226C22—2122no—
227C22—2122no—
228C22—2122no—
229C22—2122no—
230C22—2122no—
231C22—2122no—
232C22C22×D21211yescharacteristic
233C22#41×D5312yes—
234C22#1313326211yescharacteristic
235C22#1313326211yescharacteristic
236C22#1313326512yes—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—42no—
241C4—42no—
242C4—1064no—
243C4—2122no—
244C4—2122no—
245C2—4242no—
246C2#1681118511yescharacteristic
247C2#1681124312yes—
248C2#1681128512yes—
249C2#1681133511yescharacteristic
250C2#1681137211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#391C2
hypercenter#391D53
hypocenterC53#2731
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#13133215C2semidirect
3. #16810545C2semidirect
5. #16810536C2semidirect
6. #16810515C2semidirect
7. #16810514C2semidirect
8. #16810508C2semidirect
9. C53×#391C2semidirect
131. #391D53semidirect
135. C53#2731semidirect