Finite Group Databasebeta

← #20281553

Subgroups — #20281553

Order 6784.

512 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133198C211yesmaximal, characteristic
3#16810548C212yesmaximal, complemented
4#16810548C212yesmaximal, complemented
5#16810544C212yesmaximal, complemented
6#16810508C212yesmaximal, complemented
7#16810498C212yesmaximal, complemented
8#16810483C212yesmaximal, complemented
9C53×#386C211yesmaximal, characteristic
10C22×Dic106C2211yescharacteristic
11C23×Dic53C2211yescharacteristic
12#13133204—24no—
13#13133198—24no—
14C2×C4⋊Dic53C2212yes—
15C2×C4⋊Dic53C2212yes—
16C2×C4⋊Dic53C2212yes—
17C2×#11719919C2212yes—
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2211yescharacteristic
20C2×C22⋊Dic53C2212yes—
21C2×C22⋊Dic53C2212yes—
22C2×C4×Dic53C2211yescharacteristic
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28C22×D4×C53C2211yescharacteristic
29C23×C4×C53C2211yescharacteristic
30C2×C22⋊C4×C53C2212yes—
31C2×C22⋊C4×C53C2211yescharacteristic
32#95×C53C2212yes—
33C24×C53C2311yescharacteristic
34C24×C53C2311yescharacteristic
35C2×D4×C53—41no—
36C2×D4×C53—41no—
37C2×D4×C53D414yes—
38C22×C4×C53—24no—
39C22×C4×C53—41no—
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C22×Dic53—22no—
44C22×Dic53—41no—
45C22×Dic53—41no—
46C22×Dic53C2312yes—
47C22×Dic53C2312yes—
48C22×Dic53C2312yes—
49C22×Dic53C2312yes—
50C2×Dic106—22no—
51C2×Dic106—41no—
52C2×Dic106—41no—
53C22⋊C4×C53—24no—
54C22⋊C4×C53—41no—
55C22⋊Dic53—24no—
56C22⋊Dic53—22no—
57C22⋊Dic53—24no—
58C22⋊Dic53—24no—
59C4⋊Dic53—24no—
60C4⋊Dic53—42no—
61C4⋊Dic53—42no—
62#11719919—24no—
63C4×Dic53—22no—
64C23×C53—22no—
65C23×C53—22no—
66C23×C53—21no—
67C23×C53—21no—
68C23×C53—21no—
69C23×C53—21no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53C2×D412yes—
73C23×C53C2×D412yes—
74C23×C53C2411yescharacteristic
75D4×C53—44no—
76C2×C4×C53—22no—
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—44no—
84C2×C4×C53C2×D412yes—
85C2×C4×C53#4114yes—
86C2×Dic53—22no—
87C2×Dic53—22no—
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—22no—
97C2×Dic53—22no—
98C2×Dic53—44no—
99Dic106—44no—
100C22×C53—22no—
101C22×C53—22no—
102C22×C53—24no—
103C22×C53—24no—
104C22×C53—42no—
105C22×C53—42no—
106C22×C53—41no—
107C22×C53—41no—
108C22×C53—41no—
109C22×C53—41no—
110C22×C53C22×D411yescharacteristic
111C22×C53C2×#4112yes—
112C22×C53H(5,𝔽2)11yescharacteristic
113C22×C53H(5,𝔽2)11yescharacteristic
114C22×C532−1+412yes—
115C4×C53—22no—
116C4×C53—24no—
117C4×C53—42no—
118Dic53—42no—
119Dic53—42no—
120Dic53—42no—
121Dic53—42no—
122#2717—531nomaximal, Sylow
123C2×C53—42no—
124C2×C53—42no—
125C2×C53#53411yescharacteristic
126C2×C53#54212yes—
127C2×C53#54712yes—
128C2×C53#55111yescharacteristic
129C2×C53#56311yescharacteristic
130C2×#124—531no—
131C2×#123—531no—
132#397—532no—
133#397—532no—
134#395—532no—
135#387—532no—
136#386D5311yescharacteristic
137#385—532no—
138#380—532no—
139C53#271711yescharacteristic, complemented, Sylow
140C22×Q8—531no—
141C22×D4C2×D5311yescharacteristic
142C23×C4—531no—
143C23×C4C2×D5311yescharacteristic
144#124—1064no—
145#123—1064no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4—532no—
149C2×C4⋊C4—532no—
150C2×C22⋊C4—532no—
151C2×C22⋊C4—531no—
152C2×C22⋊C4—532no—
153C2×C22⋊C4—532no—
154C2×C22⋊C4C2×D5312yes—
155C2×C22⋊C4C2×D5311yescharacteristic
156C2×C42—531no—
157#95—532no—
158#95—532no—
159#95—532no—
160#95—532no—
161#95—532no—
162#95C2×D5312yes—
163C24C22×D5311yescharacteristic
164C24C22×D5311yescharacteristic
165C2×Q8—1062no—
166C2×Q8—2121no—
167C2×Q8—2121no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4D21214yes—
171C22×C4—24no—
172C22×C4—41no—
173C22×C4—532no—
174C22×C4—532no—
175C22×C4—532no—
176C22×C4—532no—
177C22×C4—1062no—
178C22×C4—2121no—
179C22×C4—2121no—
180C22×C4C22×D5312yes—
181C22×C4C22×D5311yescharacteristic
182C22×C4C22×D5312yes—
183C4⋊C4—1064no—
184C4⋊C4—1064no—
185C4⋊C4—2122no—
186C4⋊C4—2122no—
187C22⋊C4—24no—
188C22⋊C4—41no—
189C22⋊C4—1062no—
190C22⋊C4—1064no—
191C22⋊C4—1064no—
192C22⋊C4—1064no—
193C42—1062no—
194C23—22no—
195C23—22no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—22no—
201C23—22no—
202C23C2×D21212yes—
203C23C2×D21212yes—
204C23C23×D5311yescharacteristic
205Q8—2124no—
206D4—44no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—44no—
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—1062no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—2124no—
228C2×C4C2×D21212yes—
229C2×C4#1171994714yes—
230C22—24no—
231C22—24no—
232C22—22no—
233C22—22no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—42no—
239C22—42no—
240C22C22×D21211yescharacteristic
241C22C2×#1171994712yes—
242C22#1313326211yescharacteristic
243C22#1313326211yescharacteristic
244C22#1313326412yes—
245C4—24no—
246C4—22no—
247C4—42no—
248C4—2122no—
249C4—2122no—
250C4—2122no—
251C4—2122no—
252C2—42no—
253C2—42no—
254C2#1681118511yescharacteristic
255C2#1681124012yes—
256C2#1681127712yes—
257C2#1681130411yescharacteristic
258C2#1681137611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#386C2
hypercenter#386D53
hypocenterC53#2717
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
3. #16810548C2semidirect
4. #16810548C2semidirect
5. #16810544C2semidirect
6. #16810508C2semidirect
7. #16810498C2semidirect
8. #16810483C2semidirect
139. C53#2717semidirect