Finite Group Databasebeta

← #20282416

Subgroups — #20282416

Order 6784.

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

normal subgroups only

This group has 472 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
1#16810562C211yesmaximal, characteristic, complemented
2#16810546C212yesmaximal, complemented
3#16810543C211yesmaximal, characteristic, complemented
4#16810535C212yesmaximal, complemented
5#16810507C212yesmaximal, complemented
6#16810503C212yesmaximal
7#16810495C212yesmaximal, complemented
8#16810481C212yesmaximal, complemented
9C53×#390C211yesmaximal, characteristic
10C22×Dic106C2211yescharacteristic
11C23×Dic53C2212yes—
12C2×C4⋊Dic53C2211yescharacteristic
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×C22⋊Dic53C2212yes—
19#13133124C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2211yescharacteristic
25#13133124C2211yescharacteristic
26C22×D4×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2212yes—
29C2×C22⋊C4×C53C2212yes—
30C2×C42×C53C2211yescharacteristic
31C24×C53C2312yes—
32C2×D4×C53—41no—
33C2×D4×C53—41no—
34C2×D4×C53Q814yescomplemented
35C22×C4×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×Dic53—24no—
40C22×Dic53—42no—
41C22×Dic53—42no—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C2×Dic106—42no—
47C2×Dic106—41no—
48C4⋊C4×C53—41no—
49C22⋊C4×C53—42no—
50C22⋊C4×C53—42no—
51C42×C53—41no—
52C22⋊Dic53—24no—
53C22⋊Dic53—42no—
54C22⋊Dic53—42no—
55C4⋊Dic53—41no—
56#11719919—24no—
57#11719919—42no—
58#11719919—42no—
59C23×C53—22no—
60C23×C53—22no—
61C23×C53—22no—
62C23×C53—22no—
63C23×C53—22no—
64C23×C53—22no—
65C23×C53C2×Q814yes—
66C23×C53C2411yescharacteristic
67D4×C53—44no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53C2×Q812yes—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—48no—
90C2×Dic53#4114yes—
91Dic106—44no—
92C22×C53—24no—
93C22×C53—24no—
94C22×C53—24no—
95C22×C53—42no—
96C22×C53—42no—
97C22×C53—42no—
98C22×C53—42no—
99C22×C53C22×Q811yescharacteristic
100C22×C53C2×#4112yes—
101C22×C53H(5,𝔽2)11yescharacteristic
102C22×C53H(5,𝔽2)12yes—
103C22×C53H(5,𝔽2)11yescharacteristic
104C4×C53—22no—
105C4×C53—42no—
106C4×C53—41no—
107C4×C53—41no—
108Dic53—24no—
109Dic53—42no—
110Dic53—42no—
111Dic53—42no—
112#2784—531nomaximal, Sylow
113C2×C53—44no—
114C2×C53#53812yes—
115C2×C53#54311yescharacteristic
116C2×C53#55412yes—
117C2×C53#55511yescharacteristic
118C2×C53#56011yescharacteristic
119#400—531no—
120#396—532no—
121#395—531no—
122#393—532no—
123#390D5311yescharacteristic
124#387—532no—
125#386—532no—
126#385—532no—
127#380—532no—
128C53#278411yescharacteristic, complemented, Sylow
129C22×Q8—531no—
130C22×D4C2×D5311yescharacteristic
131C23×C4—532no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—532no—
135C2×C4⋊C4—531no—
136C2×C4⋊C4C2×D5311yescharacteristic
137C2×C22⋊C4—532no—
138C2×C22⋊C4—532no—
139C2×C22⋊C4—532no—
140C2×C22⋊C4C2×D5312yes—
141C2×C22⋊C4C2×D5312yes—
142C2×C42C2×D5311yescharacteristic
143#95—531no—
144#95—532no—
145#95—531no—
146#95—532no—
147#95—532no—
148#95—532no—
149#95—532no—
150C24C22×D5312yes—
151C2×Q8—2122no—
152C2×Q8—2121no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4Dic10614yescomplemented
156C22×C4—532no—
157C22×C4—532no—
158C22×C4—532no—
159C22×C4—532no—
160C22×C4—1064no—
161C22×C4—2122no—
162C22×C4—2122no—
163C22×C4C22×D5311yescharacteristic
164C22×C4C22×D5312yes—
165C22×C4C22×D5311yescharacteristic
166C22×C4C22×D5311yescharacteristic
167C4⋊C4—41no—
168C4⋊C4—1064no—
169C4⋊C4—2122no—
170C4⋊C4—2122no—
171C4⋊C4—2121no—
172C22⋊C4—42no—
173C22⋊C4—42no—
174C22⋊C4—1064no—
175C22⋊C4—2122no—
176C22⋊C4—2122no—
177C42—41no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23C2×Dic10614yes—
185C23C23×D5311yescharacteristic
186Q8—2124no—
187D4—44no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—534no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—1062no—
209C2×C4—2128no—
210C2×C4C2×Dic10612yes—
211C22—24no—
212C22—24no—
213C22—24no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22C22×Dic10611yescharacteristic
219C22#41×D5312yes—
220C22#1313326212yes—
221C22#1313326211yescharacteristic
222C22#1313326211yescharacteristic
223C4—22no—
224C4—41no—
225C4—41no—
226C4—42no—
227C4—1064no—
228C4—2122no—
229C4—2122no—
230C4—2122no—
231C2—44no—
232C2#1681120512yes—
233C2#1681124911yescharacteristic
234C2#1681133212yes—
235C2#1681133511yescharacteristic
236C2#1681136011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#390C2
hypercenter#390D53
hypocenterC53#2784
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #16810562C2semidirect
2. #16810546C2semidirect
3. #16810543C2semidirect
4. #16810535C2semidirect
5. #16810507C2semidirect
7. #16810495C2semidirect
8. #16810481C2semidirect
34. C2×D4×C53Q8semidirect
128. C53#2784semidirect
155. C2×D4Dic106semidirect