Finite Group Databasebeta

← #20280479

Subgroups — #20280479

Order 6784.

612 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133206C211yesmaximal, characteristic, complemented
2C2×#13133193C211yesmaximal, characteristic, complemented
3C2×#13133192C211yesmaximal, characteristic
4C2×#13133174C211yesmaximal, characteristic, complemented
5#16810547C214yesmaximal, complemented
6#16810541C214yesmaximal, complemented
7#16810525C211yesmaximal, characteristic, complemented
8#16810492C211yesmaximal, characteristic, complemented
9C53×#391C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×D212C2212yes—
12C22×C4×D53C2211yescharacteristic
13#13133206—24no—
14#13133193—24no—
15#13133192—24no—
16#13133174—24no—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×#11719919C2212yes—
19C2×#11719919C2212yes—
20C2×#11719919C2214yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2214yes—
24C2×#11719921C2212yes—
25C2×#11719921C2214yes—
26C2×C4×Dic53C2211yescharacteristic
27C22×Q8×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C42×C53C2211yescharacteristic
30#95×C53C2214yes—
31C23×D53C2312yes—
32C23×D53C2311yescharacteristic
33C2×Q8×C53—22no—
34C2×Q8×C53—22no—
35C2×Q8×C53—22no—
36C22×C4×C53C2312yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C22×Dic53C2311yescharacteristic
43C2×D212—24no—
44C2×D212—22no—
45C2×D212—22no—
46C2×D212—22no—
47C2×D212—44no—
48C2×C4×D53—22no—
49C2×C4×D53—22no—
50C2×C4×D53D414yes—
51C4⋊C4×C53—22no—
52C42×C53—22no—
53#11719921—24no—
54#11719921—24no—
55#11719921—24no—
56#11719921—28no—
57#11719921—44no—
58C4⋊Dic53—22no—
59#11719919—24no—
60#11719919—24no—
61#11719919—44no—
62C4×Dic53D414yescomplemented
63C23×C53C2411yescharacteristic
64C22×D53—21no—
65C22×D53—21no—
66C22×D53—22no—
67C22×D53—22no—
68C22×D53—22no—
69C22×D53—22no—
70C22×D53—22no—
71C22×D53—24no—
72C22×D53—24no—
73C22×D53C2×D412yes—
74Q8×C53—44no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—24no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53C2×D412yes—
84C2×C4×C53C2×D412yes—
85C2×C4×C53#4112yes—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—24no—
92C2×Dic53C2×D412yes—
93C2×Dic53C2×D414yes—
94D212—48no—
95D212—44no—
96C4×D53—28no—
97C22×C53C22×D411yescharacteristic
98C22×C53C22×D411yescharacteristic
99C22×C53C2×#4111yescharacteristic
100C22×C53H(5,𝔽2)12yes—
101C22×C532−1+412yes—
102C2×D53—22no—
103C2×D53—22no—
104C2×D53—22no—
105C2×D53—42no—
106C2×D53—42no—
107C2×D53—42no—
108C2×D53—42no—
109C2×D53—42no—
110C2×D53—44no—
111C2×D53—44no—
112C4×C53—42no—
113C4×C53—42no—
114C4×C53—42no—
115C4×C53#12114yes—
116Dic53—22no—
117Dic53—24no—
118Dic53—42no—
119#2620—531nomaximal, Sylow
120C2×C53C2×#12111yescharacteristic
121C2×C53#53511yescharacteristic
122C2×C53#53611yescharacteristic
123C2×C53#53712yes—
124C2×C53#54212yes—
125D53—42no—
126D53—82no—
127C2×#124—531no—
128C2×C22⋊Q8—531no—
129C2×#121—531no—
130C2×#117—531no—
131#396—534no—
132#394—534no—
133#391D5311yescharacteristic, complemented
134#390—531no—
135#384—531no—
136C53#262011yescharacteristic, complemented, Sylow
137C22×Q8C2×D5311yescharacteristic
138C22×D4—531no—
139C22×D4—532no—
140C23×C4—531no—
141#124—1064no—
142C22⋊Q8—1064no—
143#121—1064no—
144#117—1064no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—531no—
148C2×C4⋊C4—534no—
149C2×C4⋊C4C2×D5311yescharacteristic
150C2×C22⋊C4—532no—
151C2×C22⋊C4—532no—
152C2×C22⋊C4—532no—
153C2×C22⋊C4—534no—
154C2×C22⋊C4—534no—
155C2×C42—531no—
156C2×C42C2×D5311yescharacteristic
157#95C2×D5314yes—
158C24—531no—
159C24—532no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×D4—1064no—
164C2×D4—1062no—
165C2×D4—1062no—
166C2×D4—1062no—
167C2×D4—2124no—
168C22×C4—532no—
169C22×C4—531no—
170C22×C4—534no—
171C22×C4—532no—
172C22×C4—1062no—
173C22×C4—1062no—
174C22×C4C22×D5312yes—
175C22×C4C22×D5312yes—
176C22×C4C22×D5312yes—
177C22×C4C22×D5311yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—1064no—
180C4⋊C4—1064no—
181C4⋊C4—1062no—
182C4⋊C4—2124no—
183C22⋊C4—1064no—
184C22⋊C4—1064no—
185C22⋊C4—1068no—
186C22⋊C4—1064no—
187C22⋊C4—2124no—
188C42—22no—
189C42—534no—
190C23—532no—
191C23—1061no—
192C23—1061no—
193C23—1062no—
194C23—1062no—
195C23—1062no—
196C23—1062no—
197C23—1062no—
198C23—1064no—
199C23—1064no—
200C23C23×D5311yescharacteristic
201Q8—44no—
202D4—2128no—
203D4—2124no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—534no—
213C2×C4—532no—
214C2×C4—1061no—
215C2×C4—1061no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1068no—
219C2×C4—1062no—
220C2×C4—1064no—
221C2×C4#1171994512yes—
222C2×C4D4×D5312yes—
223C2×C4D4×D5312yes—
224C22—1062no—
225C22—1062no—
226C22—1062no—
227C22—2122no—
228C22—2122no—
229C22—2122no—
230C22—2124no—
231C22—2124no—
232C22—2122no—
233C22—2122no—
234C22C2×D4×D5311yescharacteristic
235C22C2×D4×D5311yescharacteristic
236C22C2×#1171994511yescharacteristic
237C22#1313326312yes—
238C22#1313326512yes—
239C4—42no—
240C4—42no—
241C4—42no—
242C4—1062no—
243C4—1064no—
244C4—2122no—
245C4#1313319014yes—
246C2—4242no—
247C2—2122no—
248C2C2×#1313319011yescharacteristic
249C2#1681119111yescharacteristic
250C2#1681119811yescharacteristic
251C2#1681120312yes—
252C2#1681124512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#391C2
hypercenter#391D53
hypocenterC53#2620
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#13133206C2semidirect
2. C2×#13133193C2semidirect
4. C2×#13133174C2semidirect
5. #16810547C2semidirect
6. #16810541C2semidirect
7. #16810525C2semidirect
8. #16810492C2semidirect
9. C53×#391C2semidirect
62. C4×Dic53D4semidirect
133. #391D53semidirect
136. C53#2620semidirect