Finite Group Databasebeta

← #20282775

Subgroups — #20282775

Order 6784.

430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13133216C211yesmaximal, characteristic, complemented
2C2×#13133212C211yesmaximal, characteristic
3#16810565C212yesmaximal, complemented
4#16810558C212yesmaximal
5#16810553C211yesmaximal, characteristic, complemented
6#16810550C211yesmaximal, characteristic
7#16810549C211yesmaximal, characteristic
8#16810543C212yesmaximal, complemented
9#16810536C212yesmaximal
10#16810490C211yesmaximal, characteristic, complemented
11C53×#398C211yesmaximal, characteristic, complemented
12C22×Dic106C2212yes—
13#13133216—24no—
14#13133212—24no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2211yescharacteristic
18C2×#11719919C2212yes—
19C2×#11719919C2212yes—
20C2×#11719921C2211yescharacteristic
21C2×#11719921C2211yescharacteristic
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×C4×Dic53C2211yescharacteristic
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2211yescharacteristic
31#13133124C2212yes—
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C23×D53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×Dic53C2312yes—
44C22×Dic53C2311yescharacteristic
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C2×Dic106—24no—
49C2×Dic106—42no—
50C2×Dic106—42no—
51C4⋊C4×C53—22no—
52C4⋊C4×C53—22no—
53C4⋊C4×C53—42no—
54#11719921—22no—
55#11719921—22no—
56#11719921—22no—
57#11719921—42no—
58#11719921—42no—
59C4⋊Dic53—22no—
60C4⋊Dic53—22no—
61#11719919—22no—
62#11719919—24no—
63#11719919—42no—
64C4×Dic53D414yes—
65C23×C53C2411yescharacteristic
66C22×D53—21no—
67C22×D53—21no—
68C22×D53—21no—
69C22×D53—22no—
70C22×D53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53C2×D412yes—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—21no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53C2×D412yes—
96C2×Dic53C2×D412yes—
97Dic106—48no—
98C22×C53C22×D411yescharacteristic
99C22×C53H(5,𝔽2)11yescharacteristic
100C22×C53H(5,𝔽2)11yescharacteristic
101C22×C532−1+412yes—
102C22×C532−1+412yes—
103C2×D53—41no—
104C2×D53—41no—
105C2×D53—41no—
106C2×D53—42no—
107C2×D53—42no—
108C4×C53—22no—
109C4×C53—42no—
110C4×C53—41no—
111C4×C53—41no—
112C4×C53—42no—
113Dic53—22no—
114Dic53—22no—
115Dic53—42no—
116Dic53—41no—
117Dic53—42no—
118#2821—531nomaximal, Sylow
119C2×C53#53511yescharacteristic
120C2×C53#53611yescharacteristic
121C2×C53#53611yescharacteristic
122C2×C53#56312yes—
123C2×C53#56312yes—
124D53—81no—
125C2×C4⋊Q8—531no—
126C2×#126—531no—
127#400—532no—
128#399—532no—
129#398—531no—
130#398D5311yescharacteristic, complemented
131#397—531no—
132#397—531no—
133#395—532no—
134#393—532no—
135#383—531no—
136C53#282111yescharacteristic, complemented, Sylow
137C22×Q8—532no—
138C4⋊Q8—1064no—
139#126—1064no—
140C2×C4⋊C4—531no—
141C2×C4⋊C4—531no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4—531no—
144C2×C4⋊C4—532no—
145C2×C4⋊C4C2×D5311yescharacteristic
146C2×C4⋊C4C2×D5311yescharacteristic
147C2×C4⋊C4C2×D5312yes—
148C2×C22⋊C4—531no—
149C2×C22⋊C4—531no—
150C2×C22⋊C4—531no—
151C2×C22⋊C4—532no—
152C2×C22⋊C4—532no—
153C2×C42—531no—
154#95—532no—
155#95—531no—
156#95—532no—
157#95—532no—
158#95—532no—
159#95—532no—
160#95C2×D5311yescharacteristic
161#95C2×D5312yes—
162C24—531no—
163C2×Q8—1064no—
164C2×Q8—2122no—
165C2×Q8—2122no—
166C22×C4—531no—
167C22×C4—531no—
168C22×C4—532no—
169C22×C4—531no—
170C22×C4—532no—
171C22×C4C22×D5312yes—
172C22×C4C22×D5311yescharacteristic
173C22×C4C22×D5312yes—
174C22×C4C22×D5311yescharacteristic
175C22×C4C22×D5311yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—1062no—
180C4⋊C4—1062no—
181C4⋊C4—1064no—
182C4⋊C4—1062no—
183C4⋊C4—2122no—
184C22⋊C4—1062no—
185C22⋊C4—1062no—
186C22⋊C4—1062no—
187C22⋊C4—2122no—
188C22⋊C4—2122no—
189C42—534no—
190C23—1061no—
191C23—1061no—
192C23—1061no—
193C23—1062no—
194C23—1062no—
195C23C23×D5311yescharacteristic
196Q8—2128no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—532no—
210C2×C4—532no—
211C2×C4—1061no—
212C2×C4—1061no—
213C2×C4—1062no—
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×C4D4×D5312yes—
223C22—2122no—
224C22—2122no—
225C22—2121no—
226C22—2121no—
227C22—2121no—
228C22C2×D4×D5311yescharacteristic
229C22#1313326211yescharacteristic
230C22#1313326311yescharacteristic
231C22#1313326412yes—
232C22#1313326512yes—
233C4—22no—
234C4—42no—
235C4—41no—
236C4—42no—
237C4—41no—
238C4—1062no—
239C4—1062no—
240C4—2122no—
241C4—2122no—
242C4—2121no—
243C2—4241no—
244C2#1681118911yescharacteristic
245C2#1681119511yescharacteristic
246C2#1681119811yescharacteristic
247C2#1681137412yes—
248C2#1681137712yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2821
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#13133216C2semidirect
3. #16810565C2semidirect
5. #16810553C2semidirect
8. #16810543C2semidirect
10. #16810490C2semidirect
11. C53×#398C2semidirect
130. #398D53semidirect
136. C53#2821semidirect