Finite Group Databasebeta

← #20282777

Subgroups — #20282777

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×#13133215C211yesmaximal, characteristic, complemented
2C2×#13133210C211yesmaximal, characteristic
3#16810565C212yesmaximal, complemented
4#16810559C212yesmaximal
5#16810557C212yesmaximal, complemented
6#16810551C212yesmaximal
7#16810544C211yesmaximal, characteristic, complemented
8#16810542C211yesmaximal, characteristic, complemented
9#16810535C211yesmaximal, characteristic
10#16810490C211yesmaximal, characteristic, complemented
11C53×#393C211yesmaximal, characteristic
12C22×Dic106C2211yescharacteristic
13#13133215—24no—
14#13133210—24no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C4⋊Dic53C2212yes—
17C2×#11719919C2211yescharacteristic
18C2×#11719919C2212yes—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2212yes—
21C2×#11719919C2212yes—
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2211yescharacteristic
24C2×C22⋊Dic53C2212yes—
25C2×C4×Dic53C2211yescharacteristic
26#13133124C2211yescharacteristic
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32C22×Q8×C53C2211yescharacteristic
33C2×C22⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2212yes—
35#95×C53C2211yescharacteristic
36#95×C53C2212yes—
37C24×C53C2311yescharacteristic
38C2×Q8×C53—22no—
39C2×Q8×C53—42no—
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×Dic53C2312yes—
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2312yes—
47C22×Dic53C2311yescharacteristic
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2311yescharacteristic
50C2×Dic106—22no—
51C2×Dic106—42no—
52C22⋊C4×C53—22no—
53C22⋊C4×C53—42no—
54C22⋊Dic53—22no—
55C22⋊Dic53—22no—
56C22⋊Dic53—42no—
57C4⋊Dic53—22no—
58C4⋊Dic53—42no—
59#11719919—22no—
60#11719919—22no—
61#11719919—24no—
62#11719919—24no—
63#11719919—42no—
64C4×Dic53D414yes—
65C23×C53—21no—
66C23×C53—22no—
67C23×C53—21no—
68C23×C53—22no—
69C23×C53—21no—
70C23×C53C2411yescharacteristic
71Q8×C53—44no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53C2×D412yes—
82C2×Dic53—22no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—22no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—21no—
93C2×Dic53—22no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53C2×D412yes—
97C2×Dic53C2×D412yes—
98Dic106—44no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—42no—
103C22×C53—42no—
104C22×C53C22×D411yescharacteristic
105C22×C53H(5,𝔽2)12yes—
106C22×C532−1+411yescharacteristic
107C22×C532−1+412yes—
108C22×C532−1+411yescharacteristic
109C4×C53—22no—
110C4×C53—42no—
111C4×C53—41no—
112C4×C53—42no—
113Dic53—22no—
114Dic53—22no—
115Dic53—42no—
116Dic53—41no—
117Dic53—41no—
118Dic53—42no—
119#2821—531nomaximal, Sylow
120C2×C53—81no—
121C2×C53#53511yescharacteristic
122C2×C53#53611yescharacteristic
123C2×C53#53611yescharacteristic
124C2×C53#56312yes—
125C2×C53#56312yes—
126C2×C4⋊Q8—531no—
127C2×#126—531no—
128#400—532no—
129#399—532no—
130#398—532no—
131#397—532no—
132#395—531no—
133#395—531no—
134#393—531no—
135#393D5311yescharacteristic
136#383—531no—
137C53#282111yescharacteristic, complemented, Sylow
138C22×Q8—531no—
139C22×Q8C2×D5311yescharacteristic
140C4⋊Q8—1064no—
141#126—1064no—
142C2×C4⋊C4—531no—
143C2×C4⋊C4—531no—
144C2×C4⋊C4—531no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4—532no—
149C2×C22⋊C4—531no—
150C2×C22⋊C4—531no—
151C2×C22⋊C4—532no—
152C2×C22⋊C4C2×D5311yescharacteristic
153C2×C22⋊C4C2×D5312yes—
154C2×C42—531no—
155#95—531no—
156#95—532no—
157#95—532no—
158#95—532no—
159#95—532no—
160#95—532no—
161#95C2×D5311yescharacteristic
162#95C2×D5312yes—
163C24C22×D5311yescharacteristic
164C2×Q8—22no—
165C2×Q8—42no—
166C2×Q8—1062no—
167C2×Q8—2122no—
168C22×C4—531no—
169C22×C4—531no—
170C22×C4—532no—
171C22×C4—531no—
172C22×C4—532no—
173C22×C4—531no—
174C22×C4C22×D5312yes—
175C22×C4C22×D5311yescharacteristic
176C22×C4C22×D5312yes—
177C22×C4C22×D5311yescharacteristic
178C4⋊C4—1062no—
179C4⋊C4—1062no—
180C4⋊C4—1064no—
181C4⋊C4—1064no—
182C4⋊C4—1062no—
183C4⋊C4—2122no—
184C4⋊C4—2122no—
185C22⋊C4—22no—
186C22⋊C4—42no—
187C22⋊C4—1062no—
188C22⋊C4—1062no—
189C22⋊C4—2122no—
190C42—534no—
191C23—21no—
192C23—22no—
193C23—21no—
194C23—22no—
195C23—21no—
196C23C23×D5311yescharacteristic
197Q8—44no—
198Q8—2124no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—532no—
209C2×C4—532no—
210C2×C4—1062no—
211C2×C4—1061no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—1061no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1061no—
222C2×C4—1062no—
223C2×C4—1061no—
224C2×C4D4×D5312yes—
225C22—42no—
226C22—42no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22C2×D4×D5311yescharacteristic
231C22#1313326212yes—
232C22#1313326411yescharacteristic
233C22#1313326412yes—
234C22#1313326511yescharacteristic
235C4—22no—
236C4—42no—
237C4—42no—
238C4—41no—
239C4—1062no—
240C4—1062no—
241C4—2122no—
242C4—2121no—
243C4—2122no—
244C4—2121no—
245C2—81no—
246C2#1681118811yescharacteristic
247C2#1681119311yescharacteristic
248C2#1681119611yescharacteristic
249C2#1681137312yes—
250C2#1681137412yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53#2821
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#13133215C2semidirect
3. #16810565C2semidirect
5. #16810557C2semidirect
7. #16810544C2semidirect
8. #16810542C2semidirect
10. #16810490C2semidirect
137. C53#2821semidirect