Finite Group Databasebeta

← #20280833

Subgroups — #20280833

Order 6784.

448 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 153 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
2C2×#13133207C211yesmaximal, characteristic
3#16810557C212yesmaximal
4#16810556C212yesmaximal
5#16810544C211yesmaximal, characteristic
6#16810543C211yesmaximal, characteristic
7#16810542C211yesmaximal, characteristic
8#16810492C212yesmaximal
9#16810487C212yesmaximal
10#16810468C211yesmaximal, characteristic
11C53×#395C211yesmaximal, characteristic
12C22×Dic106C2211yescharacteristic
13#13133215—24no—
14#13133207—24no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C4⋊Dic53C2212yes—
17C2×#11719919C2211yescharacteristic
18C2×#11719919C2212yes—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2212yes—
23C2×#11719919C2212yes—
24C2×#11719919C2212yes—
25C2×C4×Dic53C2211yescharacteristic
26C2×C4×Dic53C2212yes—
27#13133124C2211yescharacteristic
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32C22×Q8×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2212yes—
35#95×C53C2211yescharacteristic
36#95×C53C2212yes—
37C2×Q8×C53—22no—
38C2×Q8×C53—42no—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2312yes—
44C22×Dic53C2311yescharacteristic
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C22×Dic53C2312yes—
49C22×Dic53C2311yescharacteristic
50C2×Dic106—22no—
51C2×Dic106—42no—
52C4⋊C4×C53—22no—
53C4⋊C4×C53—42no—
54C4⋊Dic53—22no—
55C4⋊Dic53—42no—
56#11719919—22no—
57#11719919—22no—
58#11719919—22no—
59#11719919—22no—
60#11719919—24no—
61#11719919—24no—
62#11719919—42no—
63#11719919—42no—
64C4×Dic53D414yes—
65C4×Dic53Q818yescomplemented
66C23×C53C2411yescharacteristic
67Q8×C53—44no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53C2×D412yes—
80C2×C4×C53C2×Q814yes—
81C2×Dic53—22no—
82C2×Dic53—21no—
83C2×Dic53—21no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53C2×D412yes—
94C2×Dic53C2×D412yes—
95C2×Dic53C2×Q814yes—
96C2×Dic53C2×Q814yes—
97Dic106—44no—
98C22×C53C22×D411yescharacteristic
99C22×C53C22×Q812yes—
100C22×C53H(5,𝔽2)12yes—
101C22×C532−1+411yescharacteristic
102C22×C532−1+411yescharacteristic
103C4×C53—22no—
104C4×C53—24no—
105C4×C53—41no—
106C4×C53—42no—
107C4×C53—41no—
108Dic53—22no—
109Dic53—24no—
110Dic53—42no—
111Dic53—41no—
112Dic53—41no—
113Dic53C4⋊Q814yes—
114#2660—531nomaximal, Sylow
115C2×C53C2×C4⋊Q811yescharacteristic
116C2×C53#53511yescharacteristic
117C2×C53#53611yescharacteristic
118C2×C53#54412yes—
119C2×C53#54412yes—
120C2×C4⋊Q8—531no—
121C2×#125—531no—
122#398—532no—
123#398—532no—
124#395—531no—
125#395—531no—
126#395—531no—
127#395D5311yescharacteristic
128#384—532no—
129#382—532no—
130#376—531no—
131C53#266011yescharacteristic, complemented, Sylow
132C22×Q8—531no—
133C22×Q8C2×D5311yescharacteristic
134C4⋊Q8—1064no—
135#125—1064no—
136C2×C4⋊C4—531no—
137C2×C4⋊C4—531no—
138C2×C4⋊C4—531no—
139C2×C4⋊C4—531no—
140C2×C4⋊C4—531no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4—532no—
144C2×C4⋊C4—532no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4C2×D5311yescharacteristic
147C2×C4⋊C4C2×D5312yes—
148C2×C42—531no—
149C2×C42—532no—
150#95—531no—
151#95—532no—
152#95—532no—
153#95—532no—
154#95—532no—
155#95C2×D5311yescharacteristic
156#95C2×D5312yes—
157C2×Q8—22no—
158C2×Q8—42no—
159C2×Q8—1062no—
160C2×Q8—2122no—
161C22×C4—531no—
162C22×C4—532no—
163C22×C4—531no—
164C22×C4—531no—
165C22×C4—532no—
166C22×C4—531no—
167C22×C4C22×D5311yescharacteristic
168C22×C4C22×D5312yes—
169C22×C4C22×D5311yescharacteristic
170C22×C4C22×D5311yescharacteristic
171C22×C4C22×D5312yes—
172C4⋊C4—22no—
173C4⋊C4—42no—
174C4⋊C4—1062no—
175C4⋊C4—1062no—
176C4⋊C4—1062no—
177C4⋊C4—1062no—
178C4⋊C4—1064no—
179C4⋊C4—1064no—
180C4⋊C4—1062no—
181C4⋊C4—2122no—
182C4⋊C4—2122no—
183C4⋊C4—2122no—
184C42—534no—
185C42—538no—
186C23C23×D5311yescharacteristic
187Q8—44no—
188Q8—2124no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—532no—
201C2×C4—534no—
202C2×C4—532no—
203C2×C4—534no—
204C2×C4—1062no—
205C2×C4—1061no—
206C2×C4—1061no—
207C2×C4—1062no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—1062no—
211C2×C4—1062no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1061no—
215C2×C4—1061no—
216C2×C4D4×D5312yes—
217C2×C4Q8×D5314yes—
218C22C2×D4×D5311yescharacteristic
219C22C2×Q8×D5312yes—
220C22#1313326212yes—
221C22#1313326411yescharacteristic
222C22#1313326511yescharacteristic
223C4—22no—
224C4—24no—
225C4—41no—
226C4—42no—
227C4—41no—
228C4—534no—
229C4—1062no—
230C4—1064no—
231C4—2121no—
232C4—2122no—
233C4—2121no—
234C2C4⋊Q8×D5311yescharacteristic
235C2#1681118811yescharacteristic
236C2#1681119611yescharacteristic
237C2#1681125012yes—
238C2#1681125212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2660
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
65. C4×Dic53Q8semidirect
131. C53#2660semidirect