Finite Group Databasebeta

← #20277471

Subgroups — #20277471

Order 6784.

484 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 181 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×#13133196C211yesmaximal, characteristic
2#16810527C212yesmaximal, complemented
3#16810517C212yesmaximal
4#16810511C212yesmaximal
5#16810508C212yesmaximal, complemented
6#16810507C212yesmaximal, complemented
7#16810486C211yesmaximal, characteristic
8C4×#11719919C212yesmaximal, complemented
9C4×C22⋊C4×C53C211yesmaximal, characteristic
10C22×Dic106C2211yescharacteristic
11#13133196C418yescomplemented
12C2×C4⋊Dic53C2211yescharacteristic
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×C22⋊Dic53C2211yescharacteristic
17C2×C22⋊Dic53C2211yescharacteristic
18C2×C22⋊Dic53C2212yes—
19C2×C4×Dic53C2212yes—
20C2×C4×Dic53C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27C23×C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2212yes—
29C2×C42×C53C2211yescharacteristic
30C2×C42×C53C2211yescharacteristic
31#95×C53C2212yes—
32C24×C53C2311yescharacteristic
33C22×C4×C53—41no—
34C22×C4×C53—41no—
35C22×C4×C53C2×C414yes—
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C2×Dic106—41no—
45C2×Dic106—41no—
46C2×Dic106C2×C414yes—
47C22⋊C4×C53—24no—
48C42×C53—41no—
49C42×C53—41no—
50C22⋊Dic53—41no—
51C22⋊Dic53—41no—
52C22⋊Dic53C2×C418yes—
53C4⋊Dic53C2×C414yes—
54#11719919—24no—
55#11719919—42no—
56#11719919C2×C418yes—
57C4×Dic53—24no—
58C4×Dic53—42no—
59C23×C53—22no—
60C23×C53—21no—
61C23×C53—21no—
62C23×C53—22no—
63C23×C53C22×C412yes—
64C23×C53C2411yescharacteristic
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—44no—
75C2×C4×C53C22×C412yes—
76C2×C4×C53C22×C412yes—
77C2×C4×C53#4114yes—
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×Dic53C22×C414yes—
89C2×Dic53C22×C414yes—
90Dic106—44no—
91C22×C53—22no—
92C22×C53—22no—
93C22×C53—22no—
94C22×C53—42no—
95C22×C53—42no—
96C22×C53C23×C411yescharacteristic
97C22×C53C2×#4112yes—
98C22×C53H(5,𝔽2)11yescharacteristic
99C22×C53H(5,𝔽2)11yescharacteristic
100C22×C532−1+412yes—
101C4×C53—22no—
102C4×C53—22no—
103C4×C53—24no—
104C4×C53—42no—
105Dic53—24no—
106Dic53—24no—
107Dic53—42no—
108Dic53—42no—
109#2332—531nomaximal, Sylow
110C2×C53—42no—
111C2×C53C4×#4111yescharacteristic
112C2×C53#51811yescharacteristic
113C2×C53#51911yescharacteristic
114C2×C53#53912yes—
115C2×C53#54212yes—
116C2×C22⋊Q8—531no—
117#391—532no—
118#388—532no—
119#388—532no—
120#387—532no—
121#387—532no—
122#381—531no—
123C4×C4⋊C4—532no—
124C4×C22⋊C4D5311yescharacteristic
125C53#233211yescharacteristic, complemented, Sylow
126C22×Q8—531no—
127C23×C4C2×D5311yescharacteristic
128C22⋊Q8—538no—
129C2×C4⋊C4—532no—
130C2×C4⋊C4—531no—
131C2×C4⋊C4—532no—
132C2×C4⋊C4—532no—
133C2×C22⋊C4—532no—
134C2×C22⋊C4—531no—
135C2×C22⋊C4—531no—
136C2×C22⋊C4C2×D5312yes—
137C2×C42—532no—
138C2×C42—532no—
139C2×C42C2×D5311yescharacteristic
140C2×C42C2×D5311yescharacteristic
141#95—532no—
142#95—532no—
143#95—532no—
144#95—532no—
145#95—532no—
146#95—532no—
147#95C2×D5312yes—
148C24C22×D5311yescharacteristic
149C2×Q8—534no—
150C2×Q8—2121no—
151C2×Q8—2121no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—532no—
155C22×C4—532no—
156C22×C4—532no—
157C22×C4—532no—
158C22×C4C4×D5314yes—
159C22×C4C22×D5312yes—
160C22×C4C22×D5311yescharacteristic
161C22×C4C22×D5311yescharacteristic
162C22×C4C22×D5312yes—
163C4⋊C4—538no—
164C4⋊C4—534no—
165C4⋊C4—1064no—
166C4⋊C4—2122no—
167C22⋊C4—24no—
168C22⋊C4—538no—
169C22⋊C4—2121no—
170C22⋊C4—2121no—
171C42—41no—
172C42—41no—
173C42—1064no—
174C42—2122no—
175C23—22no—
176C23—21no—
177C23—21no—
178C23—22no—
179C23C2×C4×D5312yes—
180C23C23×D5311yescharacteristic
181Q8—2124no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—44no—
192C2×C4—534no—
193C2×C4—534no—
194C2×C4—1062no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4C2×C4×D5312yes—
205C2×C4C2×C4×D5312yes—
206C2×C4#1171994514yes—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22C22×C4×D5311yescharacteristic
213C22C2×#1171994512yes—
214C22#1313326211yescharacteristic
215C22#1313326211yescharacteristic
216C22#1313326412yes—
217C4—22no—
218C4—22no—
219C4—24no—
220C4—42no—
221C4—1064no—
222C4—1064no—
223C4—2122no—
224C4—2122no—
225C2—42no—
226C2C4×#1171994511yescharacteristic
227C2#1681109611yescharacteristic
228C2#1681110011yescharacteristic
229C2#1681121512yes—
230C2#1681123812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC4×C22⋊C4×C53C2
socleC23×C53C24
hypercenterC4×C22⋊C4D53
hypocenterC53#2332
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #16810527C2semidirect
5. #16810508C2semidirect
6. #16810507C2semidirect
8. C4×#11719919C2semidirect
11. #13133196C4semidirect
125. C53#2332semidirect