Finite Group Databasebeta

← #20281684

Subgroups — #20281684

Order 6784.

512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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×#13133193C212yesmaximal, complemented
2#16810563C212yesmaximal, complemented
3#16810553C211yesmaximal, characteristic, complemented
4#16810550C212yesmaximal, complemented
5#16810540C211yesmaximal, characteristic
6#16810514C212yesmaximal, complemented
7#16810501C212yesmaximal
8#16810497C212yesmaximal, complemented
9C53×#391C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C4×D53C2212yes—
12#13133193—28no—
13C2×C4⋊Dic53C2212yes—
14C2×C4⋊Dic53C2211yescharacteristic
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26#13133124C2211yescharacteristic
27C22×Q8×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C42×C53C2211yescharacteristic
30#95×C53C2212yes—
31#95×C53C2212yes—
32C23×D53C2312yes—
33C2×Q8×C53—41no—
34C2×Q8×C53—41no—
35C2×Q8×C53D414yescomplemented
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C2×D212—24no—
45C2×D212—41no—
46C2×C4×D53—24no—
47C2×C4×D53—42no—
48C2×C4×D53—42no—
49C4⋊C4×C53—41no—
50C42×C53—41no—
51#11719921—24no—
52#11719921—24no—
53#11719921—24no—
54#11719921—24no—
55#11719921—42no—
56C4⋊Dic53—24no—
57C4⋊Dic53—41no—
58#11719919—24no—
59#11719919—24no—
60C23×C53C2411yescharacteristic
61C22×D53—22no—
62C22×D53—22no—
63C22×D53—22no—
64C22×D53—22no—
65C22×D53—22no—
66C22×D53—22no—
67C22×D53#4114yes—
68Q8×C53—44no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53C2×D414yes—
81C2×C4×C53C2×D412yes—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91D212—44no—
92C4×D53—48no—
93C22×C53C22×D411yescharacteristic
94C22×C53C2×#4112yes—
95C22×C53H(5,𝔽2)11yescharacteristic
96C22×C53H(5,𝔽2)11yescharacteristic
97C22×C532−1+412yes—
98C2×D53—24no—
99C2×D53—24no—
100C2×D53—24no—
101C2×D53—42no—
102C2×D53—42no—
103C2×D53—42no—
104C2×D53—42no—
105C4×C53—24no—
106C4×C53—22no—
107C4×C53—42no—
108C4×C53—41no—
109C4×C53—41no—
110Dic53—42no—
111Dic53—42no—
112Dic53—42no—
113#2728—531nomaximal, Sylow
114C2×C53#53411yescharacteristic
115C2×C53#53912yes—
116C2×C53#54812yes—
117C2×C53#55111yescharacteristic
118C2×C53#56311yescharacteristic
119D53—44no—
120C2×C22⋊Q8—532no—
121#400—532no—
122#398—531no—
123#397—532no—
124#394—531no—
125#391D5311yescharacteristic, complemented
126#388—532no—
127#386—532no—
128#385—532no—
129C53#272811yescharacteristic, complemented, Sylow
130C22×Q8C2×D5311yescharacteristic
131C22×D4—531no—
132C23×C4—532no—
133C22⋊Q8—1068no—
134C2×C4⋊C4—532no—
135C2×C4⋊C4—532no—
136C2×C4⋊C4—532no—
137C2×C4⋊C4—531no—
138C2×C4⋊C4C2×D5311yescharacteristic
139C2×C22⋊C4—532no—
140C2×C22⋊C4—532no—
141C2×C22⋊C4—532no—
142C2×C22⋊C4—532no—
143C2×C22⋊C4—532no—
144C2×C42C2×D5311yescharacteristic
145#95—532no—
146#95—532no—
147#95—532no—
148#95—531no—
149#95—531no—
150#95C2×D5312yes—
151#95C2×D5312yes—
152C24—532no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8D21214yescomplemented
156C2×D4—1064no—
157C2×D4—2121no—
158C22×C4—532no—
159C22×C4—532no—
160C22×C4—532no—
161C22×C4—1064no—
162C22×C4—2122no—
163C22×C4—2122no—
164C22×C4C22×D5311yescharacteristic
165C22×C4C22×D5312yes—
166C22×C4C22×D5311yescharacteristic
167C22×C4C22×D5312yes—
168C22×C4C22×D5311yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—1064no—
171C4⋊C4—1064no—
172C4⋊C4—1064no—
173C4⋊C4—2121no—
174C22⋊C4—1064no—
175C22⋊C4—1064no—
176C22⋊C4—1064no—
177C22⋊C4—1064no—
178C22⋊C4—2122no—
179C42—41no—
180C23—534no—
181C23—1062no—
182C23—1062no—
183C23—1062no—
184C23—1062no—
185C23—1062no—
186C23—1062no—
187C23C23×D5311yescharacteristic
188Q8—44no—
189D4—2124no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—1062no—
209C2×C4—1062no—
210C2×C4—2128no—
211C2×C4C2×D21212yes—
212C2×C4C2×D21214yes—
213C22—1064no—
214C22—1064no—
215C22—1064no—
216C22—2122no—
217C22—2122no—
218C22—2122no—
219C22—2122no—
220C22C22×D21211yescharacteristic
221C22#41×D5312yes—
222C22#1313326211yescharacteristic
223C22#1313326211yescharacteristic
224C22#1313326512yes—
225C4—24no—
226C4—22no—
227C4—41no—
228C4—41no—
229C4—42no—
230C4—2122no—
231C4—2122no—
232C4—2122no—
233C2—2124no—
234C2#1681118511yescharacteristic
235C2#1681121912yes—
236C2#1681128512yes—
237C2#1681130311yescharacteristic
238C2#1681137211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#391C2
hypercenter#391D53
hypocenterC53#2728
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#13133193C2semidirect
2. #16810563C2semidirect
3. #16810553C2semidirect
4. #16810550C2semidirect
6. #16810514C2semidirect
8. #16810497C2semidirect
9. C53×#391C2semidirect
35. C2×Q8×C53D4semidirect
125. #391D53semidirect
129. C53#2728semidirect
155. C2×Q8D212semidirect