Finite Group Databasebeta

← #20282882

Subgroups — #20282882

Order 6784.

416 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#16810564C212yesmaximal, complemented
2#16810552C212yesmaximal, complemented
3#16810543C211yesmaximal, characteristic, complemented
4#16810541C212yesmaximal
5#16810540C211yesmaximal, characteristic
6#16810536C212yesmaximal, complemented
7#16810534C212yesmaximal, complemented
8#16810514C212yesmaximal, complemented
9C53×#391C211yesmaximal, characteristic, complemented
10C22×Dic106C2211yescharacteristic
11C22×D212C2211yescharacteristic
12C2×#11719919C2212yes—
13C2×#11719919C2212yes—
14C2×#11719921C2212yes—
15C2×#11719921C2212yes—
16C2×#11719921C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2211yescharacteristic
25#13133124C2211yescharacteristic
26C22×Q8×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C42×C53C2211yescharacteristic
29#95×C53C2212yes—
30#95×C53C2212yes—
31C23×D53C2312yes—
32C2×Q8×C53—22no—
33C2×Q8×C53—41no—
34C2×Q8×C53—41no—
35C22×C4×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C2×D212—42no—
44C2×D212—41no—
45C2×Dic106—42no—
46C2×Dic106—41no—
47C4⋊C4×C53—41no—
48C42×C53—41no—
49#11719921—24no—
50#11719921—42no—
51#11719921—42no—
52#11719921—42no—
53#11719921—42no—
54#11719921—42no—
55#11719919—42no—
56#11719919—42no—
57C23×C53C2411yescharacteristic
58C22×D53—22no—
59C22×D53—22no—
60C22×D53—22no—
61C22×D53—22no—
62C22×D53—22no—
63C22×D53—22no—
64C22×D53—22no—
65Q8×C53—44no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—21no—
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53#4112yes—
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—
88D212—44no—
89Dic106—44no—
90C22×C53C2×#4111yescharacteristic
91C22×C53H(5,𝔽2)11yescharacteristic
92C22×C53H(5,𝔽2)11yescharacteristic
93C22×C53H(5,𝔽2)12yes—
94C22×C532−1+412yes—
95C2×D53—42no—
96C2×D53—42no—
97C2×D53—42no—
98C2×D53—42no—
99C2×D53—42no—
100C2×D53—42no—
101C2×D53—42no—
102C4×C53—22no—
103C4×C53—42no—
104C4×C53—41no—
105C4×C53—41no—
106C4×C53—42no—
107Dic53—42no—
108Dic53—42no—
109Dic53—42no—
110#2832—531nomaximal, Sylow
111C2×C53#53811yescharacteristic
112C2×C53#54212yes—
113C2×C53H(3,𝔽4)11yescharacteristic
114C2×C53#56212yes—
115C2×C53#56311yescharacteristic
116D53—82no—
117#400—532no—
118#397—532no—
119#395—531no—
120#394—531no—
121#394—532no—
122#393—532no—
123#393—532no—
124#391D5311yescharacteristic, complemented
125#388—532no—
126C53#283211yescharacteristic, complemented, Sylow
127C22×Q8—531no—
128C22×Q8C2×D5311yescharacteristic
129C22×D4—531no—
130C2×C4⋊C4—532no—
131C2×C4⋊C4—532no—
132C2×C4⋊C4C2×D5311yescharacteristic
133C2×C22⋊C4—532no—
134C2×C22⋊C4—532no—
135C2×C22⋊C4—532no—
136C2×C22⋊C4—532no—
137C2×C22⋊C4—532no—
138C2×C22⋊C4—532no—
139C2×C42C2×D5311yescharacteristic
140#95—532no—
141#95—532no—
142#95—532no—
143#95—532no—
144#95—531no—
145#95—531no—
146#95C2×D5312yes—
147#95C2×D5312yes—
148C24—532no—
149C2×Q8—22no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—2121no—
153C2×Q8—2122no—
154C2×D4—2121no—
155C2×D4—2122no—
156C22×C4—532no—
157C22×C4—532no—
158C22×C4—532no—
159C22×C4C22×D5312yes—
160C22×C4C22×D5312yes—
161C22×C4C22×D5311yescharacteristic
162C22×C4C22×D5311yescharacteristic
163C22×C4C22×D5311yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—2122no—
166C4⋊C4—2122no—
167C22⋊C4—1064no—
168C22⋊C4—2122no—
169C22⋊C4—2122no—
170C22⋊C4—2122no—
171C22⋊C4—2122no—
172C22⋊C4—2122no—
173C42—41no—
174C23—1062no—
175C23—1062no—
176C23—1062no—
177C23—1062no—
178C23—1062no—
179C23—1062no—
180C23—1062no—
181C23C23×D5311yescharacteristic
182Q8—44no—
183Q8—2124no—
184D4—2124no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4#1171994512yes—
207C22—2122no—
208C22—2122no—
209C22—2122no—
210C22—2122no—
211C22—2122no—
212C22—2122no—
213C22—2122no—
214C22C2×#1171994511yescharacteristic
215C22#1313326211yescharacteristic
216C22#1313326211yescharacteristic
217C22#1313326312yes—
218C22#1313326512yes—
219C4—22no—
220C4—41no—
221C4—41no—
222C4—42no—
223C4—42no—
224C4—2122no—
225C4—2122no—
226C4—2122no—
227C2—4242no—
228C2#1681121111yescharacteristic
229C2#1681124512yes—
230C2#1681136511yescharacteristic
231C2#1681136912yes—
232C2#1681137211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#391C2
hypercenter#391D53
hypocenterC53#2832
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #16810564C2semidirect
2. #16810552C2semidirect
3. #16810543C2semidirect
6. #16810536C2semidirect
7. #16810534C2semidirect
8. #16810514C2semidirect
9. C53×#391C2semidirect
124. #391D53semidirect
126. C53#2832semidirect