Finite Group Databasebeta

← #20282421

Subgroups — #20282421

Order 6784.

428 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 121 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#16810543C211yesmaximal, characteristic, complemented
2#16810541C212yesmaximal
3#16810540C211yesmaximal, characteristic
4#16810536C212yesmaximal, complemented
5#16810514C212yesmaximal, complemented
6#16810513C212yesmaximal, complemented
7#16810512C212yesmaximal, complemented
8#16810487C212yesmaximal, complemented
9C53×#382C211yesmaximal, characteristic, complemented
10C22×Dic106C2211yescharacteristic
11C22×D212C2211yescharacteristic
12C2×#11719919C2212yes—
13C2×#11719919C2212yes—
14C2×#11719921C2212yes—
15C2×#11719921C2212yes—
16C2×#11719921C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×C4×Dic53C2212yes—
21#13133124C2211yescharacteristic
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26C2×C4⋊C4×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C42×C53C2211yescharacteristic
29#95×C53C2212yes—
30#95×C53C2212yes—
31C23×D53C2312yes—
32C22×C4×C53C2312yes—
33C22×C4×C53C2312yes—
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C2×D212—42no—
41C2×D212—41no—
42C2×Dic106—42no—
43C2×Dic106—41no—
44C4⋊C4×C53—41no—
45C4⋊C4×C53—41no—
46C42×C53—22no—
47#11719921—24no—
48#11719921—24no—
49#11719921—24no—
50#11719921—42no—
51#11719921—42no—
52#11719921—42no—
53#11719919—42no—
54#11719919—42no—
55C4×Dic53—24no—
56C23×C53C2411yescharacteristic
57C22×D53—22no—
58C22×D53—22no—
59C22×D53—22no—
60C22×D53—22no—
61C22×D53—22no—
62C22×D53—22no—
63C22×D53—22no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—21no—
72C2×C4×C53—22no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53#4112yes—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53#4114yes—
86D212—44no—
87Dic106—44no—
88C22×C53C2×#4112yes—
89C22×C53C2×#4111yescharacteristic
90C22×C53H(5,𝔽2)11yescharacteristic
91C22×C53H(5,𝔽2)12yes—
92C22×C532−1+411yescharacteristic
93C2×D53—42no—
94C2×D53—42no—
95C2×D53—42no—
96C2×D53—42no—
97C2×D53—42no—
98C2×D53—42no—
99C2×D53—42no—
100C4×C53—22no—
101C4×C53—42no—
102C4×C53—42no—
103C4×C53—41no—
104C4×C53—41no—
105Dic53—24no—
106Dic53—42no—
107Dic53—42no—
108#2785—531nomaximal, Sylow
109C2×C53#53911yescharacteristic
110C2×C53#53912yes—
111C2×C53#55511yescharacteristic
112C2×C53#55512yes—
113C2×C53H(3,𝔽4)11yescharacteristic
114D53—82no—
115#395—531no—
116#394—531no—
117#394—532no—
118#393—532no—
119#388—532no—
120#388—532no—
121#388—532no—
122#382—532no—
123#382D5311yescharacteristic, complemented
124C53#278511yescharacteristic, complemented, Sylow
125C22×Q8—531no—
126C22×D4—531no—
127C2×C4⋊C4—532no—
128C2×C4⋊C4—532no—
129C2×C4⋊C4C2×D5311yescharacteristic
130C2×C4⋊C4C2×D5311yescharacteristic
131C2×C22⋊C4—532no—
132C2×C22⋊C4—532no—
133C2×C22⋊C4—532no—
134C2×C22⋊C4—532no—
135C2×C22⋊C4—532no—
136C2×C22⋊C4—532no—
137C2×C42—532no—
138C2×C42C2×D5311yescharacteristic
139#95—531no—
140#95—532no—
141#95—532no—
142#95—531no—
143#95—532no—
144#95C2×D5312yes—
145#95C2×D5312yes—
146C24—532no—
147C2×Q8—2121no—
148C2×Q8—2122no—
149C2×D4—2121no—
150C2×D4—2122no—
151C22×C4—532no—
152C22×C4—532no—
153C22×C4—532no—
154C22×C4C22×D5312yes—
155C22×C4C22×D5312yes—
156C22×C4C22×D5311yescharacteristic
157C22×C4C22×D5311yescharacteristic
158C22×C4C22×D5311yescharacteristic
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—2122no—
162C4⋊C4—2122no—
163C22⋊C4—1064no—
164C22⋊C4—1064no—
165C22⋊C4—1064no—
166C22⋊C4—2122no—
167C22⋊C4—2122no—
168C22⋊C4—2122no—
169C42—22no—
170C42—1064no—
171C23—1062no—
172C23—1062no—
173C23—1062no—
174C23—1062no—
175C23—1062no—
176C23—1062no—
177C23—1062no—
178C23C23×D5311yescharacteristic
179Q8—2124no—
180D4—2124no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
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#1171994512yes—
203C22—2122no—
204C22—2122no—
205C22—2122no—
206C22—2122no—
207C22—2122no—
208C22—2122no—
209C22—2122no—
210C22#41×D5312yes—
211C22C2×#1171994511yescharacteristic
212C22#1313326211yescharacteristic
213C22#1313326312yes—
214C22#1313326511yescharacteristic
215C4—22no—
216C4—42no—
217C4—42no—
218C4—41no—
219C4—41no—
220C4—1064no—
221C4—2122no—
222C4—2122no—
223C2—4242no—
224C2#1681122011yescharacteristic
225C2#1681122112yes—
226C2#1681133511yescharacteristic
227C2#1681133612yes—
228C2#1681136511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#382C2
hypercenter#382D53
hypocenterC53#2785
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. #16810543C2semidirect
4. #16810536C2semidirect
5. #16810514C2semidirect
6. #16810513C2semidirect
7. #16810512C2semidirect
8. #16810487C2semidirect
9. C53×#382C2semidirect
123. #382D53semidirect
124. C53#2785semidirect