Finite Group Databasebeta

← #20282862

Subgroups — #20282862

Order 6784.

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

normal subgroups only

This group has 408 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#16810552C212yesmaximal, complemented
2#16810550C212yesmaximal, complemented
3#16810545C212yesmaximal, complemented
4#16810543C211yesmaximal, characteristic, complemented
5#16810540C212yesmaximal
6#16810536C212yesmaximal, complemented
7#16810524C211yesmaximal, characteristic
8#16810509C212yesmaximal, complemented
9C53×#398C211yesmaximal, characteristic, complemented
10C22×Dic106C2211yescharacteristic
11C22×D212C2211yescharacteristic
12C2×C4⋊Dic53C2212yes—
13C2×#11719919C2212yes—
14C2×#11719921C2212yes—
15C2×#11719921C2212yes—
16C2×#11719921C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×C4×Dic53C2211yescharacteristic
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26C2×C4⋊C4×C53C2212yes—
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29#95×C53C2212yes—
30#95×C53C2211yescharacteristic
31C23×D53C2312yes—
32C22×C4×C53C2312yes—
33C22×C4×C53C2311yescharacteristic
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C2×D212—22no—
41C2×D212—41no—
42C2×D212—41no—
43C2×Dic106—42no—
44C2×Dic106—41no—
45C4⋊C4×C53—24no—
46C4⋊C4×C53—41no—
47C4⋊C4×C53—41no—
48#11719921—42no—
49#11719921—42no—
50#11719921—42no—
51#11719921—42no—
52#11719921—42no—
53#11719921—42no—
54C4⋊Dic53—42no—
55#11719919—42no—
56C4×Dic53—41no—
57C23×C53C2411yescharacteristic
58C22×D53—22no—
59C22×D53—22no—
60C22×D53—22no—
61C22×D53—22no—
62C22×D53—22no—
63C22×D53—22no—
64C22×D53—22no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—21no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53#4112yes—
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—
87D212—44no—
88Dic106—44no—
89C22×C53C2×#4111yescharacteristic
90C22×C53H(5,𝔽2)11yescharacteristic
91C22×C53H(5,𝔽2)11yescharacteristic
92C22×C53H(5,𝔽2)12yes—
93C22×C532−1+412yes—
94C2×D53—42no—
95C2×D53—42no—
96C2×D53—42no—
97C2×D53—42no—
98C2×D53—42no—
99C2×D53—42no—
100C2×D53—42no—
101C4×C53—22no—
102C4×C53—42no—
103C4×C53—42no—
104C4×C53—41no—
105C4×C53—41no—
106Dic53—42no—
107Dic53—42no—
108Dic53—42no—
109#2830—531nomaximal, Sylow
110C2×C53#53811yescharacteristic
111C2×C53#54212yes—
112C2×C53#56011yescharacteristic
113C2×C53#56212yes—
114C2×C53#56311yescharacteristic
115D53—82no—
116#398D5311yescharacteristic, complemented
117#397—532no—
118#397—532no—
119#396—532no—
120#395—531no—
121#394—532no—
122#393—532no—
123#390—531no—
124#387—532no—
125C53#283011yescharacteristic, complemented, Sylow
126C22×Q8—531no—
127C22×D4—531no—
128C2×C4⋊C4—532no—
129C2×C4⋊C4—532no—
130C2×C4⋊C4C2×D5312yes—
131C2×C4⋊C4C2×D5311yescharacteristic
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×C42—531no—
140#95—532no—
141#95—532no—
142#95—532no—
143#95—532no—
144#95—531no—
145#95C2×D5312yes—
146#95C2×D5311yescharacteristic
147C24—532no—
148C2×Q8—2121no—
149C2×Q8—2122no—
150C2×D4—1062no—
151C2×D4—2121no—
152C2×D4—2121no—
153C22×C4—532no—
154C22×C4—532no—
155C22×C4—532no—
156C22×C4C22×D5312yes—
157C22×C4C22×D5311yescharacteristic
158C22×C4C22×D5312yes—
159C22×C4C22×D5311yescharacteristic
160C22×C4C22×D5311yescharacteristic
161C4⋊C4—24no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—2122no—
165C4⋊C4—2122no—
166C22⋊C4—2122no—
167C22⋊C4—2122no—
168C22⋊C4—2122no—
169C22⋊C4—2122no—
170C22⋊C4—2122no—
171C22⋊C4—2122no—
172C42—2121no—
173C23—1062no—
174C23—1062no—
175C23—1062no—
176C23—1062no—
177C23—1062no—
178C23—1062no—
179C23—1062no—
180C23C23×D5311yescharacteristic
181Q8—2124no—
182D4—2124no—
183C2×C4—21no—
184C2×C4—21no—
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—22no—
194C2×C4—22no—
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×C4#1171994712yes—
205C22—2122no—
206C22—2122no—
207C22—2122no—
208C22—2122no—
209C22—2122no—
210C22—2122no—
211C22—2122no—
212C22C2×#1171994711yescharacteristic
213C22#1313326211yescharacteristic
214C22#1313326312yes—
215C22#1313326311yescharacteristic
216C22#1313326512yes—
217C4—22no—
218C4—41no—
219C4—41no—
220C4—42no—
221C4—42no—
222C4—2122no—
223C4—2122no—
224C4—2122no—
225C2—4242no—
226C2#1681121011yescharacteristic
227C2#1681124612yes—
228C2#1681136311yescharacteristic
229C2#1681137112yes—
230C2#1681137211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2830
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. #16810552C2semidirect
2. #16810550C2semidirect
3. #16810545C2semidirect
4. #16810543C2semidirect
6. #16810536C2semidirect
8. #16810509C2semidirect
9. C53×#398C2semidirect
116. #398D53semidirect
125. C53#2830semidirect