Finite Group Databasebeta

← #20282393

Subgroups — #20282393

Order 6784.

448 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 125 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#16810563C212yesmaximal, complemented
2#16810562C211yesmaximal, characteristic, complemented
3#16810551C212yesmaximal, complemented
4#16810511C212yesmaximal, complemented
5#16810503C212yesmaximal
6#16810498C212yesmaximal, complemented
7#16810498C212yesmaximal, complemented
8#16810488C211yesmaximal, characteristic, complemented
9C53×#394C211yesmaximal, characteristic
10C23×Dic53C2212yes—
11C2×C4⋊Dic53C2211yescharacteristic
12C2×C4⋊Dic53C2212yes—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919C2212yes—
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×C4×Dic53C2211yescharacteristic
19#13133124C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26C22×D4×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2212yes—
29C2×C22⋊C4×C53C2212yes—
30#95×C53C2211yescharacteristic
31C24×C53C2312yes—
32C2×D4×C53—24no—
33C2×D4×C53—41no—
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×Dic53—42no—
39C22×Dic53—42no—
40C22×Dic53—42no—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C4⋊C4×C53—41no—
46C22⋊C4×C53—24no—
47C22⋊C4×C53—24no—
48C22⋊Dic53—24no—
49C22⋊Dic53—42no—
50C22⋊Dic53—42no—
51C4⋊Dic53—41no—
52C4⋊Dic53—42no—
53C4⋊Dic53—41no—
54#11719919—42no—
55C4×Dic53—22no—
56C23×C53—22no—
57C23×C53—22no—
58C23×C53—22no—
59C23×C53—22no—
60C23×C53—22no—
61C23×C53—22no—
62C23×C53#4114yes—
63C23×C53C2411yescharacteristic
64D4×C53—44no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
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#4112yes—
75C2×Dic53—22no—
76C2×Dic53—22no—
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—22no—
86C2×Dic53—22no—
87C2×Dic53—48no—
88C22×C53—24no—
89C22×C53—24no—
90C22×C53—24no—
91C22×C53—42no—
92C22×C53—42no—
93C22×C53—42no—
94C22×C53—42no—
95C22×C53C2×#4112yes—
96C22×C53C2×#4111yescharacteristic
97C22×C53H(5,𝔽2)11yescharacteristic
98C22×C53H(5,𝔽2)12yes—
99C22×C532−1+411yescharacteristic
100C4×C53—22no—
101C4×C53—41no—
102C4×C53—41no—
103C4×C53—42no—
104Dic53—42no—
105Dic53—42no—
106Dic53—42no—
107Dic53—42no—
108#2782—531nomaximal, Sylow
109C2×C53—44no—
110C2×C53#53912yes—
111C2×C53#53911yescharacteristic
112C2×C53#55111yescharacteristic
113C2×C53#55312yes—
114C2×C53H(3,𝔽4)11yescharacteristic
115#400—531no—
116#400—532no—
117#397—532no—
118#394D5311yescharacteristic
119#388—532no—
120#386—532no—
121#385—532no—
122#385—532no—
123#382—531no—
124C53#278211yescharacteristic, complemented, Sylow
125C22×D4C2×D5311yescharacteristic
126C23×C4—532no—
127C2×C4⋊C4—532no—
128C2×C4⋊C4—531no—
129C2×C4⋊C4—531no—
130C2×C4⋊C4—532no—
131C2×C4⋊C4C2×D5311yescharacteristic
132C2×C22⋊C4—532no—
133C2×C22⋊C4—532no—
134C2×C22⋊C4—532no—
135C2×C22⋊C4C2×D5312yes—
136C2×C22⋊C4C2×D5312yes—
137C2×C42—531no—
138#95—532no—
139#95—532no—
140#95—532no—
141#95—532no—
142#95—532no—
143#95—532no—
144#95—531no—
145#95C2×D5311yescharacteristic
146C24C22×D5312yes—
147C2×D4—24no—
148C2×D4—41no—
149C22×C4—532no—
150C22×C4—532no—
151C22×C4—532no—
152C22×C4—532no—
153C22×C4—2122no—
154C22×C4—2122no—
155C22×C4—2122no—
156C22×C4C22×D5311yescharacteristic
157C22×C4C22×D5311yescharacteristic
158C22×C4C22×D5312yes—
159C22×C4C22×D5311yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—2122no—
162C4⋊C4—2121no—
163C4⋊C4—2121no—
164C4⋊C4—2122no—
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—1064no—
168C22⋊C4—2122no—
169C22⋊C4—2122no—
170C42—1062no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23#1171994714yes—
178C23C23×D5311yescharacteristic
179D4—44no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—2128no—
202C2×C4#1171994712yes—
203C22—24no—
204C22—24no—
205C22—24no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22C2×#1171994712yes—
211C22C2×#1171994711yescharacteristic
212C22#1313326211yescharacteristic
213C22#1313326312yes—
214C22#1313326411yescharacteristic
215C4—22no—
216C4—41no—
217C4—41no—
218C4—42no—
219C4—2122no—
220C4—2122no—
221C4—2122no—
222C4—2122no—
223C2—44no—
224C2#1681121712yes—
225C2#1681121811yescharacteristic
226C2#1681130411yescharacteristic
227C2#1681132512yes—
228C2#1681136511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2782
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #16810563C2semidirect
2. #16810562C2semidirect
3. #16810551C2semidirect
4. #16810511C2semidirect
6. #16810498C2semidirect
7. #16810498C2semidirect
8. #16810488C2semidirect
124. C53#2782semidirect