Finite Group Databasebeta

← #20282900

Subgroups — #20282900

Order 6784.

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

normal subgroups only

This group has 372 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
1C53⋊#401C212yesmaximal, complemented
2#16810559C211yesmaximal, characteristic
3#16810557C212yesmaximal, complemented
4#16810555C212yesmaximal, complemented
5#16810553C212yesmaximal, complemented
6#16810551C212yesmaximal
7#16810498C212yesmaximal
8#16810495C211yesmaximal, characteristic
9C53×#397C211yesmaximal, characteristic
10C23×Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2212yes—
12C2×C4⋊Dic53C2212yes—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×C22⋊Dic53C2211yescharacteristic
18C2×C22⋊Dic53C2212yes—
19#13133124C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26C2×C4⋊C4×C53C2211yescharacteristic
27C2×C22⋊C4×C53C2212yes—
28C2×C22⋊C4×C53C2211yescharacteristic
29#95×C53C2212yes—
30#95×C53C2211yescharacteristic
31C24×C53C2311yescharacteristic
32C22×C4×C53C2311yescharacteristic
33C22×C4×C53C2312yes—
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×Dic53—42no—
37C22×Dic53—41no—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C4⋊C4×C53—41no—
43C22⋊C4×C53—24no—
44C22⋊C4×C53—41no—
45C22⋊Dic53—22no—
46C22⋊Dic53—42no—
47C4⋊Dic53—42no—
48C4⋊Dic53—42no—
49C4⋊Dic53—41no—
50#11719919—42no—
51#11719919—42no—
52#11719919—42no—
53C23×C53—22no—
54C23×C53—21no—
55C23×C53—22no—
56C23×C53—21no—
57C23×C53#4112yes—
58C23×C53C2411yescharacteristic
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—21no—
64C2×C4×C53—22no—
65C2×C4×C53—21no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×Dic53—22no—
70C2×Dic53—22no—
71C2×Dic53—22no—
72C2×Dic53—22no—
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—44no—
82C22×C53—24no—
83C22×C53—22no—
84C22×C53—41no—
85C22×C53—41no—
86C22×C53—42no—
87C22×C53C2×#4111yescharacteristic
88C22×C53H(5,𝔽2)11yescharacteristic
89C22×C53H(5,𝔽2)12yes—
90C22×C532−1+412yes—
91C22×C532−1+411yescharacteristic
92C4×C53—42no—
93C4×C53—42no—
94C4×C53—41no—
95C4×C53—41no—
96Dic53—42no—
97Dic53—42no—
98Dic53—42no—
99Dic53—42no—
100#2834—531nomaximal, Sylow
101C2×C53—42no—
102C2×C53#53912yes—
103C2×C53#53911yescharacteristic
104C2×C53#56011yescharacteristic
105C2×C53#56311yescharacteristic
106C2×C53#56312yes—
107#401—532no—
108#399—531no—
109#398—532no—
110#398—532no—
111#398—532no—
112#397—532no—
113#397D5311yescharacteristic
114#385—532no—
115#385—531no—
116C53#283411yescharacteristic, complemented, Sylow
117C23×C4—531no—
118C2×C4⋊C4—532no—
119C2×C4⋊C4—532no—
120C2×C4⋊C4—532no—
121C2×C4⋊C4—532no—
122C2×C4⋊C4—532no—
123C2×C4⋊C4—531no—
124C2×C4⋊C4C2×D5311yescharacteristic
125C2×C22⋊C4—531no—
126C2×C22⋊C4—532no—
127C2×C22⋊C4C2×D5312yes—
128C2×C22⋊C4C2×D5311yescharacteristic
129#95—532no—
130#95—532no—
131#95—532no—
132#95—532no—
133#95—532no—
134#95—532no—
135#95—531no—
136#95C2×D5311yescharacteristic
137#95C2×D5312yes—
138C24C22×D5311yescharacteristic
139C22×C4—532no—
140C22×C4—532no—
141C22×C4—532no—
142C22×C4—532no—
143C22×C4—2121no—
144C22×C4—2122no—
145C22×C4C22×D5311yescharacteristic
146C22×C4C22×D5312yes—
147C22×C4C22×D5312yes—
148C22×C4C22×D5311yescharacteristic
149C4⋊C4—41no—
150C4⋊C4—2122no—
151C4⋊C4—2122no—
152C4⋊C4—2122no—
153C4⋊C4—2121no—
154C4⋊C4—2122no—
155C4⋊C4—2122no—
156C22⋊C4—24no—
157C22⋊C4—41no—
158C22⋊C4—1062no—
159C22⋊C4—2122no—
160C23—21no—
161C23—22no—
162C23—22no—
163C23—21no—
164C23#1171994712yes—
165C23C23×D5311yescharacteristic
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—1062no—
177C2×C4—1062no—
178C2×C4—1062no—
179C2×C4—1062no—
180C2×C4—1062no—
181C2×C4—1062no—
182C2×C4—1062no—
183C2×C4—1062no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—2124no—
189C22—24no—
190C22—22no—
191C22—41no—
192C22—41no—
193C22—42no—
194C22C2×#1171994711yescharacteristic
195C22#1313326211yescharacteristic
196C22#1313326312yes—
197C22#1313326412yes—
198C22#1313326411yescharacteristic
199C4—41no—
200C4—41no—
201C4—42no—
202C4—42no—
203C4—2122no—
204C4—2122no—
205C4—2122no—
206C4—2122no—
207C2—42no—
208C2#1681121712yes—
209C2#1681121811yescharacteristic
210C2#1681136311yescharacteristic
211C2#1681137311yescharacteristic
212C2#1681137512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2834
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C53⋊#401C2semidirect
3. #16810557C2semidirect
4. #16810555C2semidirect
5. #16810553C2semidirect
116. C53#2834semidirect