Finite Group Databasebeta

← #20282230

Subgroups — #20282230

Order 6784.

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

normal subgroups only

This group has 392 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#16810565C212yesmaximal, complemented
2#16810564C212yesmaximal, complemented
3#16810559C211yesmaximal, characteristic
4#16810548C211yesmaximal, characteristic
5#16810520C212yesmaximal, complemented
6#16810511C212yesmaximal
7#16810505C212yesmaximal, complemented
8#16810496C212yesmaximal
9C53×#381C211yesmaximal, characteristic
10C2×C4⋊Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12C2×#11719919C2212yes—
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×C4×Dic53C2212yes—
19#13133124C2212yes—
20#13133124C2212yes—
21#13133124C2211yescharacteristic
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26C23×C4×C53C2211yescharacteristic
27C2×C22⋊C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2211yescharacteristic
29#95×C53C2212yes—
30#95×C53C2212yes—
31C24×C53C2311yescharacteristic
32C22×C4×C53—22no—
33C22×C4×C53—41no—
34C22×C4×C53—41no—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C22⋊C4×C53—41no—
44C22⋊C4×C53—41no—
45C22⋊Dic53—24no—
46C22⋊Dic53—24no—
47C4⋊Dic53—41no—
48C4⋊Dic53—41no—
49#11719919—24no—
50#11719919—24no—
51#11719919—42no—
52#11719919—42no—
53C4×Dic53—42no—
54C23×C53—22no—
55C23×C53—21no—
56C23×C53—21no—
57C23×C53—22no—
58C23×C53#4112yes—
59C23×C53C2411yescharacteristic
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—44no—
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—22no—
82C2×Dic53#4114yes—
83C22×C53—22no—
84C22×C53—24no—
85C22×C53—42no—
86C22×C53—41no—
87C22×C53—41no—
88C22×C53C2×#4112yes—
89C22×C53C2×#4111yescharacteristic
90C22×C53H(5,𝔽2)11yescharacteristic
91C22×C53H(5,𝔽2)11yescharacteristic
92C22×C532−1+412yes—
93C4×C53—42no—
94C4×C53—42no—
95C4×C53—41no—
96C4×C53—41no—
97Dic53—24no—
98Dic53—42no—
99Dic53—42no—
100Dic53—42no—
101#2771—531nomaximal, Sylow
102C2×C53—42no—
103C2×C53#53811yescharacteristic
104C2×C53#53912yes—
105C2×C53#55212yes—
106C2×C53#55911yescharacteristic
107C2×C53#56311yescharacteristic
108#400—532no—
109#400—532no—
110#399—531no—
111#397—531no—
112#389—532no—
113#388—532no—
114#387—532no—
115#385—532no—
116#381D5311yescharacteristic
117C53#277111yescharacteristic, complemented, Sylow
118C23×C4C2×D5311yescharacteristic
119C2×C4⋊C4—532no—
120C2×C4⋊C4—532no—
121C2×C4⋊C4—532no—
122C2×C4⋊C4—532no—
123C2×C4⋊C4—531no—
124C2×C4⋊C4—531no—
125C2×C22⋊C4—532no—
126C2×C22⋊C4—532no—
127C2×C22⋊C4C2×D5311yescharacteristic
128C2×C22⋊C4C2×D5311yescharacteristic
129C2×C42—532no—
130#95—532no—
131#95—532no—
132#95—531no—
133#95—532no—
134#95—532no—
135#95—532no—
136#95—531no—
137#95C2×D5312yes—
138#95C2×D5312yes—
139C24C22×D5311yescharacteristic
140C22×C4—22no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—532no—
144C22×C4—532no—
145C22×C4—532no—
146C22×C4—532no—
147C22×C4C22×D5311yescharacteristic
148C22×C4C22×D5312yes—
149C22×C4C22×D5312yes—
150C22×C4C22×D5311yescharacteristic
151C4⋊C4—1064no—
152C4⋊C4—1064no—
153C4⋊C4—2122no—
154C4⋊C4—2122no—
155C4⋊C4—2121no—
156C4⋊C4—2121no—
157C22⋊C4—41no—
158C22⋊C4—41no—
159C22⋊C4—1064no—
160C22⋊C4—1064no—
161C42—2122no—
162C23—22no—
163C23—21no—
164C23—21no—
165C23—22no—
166C23#1171994512yes—
167C23C23×D5311yescharacteristic
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—44no—
179C2×C4—534no—
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—1062no—
189C2×C4—1062no—
190C2×C4—1062no—
191C22—24no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—42no—
196C22#41×D5312yes—
197C22C2×#1171994511yescharacteristic
198C22#1313326211yescharacteristic
199C22#1313326211yescharacteristic
200C22#1313326412yes—
201C4—41no—
202C4—41no—
203C4—42no—
204C4—42no—
205C4—1064no—
206C4—2122no—
207C4—2122no—
208C4—2122no—
209C2—42no—
210C2#1681121111yescharacteristic
211C2#1681121612yes—
212C2#1681130612yes—
213C2#1681135711yescharacteristic
214C2#1681137611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#381C2
hypercenter#381D53
hypocenterC53#2771
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. #16810565C2semidirect
2. #16810564C2semidirect
5. #16810520C2semidirect
7. #16810505C2semidirect
117. C53#2771semidirect