Finite Group Databasebeta

← #20277311

Subgroups — #20277311

Order 6784.

672 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 189 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
1C22×C22⋊Dic53C211yesmaximal, characteristic
2#16810517C214yesmaximal, complemented
3#16810511C214yesmaximal, complemented
4#16810481C212yesmaximal, complemented
5#16810479C211yesmaximal, characteristic
6C22⋊C4×Dic53C212yesmaximal, complemented
7C53×#379C211yesmaximal, characteristic
8C23×Dic53C2211yescharacteristic
9C23×Dic53C2211yescharacteristic
10C2×#11719919C2214yescomplemented
11C2×C22⋊Dic53C418yescomplemented
12C2×C22⋊Dic53C2212yes—
13C2×C22⋊Dic53C2212yes—
14C2×C22⋊Dic53C2212yes—
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×C4×Dic53C2214yescomplemented
18#13133124C2214yescomplemented
19#13133124C2214yescomplemented
20C25×C53C2211yescharacteristic
21C2×C22⋊C4×C53C2212yes—
22C2×C22⋊C4×C53C2212yes—
23C2×C22⋊C4×C53C2212yes—
24C24×C53—24no—
25C24×C53—42no—
26C24×C53—42no—
27C24×C53C2×C414yes—
28C24×C53C2311yescharacteristic
29C24×C53C2311yescharacteristic
30C24×C53C2311yescharacteristic
31C22×C4×C53C2314yes—
32C22×Dic53—42no—
33C22×Dic53—42no—
34C22×Dic53C2×C414yes—
35C22×Dic53C2×C414yes—
36C22×Dic53C2314yes—
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C22⋊C4×C53—24no—
40C22⋊C4×C53—42no—
41C22⋊C4×C53—42no—
42C22⋊Dic53—24no—
43C22⋊Dic53—42no—
44C22⋊Dic53—42no—
45C22⋊Dic53C2×C418yes—
46C22⋊Dic53C2×C418yes—
47#11719919—44no—
48C4×Dic53—44no—
49C23×C53—22no—
50C23×C53—22no—
51C23×C53—28no—
52C23×C53—22no—
53C23×C53—22no—
54C23×C53—22no—
55C23×C53—22no—
56C23×C53—22no—
57C23×C53—22no—
58C23×C53—44no—
59C23×C53—44no—
60C23×C53—416no—
61C23×C53C22×C412yes—
62C23×C53C22×C412yes—
63C23×C53C22×C412yes—
64C23×C53#4114yes—
65C23×C53C2411yescharacteristic
66C2×C4×C53—24no—
67C2×C4×C53—24no—
68C2×C4×C53—24no—
69C2×Dic53—24no—
70C2×Dic53—24no—
71C2×Dic53—24no—
72C2×Dic53—24no—
73C2×Dic53—24no—
74C2×Dic53—44no—
75C2×Dic53—44no—
76C2×Dic53C22×C414yes—
77C2×Dic53C22×C414yes—
78C22×C53—22no—
79C22×C53—22no—
80C22×C53—28no—
81C22×C53—22no—
82C22×C53—22no—
83C22×C53—22no—
84C22×C53—22no—
85C22×C53—22no—
86C22×C53—22no—
87C22×C53—416no—
88C22×C53—44no—
89C22×C53—44no—
90C22×C53#11714yes—
91C22×C53C23×C411yescharacteristic
92C22×C53C2×#4112yes—
93C22×C53H(5,𝔽2)12yes—
94C22×C53H(5,𝔽2)12yes—
95C4×C53—44no—
96Dic53—24no—
97Dic53—24no—
98Dic53—44no—
99#2312—531nomaximal, Sylow
100C2×C53—24no—
101C2×C53—42no—
102C2×C53—42no—
103C2×C53C2×#11711yescharacteristic
104C2×C53#51811yescharacteristic
105C2×C53#51811yescharacteristic
106C2×C53#53814yes—
107C22×C22⋊C4—531no—
108#388—534no—
109#388—534no—
110#380—532no—
111#379—531no—
112#379D5311yescharacteristic
113C4×C22⋊C4—532no—
114C53#231211yescharacteristic, complemented, Sylow
115C25C2×D5311yescharacteristic
116C23×C4—531no—
117C23×C4—531no—
118C2×C4⋊C4—534no—
119C2×C22⋊C4—538no—
120C2×C22⋊C4—532no—
121C2×C22⋊C4—532no—
122C2×C22⋊C4—532no—
123C2×C22⋊C4—532no—
124C2×C22⋊C4—532no—
125C2×C22⋊C4C2×D5312yes—
126C2×C22⋊C4C2×D5312yes—
127C2×C22⋊C4C2×D5312yes—
128C2×C42—534no—
129#95—534no—
130#95—534no—
131C24—24no—
132C24—42no—
133C24—42no—
134C24C4×D5314yes—
135C24C22×D5311yescharacteristic
136C24C22×D5311yescharacteristic
137C24C22×D5311yescharacteristic
138C22×C4—532no—
139C22×C4—532no—
140C22×C4—534no—
141C22×C4—534no—
142C22×C4—534no—
143C22×C4—2122no—
144C22×C4—2122no—
145C22×C4C22×D5314yes—
146C4⋊C4—2124no—
147C22⋊C4—24no—
148C22⋊C4—42no—
149C22⋊C4—42no—
150C22⋊C4—538no—
151C22⋊C4—538no—
152C22⋊C4—1064no—
153C22⋊C4—2122no—
154C22⋊C4—2122no—
155C42—2124no—
156C23—22no—
157C23—22no—
158C23—28no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—22no—
163C23—22no—
164C23—22no—
165C23—44no—
166C23—44no—
167C23—416no—
168C23C2×C4×D5312yes—
169C23C2×C4×D5312yes—
170C23C2×C4×D5312yes—
171C23#1171994714yes—
172C23C23×D5311yescharacteristic
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—534no—
177C2×C4—534no—
178C2×C4—1064no—
179C2×C4—1064no—
180C2×C4—1064no—
181C2×C4—1064no—
182C2×C4—1064no—
183C2×C4—2124no—
184C2×C4—2124no—
185C22—22no—
186C22—22no—
187C22—28no—
188C22—22no—
189C22—22no—
190C22—22no—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—416no—
195C22—44no—
196C22—44no—
197C22#1313317314yes—
198C22C22×C4×D5311yescharacteristic
199C22C2×#1171994712yes—
200C22#1313326212yes—
201C22#1313326212yes—
202C4—44no—
203C4—1064no—
204C4—1064no—
205C4—2124no—
206C2—24no—
207C2—42no—
208C2—42no—
209C2C2×#1313317311yescharacteristic
210C2#1681109611yescharacteristic
211C2#1681109611yescharacteristic
212C2#1681120814yes—
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#379C2
socleC23×C53C24
hypercenter#379D53
hypocenterC53#2312
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #16810517C2semidirect
3. #16810511C2semidirect
4. #16810481C2semidirect
6. C22⋊C4×Dic53C2semidirect
10. C2×#11719919C22semidirect
11. C2×C22⋊Dic53C4semidirect
17. C2×C4×Dic53C22semidirect
18. #13133124C22semidirect
19. #13133124C22semidirect
114. C53#2312semidirect