Finite Group Databasebeta

← #20279002

Subgroups — #20279002

Order 6784.

512 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C2×#13133176C211yesmaximal, characteristic
2#16810562C212yesmaximal, complemented
3#16810553C212yesmaximal, complemented
4#16810504C214yesmaximal, complemented
5#16810496C212yesmaximal
6#16810486C212yesmaximal
7#16810486C211yesmaximal, characteristic
8C22×C4⋊C4×C53C211yesmaximal, characteristic
9#13133176—24no—
10C2×C4⋊Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12C2×#11719919C2214yes—
13C2×C22⋊Dic53C2212yes—
14C2×C22⋊Dic53C2212yes—
15C2×C4×Dic53C2212yes—
16#13133124C2214yes—
17#13133124C2212yes—
18#13133124C2214yes—
19#13133124C2214yes—
20#13133124C2212yes—
21C23×C4×C53C2211yescharacteristic
22C23×C4×C53C2211yescharacteristic
23C23×C4×C53C2211yescharacteristic
24C2×C4⋊C4×C53—24no—
25C2×C4⋊C4×C53C2212yes—
26C2×C4⋊C4×C53C2212yes—
27C24×C53C2311yescharacteristic
28C22×C4×C53—22no—
29C22×C4×C53—24no—
30C22×C4×C53—24no—
31C22×C4×C53—22no—
32C22×C4×C53—24no—
33C22×C4×C53D414yes—
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2312yes—
38C22×Dic53C2314yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C4⋊C4×C53—24no—
42C4⋊C4×C53—24no—
43C22⋊Dic53—24no—
44C22⋊Dic53—24no—
45C4⋊Dic53—22no—
46C4⋊Dic53—22no—
47#11719919—28no—
48C4×Dic53—24no—
49C23×C53—22no—
50C23×C53—22no—
51C23×C53C2×D412yes—
52C23×C53#4114yes—
53C23×C53C2411yescharacteristic
54C2×C4×C53—28no—
55C2×C4×C53—28no—
56C2×C4×C53—22no—
57C2×C4×C53—22no—
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—24no—
62C2×C4×C53—44no—
63C2×C4×C53C2×D412yes—
64C2×C4×C53C2×D412yes—
65C2×C4×C53#4114yes—
66C2×Dic53—22no—
67C2×Dic53—22no—
68C2×Dic53—24no—
69C2×Dic53—24no—
70C2×Dic53—24no—
71C2×Dic53—24no—
72C2×Dic53—24no—
73C22×C53—22no—
74C22×C53—22no—
75C22×C53—28no—
76C22×C53#12314yes—
77C22×C53C22×D411yescharacteristic
78C22×C53C2×#4112yes—
79C22×C53C2×#4112yes—
80C22×C532−1+412yes—
81C4×C53—22no—
82C4×C53—22no—
83C4×C53—24no—
84C4×C53—42no—
85Dic53—42no—
86Dic53—42no—
87Dic53—44no—
88#2484—531nomaximal, Sylow
89C2×C53—24no—
90C2×C53C2×#12311yescharacteristic
91C2×C53#52511yescharacteristic
92C2×C53#53611yescharacteristic
93C2×C53#55214yes—
94C2×#117—531no—
95C22×C4⋊C4D5311yescharacteristic
96#400—532no—
97#398—532no—
98#387—534no—
99#385—532no—
100#381—531no—
101#381—532no—
102C53#248411yescharacteristic, complemented, Sylow
103C23×C4C2×D5311yescharacteristic
104C23×C4C2×D5311yescharacteristic
105C23×C4C2×D5311yescharacteristic
106#117—1064no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—531no—
109C2×C4⋊C4—531no—
110C2×C4⋊C4—534no—
111C2×C4⋊C4C2×D5312yes—
112C2×C4⋊C4C2×D5312yes—
113C2×C22⋊C4—532no—
114C2×C22⋊C4—532no—
115C2×C42—532no—
116#95—532no—
117#95—534no—
118#95—534no—
119#95—532no—
120#95—534no—
121C24C22×D5311yescharacteristic
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—24no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—534no—
128C22×C4—532no—
129C22×C4—532no—
130C22×C4C22⋊D5314yes—
131C22×C4C22×D5312yes—
132C22×C4C22×D5312yes—
133C22×C4C22×D5311yescharacteristic
134C22×C4C22×D5311yescharacteristic
135C4⋊C4—24no—
136C4⋊C4—24no—
137C4⋊C4—1068no—
138C4⋊C4—1062no—
139C4⋊C4—1062no—
140C22⋊C4—1064no—
141C22⋊C4—1064no—
142C42—1064no—
143C23—22no—
144C23—22no—
145C23#1171994714yes—
146C23C2×C22⋊D5312yes—
147C23C23×D5311yescharacteristic
148C2×C4—28no—
149C2×C4—28no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—44no—
157C2×C4—1062no—
158C2×C4—1062no—
159C2×C4—1064no—
160C2×C4—1064no—
161C2×C4—1064no—
162C2×C4—1064no—
163C2×C4—1064no—
164C2×C4#1171994514yes—
165C2×C4C2×C22⋊D5312yes—
166C2×C4C2×C22⋊D5312yes—
167C22—28no—
168C22—22no—
169C22—22no—
170C22#1313319814yes—
171C22C22×C22⋊D5311yescharacteristic
172C22C2×#1171994712yes—
173C22C2×#1171994512yes—
174C22#1313326512yes—
175C4—22no—
176C4—22no—
177C4—24no—
178C4—42no—
179C4—2124no—
180C4—2122no—
181C4—2122no—
182C2—24no—
183C2C2×#1313319811yescharacteristic
184C2#1681113411yescharacteristic
185C2#1681119711yescharacteristic
186C2#1681130814yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC22×C4⋊C4×C53C2
hypercenterC22×C4⋊C4D53
hypocenterC53#2484
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #16810562C2semidirect
3. #16810553C2semidirect
4. #16810504C2semidirect
102. C53#2484semidirect