Finite Group Databasebeta

← #20282742

Subgroups — #20282742

Order 6784.

528 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133191C211yesmaximal, characteristic
2#16810558C212yesmaximal, complemented
3#16810553C212yesmaximal, complemented
4#16810540C212yesmaximal, complemented
5#16810538C214yesmaximal, complemented
6#16810534C212yesmaximal, complemented
7#16810486C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
9C22×D212C2211yescharacteristic
10C22×C22⋊D53C2212yes—
11#13133191—24no—
12C2×C4⋊Dic53C2211yescharacteristic
13C2×#11719919C2214yes—
14C2×C22⋊Dic53C2212yes—
15C2×#11719921C2212yes—
16C2×#11719921C2214yes—
17C2×#11719921C2214yes—
18#13133124C2214yes—
19#13133124C2212yes—
20#13133124C2212yes—
21C22×Q8×C53C2211yescharacteristic
22C23×C4×C53C2211yescharacteristic
23C22⋊Q8×C53—24no—
24C2×C4⋊C4×C53C2212yes—
25C2×C4⋊C4×C53C2211yescharacteristic
26C2×C22⋊C4×C53C2212yes—
27C24×C53C2311yescharacteristic
28C23×D53C2312yes—
29C2×Q8×C53—22no—
30C2×Q8×C53—42no—
31C22×C4×C53—42no—
32C22×C4×C53D414yes—
33C22×C4×C53C2312yes—
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C2×C22⋊D53—24no—
38C2×C22⋊D53—44no—
39C22×Dic53C2314yes—
40C22×Dic53C2312yes—
41C2×D212—22no—
42C2×D212—42no—
43C4⋊C4×C53—24no—
44C4⋊C4×C53—22no—
45C22⋊C4×C53—24no—
46C22⋊Dic53—42no—
47#11719921—24no—
48#11719921—44no—
49#11719921—44no—
50C4⋊Dic53—22no—
51#11719919—44no—
52C23×C53—22no—
53C23×C53—22no—
54C23×C53—22no—
55C23×C53C2×D412yes—
56C23×C53C2411yescharacteristic
57C22×D53—22no—
58C22×D53—24no—
59C22×D53—24no—
60C22×D53—24no—
61Q8×C53—44no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—24no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—44no—
70C2×C4×C53C2×D412yes—
71C2×C4×C53C2×D412yes—
72C22⋊D53—48no—
73C2×Dic53—22no—
74C2×Dic53—24no—
75C2×Dic53—24no—
76C2×Dic53—24no—
77C2×Dic53—24no—
78D212—44no—
79C22×C53—22no—
80C22×C53—22no—
81C22×C53—22no—
82C22×C53—44no—
83C22×C53C22×D411yescharacteristic
84C22×C53H(5,𝔽2)12yes—
85C22×C53H(5,𝔽2)12yes—
86C22×C532−1+412yes—
87C2×D53—42no—
88C2×D53—42no—
89C2×D53—42no—
90C2×D53—48no—
91C4×C53—22no—
92C4×C53—22no—
93C4×C53—42no—
94C4×C53—42no—
95Dic53—42no—
96Dic53—44no—
97#2816—531nomaximal, Sylow
98C2×C53—42no—
99C2×C53#53411yescharacteristic
100C2×C53#53511yescharacteristic
101C2×C53#53611yescharacteristic
102C2×C53#56214yes—
103D53—82no—
104C2×C22⋊Q8D5311yescharacteristic, complemented
105C2×#121—531no—
106#399—532no—
107#398—532no—
108#394—532no—
109#394—534no—
110#393—532no—
111#381—531no—
112C53#281611yescharacteristic, complemented, Sylow
113C22×Q8C2×D5311yescharacteristic
114C22×D4—532no—
115C22×D4—531no—
116C23×C4C2×D5311yescharacteristic
117C22⋊Q8—24no—
118#121—1064no—
119C2×C4⋊C4—531no—
120C2×C4⋊C4—534no—
121C2×C4⋊C4C2×D5312yes—
122C2×C4⋊C4C2×D5311yescharacteristic
123C2×C22⋊C4—532no—
124C2×C22⋊C4—532no—
125C2×C22⋊C4—534no—
126C2×C22⋊C4—534no—
127C2×C22⋊C4C2×D5312yes—
128#95—532no—
129#95—532no—
130#95—534no—
131C24—532no—
132C24C22×D5311yescharacteristic
133C2×Q8—22no—
134C2×Q8—42no—
135C2×D4—1064no—
136C2×D4—1062no—
137C2×D4—2122no—
138C2×D4—2124no—
139C22×C4—42no—
140C22×C4—534no—
141C22×C4—532no—
142C22×C4C22⋊D5314yes—
143C22×C4C22×D5312yes—
144C22×C4C22×D5312yes—
145C22×C4C22×D5311yescharacteristic
146C22×C4C22×D5311yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—24no—
149C4⋊C4—1062no—
150C4⋊C4—2124no—
151C22⋊C4—24no—
152C22⋊C4—1064no—
153C22⋊C4—2124no—
154C22⋊C4—2124no—
155C22⋊C4—2122no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—1062no—
160C23—1064no—
161C23—1064no—
162C23—1064no—
163C23C2×C22⋊D5312yes—
164C23C23×D5311yescharacteristic
165Q8—44no—
166D4—2124no—
167D4—2128no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—44no—
176C2×C4—1062no—
177C2×C4—1064no—
178C2×C4—1064no—
179C2×C4—1064no—
180C2×C4—1064no—
181C2×C4C2×C22⋊D5312yes—
182C2×C4C2×C22⋊D5312yes—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—44no—
187C22—2128no—
188C22—2122no—
189C22—2122no—
190C22—2122no—
191C22C22×C22⋊D5311yescharacteristic
192C22#1313326212yes—
193C22#1313326312yes—
194C22#1313326512yes—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—2124no—
200C4—2122no—
201C2—42no—
202C2—4242no—
203C2#1681118211yescharacteristic
204C2#1681118711yescharacteristic
205C2#1681119711yescharacteristic
206C2#1681136914yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C22⋊Q8×C53C2
hypercenterC2×C22⋊Q8D53
hypocenterC53#2816
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16810558C2semidirect
3. #16810553C2semidirect
4. #16810540C2semidirect
5. #16810538C2semidirect
6. #16810534C2semidirect
7. #16810486C2semidirect
8. C2×C22⋊Q8×C53C2semidirect
104. C2×C22⋊Q8D53semidirect
112. C53#2816semidirect