Finite Group Databasebeta

← #20282773

Subgroups — #20282773

Order 6784.

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

normal subgroups only

This group has 430 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×#13133217C211yesmaximal, characteristic, complemented
2#16810562C212yesmaximal, complemented
3#16810559C212yesmaximal
4#16810553C212yesmaximal, complemented
5#16810548C212yesmaximal
6#16810543C212yesmaximal, complemented
7#16810535C212yesmaximal
8#16810491C211yesmaximal, characteristic, complemented
9C2×#126×C53C211yesmaximal, characteristic
10C22×Dic106C2212yes—
11#13133217—24no—
12C2×C4⋊Dic53C2212yes—
13C2×C4⋊Dic53C2212yes—
14C2×#11719919C2214yes—
15C2×C22⋊Dic53C2214yes—
16#13133124C2212yes—
17#13133124C2214yes—
18#13133124C2214yes—
19#13133124C2212yes—
20#13133124C2212yes—
21#126×C53—24no—
22C2×C4⋊C4×C53C2211yescharacteristic
23C2×C4⋊C4×C53C2211yescharacteristic
24C2×C4⋊C4×C53C2211yescharacteristic
25C2×C22⋊C4×C53C2211yescharacteristic
26C2×C22⋊C4×C53C2211yescharacteristic
27C2×C22⋊C4×C53C2211yescharacteristic
28C2×C42×C53C2211yescharacteristic
29C24×C53C2311yescharacteristic
30C22×C4×C53C2311yescharacteristic
31C22×C4×C53C2311yescharacteristic
32C22×C4×C53C2311yescharacteristic
33C22×C4×C53C2311yescharacteristic
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×Dic53C2314yes—
37C22×Dic53C2314yes—
38C2×Dic106—24no—
39C2×Dic106—44no—
40C4⋊C4×C53—22no—
41C4⋊C4×C53—22no—
42C4⋊C4×C53—22no—
43C22⋊C4×C53—22no—
44C22⋊C4×C53—22no—
45C22⋊C4×C53—22no—
46C42×C53D414yes—
47C22⋊Dic53—44no—
48C4⋊Dic53—24no—
49C4⋊Dic53—24no—
50#11719919—44no—
51C23×C53—21no—
52C23×C53—22no—
53C23×C53—22no—
54C23×C53—22no—
55C23×C53C2411yescharacteristic
56C2×C4×C53—22no—
57C2×C4×C53—22no—
58C2×C4×C53—22no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53C2×D412yes—
66C2×C4×C53C2×D412yes—
67C2×C4×C53C2×D412yes—
68C2×Dic53—24no—
69C2×Dic53—24no—
70C2×Dic53—24no—
71C2×Dic53—24no—
72C2×Dic53—24no—
73C2×Dic53—24no—
74Dic106—48no—
75C22×C53—41no—
76C22×C53—41no—
77C22×C53—41no—
78C22×C53—44no—
79C22×C53C22×D411yescharacteristic
80C22×C53H(5,𝔽2)12yes—
81C22×C532−1+412yes—
82C22×C532−1+412yes—
83C4×C53—22no—
84C4×C53—22no—
85C4×C53—22no—
86C4×C53—41no—
87C4×C53—41no—
88C4×C53—41no—
89Dic53—44no—
90Dic53—44no—
91#2821—531nomaximal, Sylow
92C2×C53—81no—
93C2×C53#53511yescharacteristic
94C2×C53#53611yescharacteristic
95C2×C53#53611yescharacteristic
96C2×C53#56314yes—
97C2×C4⋊Q8—531no—
98C2×#126D5311yescharacteristic
99#400—532no—
100#399—532no—
101#398—532no—
102#397—532no—
103#395—532no—
104#393—532no—
105#383—531no—
106C53#282111yescharacteristic, complemented, Sylow
107C22×Q8—532no—
108C4⋊Q8—1064no—
109#126—24no—
110C2×C4⋊C4—532no—
111C2×C4⋊C4—532no—
112C2×C4⋊C4—534no—
113C2×C4⋊C4C2×D5311yescharacteristic
114C2×C4⋊C4C2×D5311yescharacteristic
115C2×C4⋊C4C2×D5311yescharacteristic
116C2×C22⋊C4—534no—
117C2×C22⋊C4C2×D5311yescharacteristic
118C2×C22⋊C4C2×D5311yescharacteristic
119C2×C22⋊C4C2×D5311yescharacteristic
120C2×C42C2×D5311yescharacteristic
121#95—532no—
122#95—532no—
123#95—532no—
124#95—534no—
125#95—534no—
126C24C22×D5311yescharacteristic
127C2×Q8—1064no—
128C2×Q8—2124no—
129C22×C4—534no—
130C22×C4—534no—
131C22×C4C22×D5311yescharacteristic
132C22×C4C22×D5311yescharacteristic
133C22×C4C22×D5311yescharacteristic
134C22×C4C22×D5311yescharacteristic
135C22×C4C22×D5311yescharacteristic
136C22×C4C22×D5311yescharacteristic
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—1064no—
141C4⋊C4—1064no—
142C4⋊C4—2124no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—2124no—
147C42C22⋊D5314yes—
148C23—21no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23C23×D5311yescharacteristic
153Q8—2128no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—1064no—
164C2×C4—1064no—
165C2×C4—1064no—
166C2×C4—1064no—
167C2×C4—1064no—
168C2×C4—1064no—
169C2×C4C2×C22⋊D5312yes—
170C2×C4C2×C22⋊D5312yes—
171C2×C4C2×C22⋊D5312yes—
172C22—44no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22C22×C22⋊D5311yescharacteristic
177C22#1313326312yes—
178C22#1313326412yes—
179C22#1313326412yes—
180C4—22no—
181C4—22no—
182C4—22no—
183C4—41no—
184C4—41no—
185C4—41no—
186C4—2124no—
187C4—2124no—
188C2—81no—
189C2#1681118711yescharacteristic
190C2#1681119411yescharacteristic
191C2#1681119411yescharacteristic
192C2#1681137514yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#126×C53C2
hypercenterC2×#126D53
hypocenterC53#2821
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#13133217C2semidirect
2. #16810562C2semidirect
4. #16810553C2semidirect
6. #16810543C2semidirect
8. #16810491C2semidirect
106. C53#2821semidirect