Finite Group Databasebeta

← #20282774

Subgroups — #20282774

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×#13133211C211yesmaximal, characteristic
2#16810561C212yesmaximal, complemented
3#16810558C212yesmaximal
4#16810554C212yesmaximal, complemented
5#16810549C212yesmaximal
6#16810542C212yesmaximal, complemented
7#16810534C212yesmaximal
8#16810491C211yesmaximal, characteristic, complemented
9C2×C4⋊Q8×C53C211yesmaximal, characteristic, complemented
10#13133211—24no—
11C2×#11719919C2211yescharacteristic
12C2×#11719919C2211yescharacteristic
13C2×#11719919C2211yescharacteristic
14C2×#11719919C2214yes—
15C2×#11719921C2211yescharacteristic
16C2×#11719921C2211yescharacteristic
17C2×#11719921C2211yescharacteristic
18C2×#11719921C2214yes—
19#13133124C2212yes—
20#13133124C2214yes—
21#13133124C2214yes—
22#13133124C2212yes—
23#13133124C2212yes—
24C22×Q8×C53C2212yes—
25C4⋊Q8×C53—24no—
26C2×C4⋊C4×C53C2212yes—
27C2×C4⋊C4×C53C2212yes—
28C2×C42×C53C2211yescharacteristic
29C23×D53C2311yescharacteristic
30C2×Q8×C53—24no—
31C2×Q8×C53—44no—
32C22×C4×C53C2314yes—
33C22×C4×C53C2311yescharacteristic
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×Dic53C2314yes—
37C22×Dic53C2311yescharacteristic
38C22×Dic53C2311yescharacteristic
39C22×Dic53C2311yescharacteristic
40C4⋊C4×C53—24no—
41C4⋊C4×C53—24no—
42C42×C53D414yes—
43#11719921—22no—
44#11719921—22no—
45#11719921—22no—
46#11719921—44no—
47#11719919—22no—
48#11719919—22no—
49#11719919—22no—
50#11719919—44no—
51C23×C53C2411yescharacteristic
52C22×D53—21no—
53C22×D53—22no—
54C22×D53—22no—
55C22×D53—22no—
56Q8×C53—48no—
57C2×C4×C53—22no—
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53—24no—
61C2×C4×C53—24no—
62C2×C4×C53—24no—
63C2×C4×C53C2×D412yes—
64C2×C4×C53C2×D412yes—
65C2×C4×C53C2×D412yes—
66C2×Dic53—21no—
67C2×Dic53—21no—
68C2×Dic53—21no—
69C2×Dic53—22no—
70C2×Dic53—24no—
71C2×Dic53—22no—
72C2×Dic53—22no—
73C2×Dic53—24no—
74C2×Dic53—24no—
75C22×C53C22×D411yescharacteristic
76C22×C53H(5,𝔽2)12yes—
77C22×C532−1+412yes—
78C22×C532−1+412yes—
79C2×D53—41no—
80C2×D53—41no—
81C2×D53—41no—
82C2×D53—44no—
83C4×C53—22no—
84C4×C53—22no—
85C4×C53—22no—
86C4×C53—44no—
87Dic53—41no—
88Dic53—41no—
89Dic53—41no—
90Dic53—44no—
91#2821—531nomaximal, Sylow
92C2×C53#53511yescharacteristic
93C2×C53#53611yescharacteristic
94C2×C53#53611yescharacteristic
95C2×C53#56314yes—
96D53—81no—
97C2×C4⋊Q8D5311yescharacteristic, complemented
98C2×#126—531no—
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×Q8C2×D5312yes—
108C4⋊Q8—24no—
109#126—1064no—
110C2×C4⋊C4—531no—
111C2×C4⋊C4—531no—
112C2×C4⋊C4—531no—
113C2×C4⋊C4—534no—
114C2×C4⋊C4C2×D5312yes—
115C2×C4⋊C4C2×D5312yes—
116C2×C22⋊C4—531no—
117C2×C22⋊C4—531no—
118C2×C22⋊C4—531no—
119C2×C22⋊C4—534no—
120C2×C42C2×D5311yescharacteristic
121#95—532no—
122#95—532no—
123#95—532no—
124#95—534no—
125#95—534no—
126C24—531no—
127C2×Q8—24no—
128C2×Q8—44no—
129C22×C4—534no—
130C22×C4—531no—
131C22×C4—531no—
132C22×C4—531no—
133C22×C4C22×D5314yes—
134C22×C4C22×D5311yescharacteristic
135C22×C4C22×D5311yescharacteristic
136C22×C4C22×D5311yescharacteristic
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—1062no—
140C4⋊C4—1062no—
141C4⋊C4—1062no—
142C4⋊C4—2124no—
143C22⋊C4—1062no—
144C22⋊C4—1062no—
145C22⋊C4—1062no—
146C22⋊C4—2124no—
147C42C22⋊D5314yes—
148C23—1061no—
149C23—1062no—
150C23—1062no—
151C23—1062no—
152C23C23×D5311yescharacteristic
153Q8—48no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—1061no—
161C2×C4—1061no—
162C2×C4—1061no—
163C2×C4—1064no—
164C2×C4—1062no—
165C2×C4—1064no—
166C2×C4—1064no—
167C2×C4—1062no—
168C2×C4—1062no—
169C2×C4C2×C22⋊D5312yes—
170C2×C4C2×C22⋊D5312yes—
171C2×C4C2×C22⋊D5312yes—
172C22—2124no—
173C22—2121no—
174C22—2121no—
175C22—2121no—
176C22C22×C22⋊D5311yescharacteristic
177C22#1313326212yes—
178C22#1313326512yes—
179C22#1313326512yes—
180C4—22no—
181C4—22no—
182C4—22no—
183C4—44no—
184C4—2124no—
185C4—2121no—
186C4—2121no—
187C4—2121no—
188C2—4241no—
189C2#1681119011yescharacteristic
190C2#1681119711yescharacteristic
191C2#1681119711yescharacteristic
192C2#1681137214yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C4⋊Q8×C53C2
hypercenterC2×C4⋊Q8D53
hypocenterC53#2821
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16810561C2semidirect
4. #16810554C2semidirect
6. #16810542C2semidirect
8. #16810491C2semidirect
9. C2×C4⋊Q8×C53C2semidirect
97. C2×C4⋊Q8D53semidirect
106. C53#2821semidirect