Finite Group Databasebeta

← #20282981

Subgroups — #20282981

Order 6784.

348 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#16810564C212yesmaximal
2#16810562C211yesmaximal, characteristic
3#16810557C212yesmaximal
4#16810556C212yesmaximal
5#16810554C212yesmaximal
6#16810542C212yesmaximal
7#16810492C211yesmaximal, characteristic
8#16810490C212yesmaximal
9C53×#395C211yesmaximal, characteristic
10C2×C4⋊Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12C2×#11719919C2212yes—
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×C4×Dic53C2211yescharacteristic
18#13133124C2212yes—
19#13133124C2212yes—
20#13133124C2212yes—
21#13133124C2211yescharacteristic
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26C22×Q8×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2212yes—
28C2×C4⋊C4×C53C2211yescharacteristic
29#95×C53C2212yes—
30#95×C53C2211yescharacteristic
31C2×Q8×C53—41no—
32C2×Q8×C53—42no—
33C22×C4×C53C2312yes—
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C4⋊C4×C53—41no—
43C4⋊C4×C53—42no—
44C4⋊Dic53—41no—
45C4⋊Dic53—41no—
46#11719919—42no—
47#11719919—42no—
48#11719919—42no—
49#11719919—42no—
50#11719919—42no—
51C4×Dic53Q814yescomplemented
52C23×C53C2411yescharacteristic
53Q8×C53—44no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—22no—
57C2×C4×C53—22no—
58C2×C4×C53—22no—
59C2×C4×C53—21no—
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—21no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53C2×Q812yes—
67C2×Dic53—22no—
68C2×Dic53—22no—
69C2×Dic53—22no—
70C2×Dic53—22no—
71C2×Dic53—22no—
72C2×Dic53—22no—
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—24no—
77C2×Dic53C2×Q814yes—
78C22×C53C22×Q811yescharacteristic
79C22×C53H(5,𝔽2)11yescharacteristic
80C22×C53H(5,𝔽2)12yes—
81C22×C532−1+411yescharacteristic
82C22×C532−1+412yes—
83C4×C53—22no—
84C4×C53—41no—
85C4×C53—41no—
86C4×C53—42no—
87C4×C53—42no—
88Dic53—24no—
89Dic53—42no—
90Dic53—42no—
91Dic53—42no—
92#2843—531nomaximal, Sylow
93C2×C53#54412yes—
94C2×C53#54411yescharacteristic
95C2×C53#56011yescharacteristic
96C2×C53#56311yescharacteristic
97C2×C53#56312yes—
98#400—531no—
99#400—532no—
100#398—532no—
101#398—532no—
102#398—532no—
103#395—532no—
104#395D5311yescharacteristic
105#384—531no—
106#383—532no—
107C53#284311yescharacteristic, complemented, Sylow
108C22×Q8C2×D5311yescharacteristic
109C2×C4⋊C4—532no—
110C2×C4⋊C4—532no—
111C2×C4⋊C4—532no—
112C2×C4⋊C4—532no—
113C2×C4⋊C4—532no—
114C2×C4⋊C4—531no—
115C2×C4⋊C4—531no—
116C2×C4⋊C4C2×D5311yescharacteristic
117C2×C4⋊C4C2×D5312yes—
118C2×C42—531no—
119#95—532no—
120#95—532no—
121#95—532no—
122#95—532no—
123#95—532no—
124#95—532no—
125#95—531no—
126#95—532no—
127#95C2×D5312yes—
128#95C2×D5311yescharacteristic
129C2×Q8—41no—
130C2×Q8—42no—
131C22×C4—532no—
132C22×C4—532no—
133C22×C4—532no—
134C22×C4—532no—
135C22×C4C22×D5312yes—
136C22×C4C22×D5312yes—
137C22×C4C22×D5311yescharacteristic
138C22×C4C22×D5311yescharacteristic
139C22×C4C22×D5311yescharacteristic
140C4⋊C4—41no—
141C4⋊C4—42no—
142C4⋊C4—2122no—
143C4⋊C4—2122no—
144C4⋊C4—2122no—
145C4⋊C4—2122no—
146C4⋊C4—2122no—
147C4⋊C4—2121no—
148C4⋊C4—2121no—
149C42—534no—
150C23C23×D5311yescharacteristic
151Q8—44no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—534no—
165C2×C4—1062no—
166C2×C4—1062no—
167C2×C4—1062no—
168C2×C4—1062no—
169C2×C4—1062no—
170C2×C4—1062no—
171C2×C4—1062no—
172C2×C4—1062no—
173C2×C4—1064no—
174C2×C4—1062no—
175C2×C4Q8×D5312yes—
176C22C2×Q8×D5311yescharacteristic
177C22#1313326211yescharacteristic
178C22#1313326212yes—
179C22#1313326411yescharacteristic
180C22#1313326512yes—
181C4—22no—
182C4—41no—
183C4—41no—
184C4—42no—
185C4—42no—
186C4—1064no—
187C4—2122no—
188C4—2122no—
189C4—2122no—
190C2#1681125012yes—
191C2#1681125211yescharacteristic
192C2#1681136211yescharacteristic
193C2#1681137211yescharacteristic
194C2#1681137412yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2843
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
51. C4×Dic53Q8semidirect
107. C53#2843semidirect