Finite Group Databasebeta

← #20282211

Subgroups — #20282211

Order 6784.

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

normal subgroups only

This group has 344 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#16810562C212yesmaximal
2#16810562C211yesmaximal, characteristic
3#16810555C212yesmaximal
4#16810522C212yesmaximal
5#16810508C212yesmaximal
6#16810504C212yesmaximal
7#16810490C212yesmaximal
8#16810488C211yesmaximal, characteristic
9C53×#398C211yesmaximal, characteristic
10C2×C4⋊Dic53C2212yes—
11C2×C4⋊Dic53C2211yescharacteristic
12C2×C4⋊Dic53C2212yes—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919C2212yes—
15C2×C4×Dic53C2211yescharacteristic
16C2×C4×Dic53C2212yes—
17#13133124C2212yes—
18#13133124C2212yes—
19#13133124C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26C2×C4⋊C4×C53C2212yes—
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29#95×C53C2211yescharacteristic
30#95×C53C2212yes—
31C22×C4×C53C2311yescharacteristic
32C22×C4×C53C2311yescharacteristic
33C22×C4×C53C2312yes—
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×Dic53C2312yes—
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C4⋊C4×C53—24no—
41C4⋊C4×C53—41no—
42C4⋊C4×C53—41no—
43C4⋊Dic53—24no—
44C4⋊Dic53—41no—
45C4⋊Dic53—42no—
46C4⋊Dic53—41no—
47#11719919—24no—
48C4×Dic53—22no—
49C4×Dic53—24no—
50C23×C53C2411yescharacteristic
51C2×C4×C53—22no—
52C2×C4×C53—21no—
53C2×C4×C53—21no—
54C2×C4×C53—22no—
55C2×C4×C53—22no—
56C2×C4×C53—22no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53#4112yes—
63C2×C4×C53#4114yes—
64C2×Dic53—22no—
65C2×Dic53—22no—
66C2×Dic53—22no—
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—
76C22×C53C2×#4111yescharacteristic
77C22×C53C2×#4112yes—
78C22×C53H(5,𝔽2)11yescharacteristic
79C22×C532−1+411yescharacteristic
80C22×C532−1+412yes—
81C4×C53—22no—
82C4×C53—24no—
83C4×C53—42no—
84C4×C53—41no—
85C4×C53—41no—
86Dic53—42no—
87Dic53—42no—
88Dic53—42no—
89Dic53—42no—
90#2769—531nomaximal, Sylow
91C2×C53#53911yescharacteristic
92C2×C53#54112yes—
93C2×C53#55212yes—
94C2×C53#55511yescharacteristic
95C2×C53#56311yescharacteristic
96#400—532no—
97#400—531no—
98#398—532no—
99#398D5311yescharacteristic
100#389—532no—
101#387—532no—
102#387—532no—
103#383—532no—
104#382—531no—
105C53#276911yescharacteristic, complemented, Sylow
106C2×C4⋊C4—532no—
107C2×C4⋊C4—532no—
108C2×C4⋊C4—531no—
109C2×C4⋊C4—532no—
110C2×C4⋊C4—531no—
111C2×C4⋊C4C2×D5312yes—
112C2×C4⋊C4C2×D5311yescharacteristic
113C2×C4⋊C4C2×D5311yescharacteristic
114C2×C42—531no—
115C2×C42—532no—
116#95—532no—
117#95—532no—
118#95—531no—
119#95—532no—
120#95—532no—
121#95—532no—
122#95—532no—
123#95—532no—
124#95—532no—
125#95C2×D5312yes—
126#95C2×D5311yescharacteristic
127C22×C4—532no—
128C22×C4—532no—
129C22×C4—532no—
130C22×C4—532no—
131C22×C4C22×D5311yescharacteristic
132C22×C4C22×D5311yescharacteristic
133C22×C4C22×D5312yes—
134C22×C4C22×D5311yescharacteristic
135C22×C4C22×D5312yes—
136C4⋊C4—24no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—1064no—
140C4⋊C4—1064no—
141C4⋊C4—2121no—
142C4⋊C4—2122no—
143C4⋊C4—2121no—
144C42—1062no—
145C42—1064no—
146C23C23×D5311yescharacteristic
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—1062no—
159C2×C4—1062no—
160C2×C4—1062no—
161C2×C4—1062no—
162C2×C4—1062no—
163C2×C4—1062no—
164C2×C4—1062no—
165C2×C4—1062no—
166C2×C4—1062no—
167C2×C4—1062no—
168C2×C4—1062no—
169C2×C4—1062no—
170C2×C4#1171994712yes—
171C2×C4#1171994914yes—
172C22C2×#1171994711yescharacteristic
173C22C2×#1171994912yes—
174C22#1313326211yescharacteristic
175C22#1313326412yes—
176C22#1313326411yescharacteristic
177C4—24no—
178C4—22no—
179C4—41no—
180C4—41no—
181C4—42no—
182C4—2122no—
183C4—2122no—
184C4—2122no—
185C4—2122no—
186C2#1681121811yescharacteristic
187C2#1681123612yes—
188C2#1681131512yes—
189C2#1681133911yescharacteristic
190C2#1681137611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2769
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
105. C53#2769semidirect