Finite Group Databasebeta

← #20280423

Subgroups — #20280423

Order 6784.

388 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 141 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#16810562C211yesmaximal, characteristic
2#16810556C212yesmaximal
3#16810543C211yesmaximal, characteristic
4#16810527C212yesmaximal
5#16810520C212yesmaximal
6#16810510C212yesmaximal
7#16810506C212yesmaximal
8#16810469C212yesmaximal
9C53×#382C211yesmaximal, characteristic
10C22×Dic106C2211yescharacteristic
11C2×C4⋊Dic53C2212yes—
12C2×C4⋊Dic53C2211yescharacteristic
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×C4×Dic53C2212yes—
17C2×C4×Dic53C2212yes—
18#13133124C2211yescharacteristic
19#13133124C2212yes—
20#13133124C2212yes—
21#13133124C2211yescharacteristic
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26C2×C4⋊C4×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C42×C53C2211yescharacteristic
29#95×C53C2212yes—
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—
40C2×Dic106—24no—
41C2×Dic106—41no—
42C4⋊C4×C53—41no—
43C4⋊C4×C53—41no—
44C42×C53Q814yescomplemented
45C4⋊Dic53—24no—
46C4⋊Dic53—41no—
47#11719919—24no—
48#11719919—24no—
49#11719919—42no—
50C4×Dic53—24no—
51C4×Dic53—24no—
52C23×C53C2411yescharacteristic
53C2×C4×C53—22no—
54C2×C4×C53—22no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—22no—
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—21no—
62C2×C4×C53—22no—
63C2×C4×C53—21no—
64C2×C4×C53C2×Q814yes—
65C2×C4×C53C2×Q812yes—
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—
76C2×Dic53#4114yes—
77C2×Dic53#4114yes—
78Dic106—44no—
79C22×C53C22×Q811yescharacteristic
80C22×C53C2×#4112yes—
81C22×C53C2×#4112yes—
82C22×C53H(5,𝔽2)11yescharacteristic
83C22×C532−1+411yescharacteristic
84C4×C53—24no—
85C4×C53—22no—
86C4×C53—41no—
87C4×C53—41no—
88C4×C53—42no—
89Dic53—24no—
90Dic53—24no—
91Dic53—42no—
92Dic53—42no—
93#2612—531nomaximal, Sylow
94C2×C53#53312yes—
95C2×C53#54411yescharacteristic
96C2×C53#55111yescharacteristic
97C2×C53#55212yes—
98C2×C53#55511yescharacteristic
99#400—531no—
100#398—532no—
101#395—531no—
102#391—532no—
103#389—532no—
104#387—532no—
105#387—532no—
106#382D5311yescharacteristic
107#376—532no—
108C53#261211yescharacteristic, complemented, Sylow
109C22×Q8—531no—
110C2×C4⋊C4—532no—
111C2×C4⋊C4—532no—
112C2×C4⋊C4—532no—
113C2×C4⋊C4—532no—
114C2×C4⋊C4—531no—
115C2×C4⋊C4C2×D5311yescharacteristic
116C2×C4⋊C4C2×D5311yescharacteristic
117C2×C42—532no—
118C2×C42—532no—
119C2×C42C2×D5311yescharacteristic
120#95—532no—
121#95—532no—
122#95—532no—
123#95—532no—
124#95—531no—
125#95—531no—
126#95—532no—
127#95—532no—
128#95C2×D5312yes—
129#95C2×D5312yes—
130C2×Q8—1064no—
131C2×Q8—2121no—
132C22×C4—532no—
133C22×C4—532no—
134C22×C4—532no—
135C22×C4—532no—
136C22×C4C22×D5311yescharacteristic
137C22×C4C22×D5311yescharacteristic
138C22×C4C22×D5312yes—
139C22×C4C22×D5311yescharacteristic
140C22×C4C22×D5312yes—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—1064no—
144C4⋊C4—1064no—
145C4⋊C4—1064no—
146C4⋊C4—2122no—
147C4⋊C4—2121no—
148C42—1064no—
149C42—1064no—
150C42Dic10614yescomplemented
151C23C23×D5311yescharacteristic
152Q8—2124no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—534no—
165C2×C4—534no—
166C2×C4—1062no—
167C2×C4—1062no—
168C2×C4—1062no—
169C2×C4—1062no—
170C2×C4—1062no—
171C2×C4—1062no—
172C2×C4—1062no—
173C2×C4—1062no—
174C2×C4—1062no—
175C2×C4—1062no—
176C2×C4C2×Dic10614yes—
177C2×C4C2×Dic10612yes—
178C22C22×Dic10611yescharacteristic
179C22#41×D5312yes—
180C22#41×D5312yes—
181C22#1313326211yescharacteristic
182C22#1313326511yescharacteristic
183C4—24no—
184C4—22no—
185C4—41no—
186C4—41no—
187C4—42no—
188C4—1064no—
189C4—1064no—
190C4—2122no—
191C4—2122no—
192C2#1681117612yes—
193C2#1681125411yescharacteristic
194C2#1681130311yescharacteristic
195C2#1681130512yes—
196C2#1681133511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#382C2
hypercenter#382D53
hypocenterC53#2612
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
44. C42×C53Q8semidirect
108. C53#2612semidirect
150. C42Dic106semidirect