Finite Group Databasebeta

← #20280386

Subgroups — #20280386

Order 6784.

540 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 137 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×#13133173C212yesmaximal, complemented
2#16810553C211yesmaximal, characteristic, complemented
3#16810546C212yesmaximal, complemented
4#16810516C212yesmaximal, complemented
5#16810505C212yesmaximal, complemented
6#16810503C212yesmaximal
7#16810495C212yesmaximal, complemented
8#16810488C211yesmaximal, characteristic, complemented
9C53×#394C211yesmaximal, characteristic
10C23×Dic53C2212yes—
11#13133173—28no—
12C2×C4⋊Dic53C2211yescharacteristic
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2212yes—
20C2×C4×Dic53C2212yes—
21C2×C4×Dic53C2211yescharacteristic
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2211yescharacteristic
27C22×D4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2212yes—
30C2×C22⋊C4×C53C2212yes—
31#95×C53C2211yescharacteristic
32C24×C53C2312yes—
33C2×D4×C53—24no—
34C2×D4×C53—41no—
35C22×C4×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×Dic53—24no—
40C22×Dic53—24no—
41C22×Dic53—24no—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C4⋊C4×C53—41no—
47C22⋊C4×C53—24no—
48C22⋊C4×C53—24no—
49C22⋊Dic53—24no—
50C22⋊Dic53—24no—
51C22⋊Dic53—42no—
52C4⋊Dic53—41no—
53C4⋊Dic53—41no—
54#11719919—24no—
55#11719919—24no—
56#11719919—24no—
57C4×Dic53—24no—
58C4×Dic53D414yescomplemented
59C23×C53—22no—
60C23×C53—22no—
61C23×C53—22no—
62C23×C53—22no—
63C23×C53—22no—
64C23×C53—22no—
65C23×C53#4114yes—
66C23×C53C2411yescharacteristic
67D4×C53—44no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53C2×D412yes—
78C2×Dic53—216no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53C2×D414yes—
90C2×Dic53#4114yes—
91C22×C53—24no—
92C22×C53—24no—
93C22×C53—24no—
94C22×C53—42no—
95C22×C53—42no—
96C22×C53—42no—
97C22×C53—42no—
98C22×C53C22×D411yescharacteristic
99C22×C53C2×#4112yes—
100C22×C53C2×#4112yes—
101C22×C53H(5,𝔽2)11yescharacteristic
102C22×C532−1+411yescharacteristic
103C4×C53—22no—
104C4×C53—42no—
105C4×C53—41no—
106C4×C53—41no—
107Dic53—24no—
108Dic53—24no—
109Dic53—42no—
110Dic53—42no—
111#2609—531nomaximal, Sylow
112C2×C53—44no—
113C2×C53#52512yes—
114C2×C53#53711yescharacteristic
115C2×C53#55111yescharacteristic
116C2×C53#55212yes—
117C2×C53#55511yescharacteristic
118C2×#117—532no—
119#398—531no—
120#396—532no—
121#394D5311yescharacteristic
122#388—532no—
123#387—532no—
124#386—532no—
125#385—532no—
126#382—531no—
127C53#260911yescharacteristic, complemented, Sylow
128C22×D4C2×D5311yescharacteristic
129C23×C4—532no—
130#117—1068no—
131C2×C4⋊C4—532no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—531no—
135C2×C4⋊C4—531no—
136C2×C4⋊C4C2×D5311yescharacteristic
137C2×C22⋊C4—532no—
138C2×C22⋊C4—532no—
139C2×C22⋊C4—532no—
140C2×C22⋊C4C2×D5312yes—
141C2×C22⋊C4C2×D5312yes—
142C2×C42—531no—
143C2×C42—532no—
144#95—532no—
145#95—532no—
146#95—531no—
147#95—532no—
148#95—532no—
149#95C2×D5311yescharacteristic
150C24C22×D5312yes—
151C2×D4—24no—
152C2×D4—41no—
153C22×C4—532no—
154C22×C4—532no—
155C22×C4—532no—
156C22×C4—532no—
157C22×C4—1064no—
158C22×C4—1064no—
159C22×C4—1064no—
160C22×C4C22×D5311yescharacteristic
161C22×C4C22×D5311yescharacteristic
162C22×C4C22×D5312yes—
163C22×C4C22×D5311yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—1064no—
166C4⋊C4—1064no—
167C4⋊C4—1064no—
168C4⋊C4—2121no—
169C4⋊C4—2121no—
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—1064no—
173C22⋊C4—1064no—
174C22⋊C4—2122no—
175C42—534no—
176C42—1064no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23#1171994714yes—
184C23C23×D5311yescharacteristic
185D4—44no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—534no—
196C2×C4—534no—
197C2×C4—10616no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4D4×D5312yes—
209C22—24no—
210C22—24no—
211C22—24no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22C2×D4×D5311yescharacteristic
217C22#41×D5312yes—
218C22C2×#1171994712yes—
219C22#1313326211yescharacteristic
220C22#1313326411yescharacteristic
221C4—22no—
222C4—41no—
223C4—41no—
224C4—42no—
225C4—1064no—
226C4—1064no—
227C4—2122no—
228C4—2122no—
229C2—44no—
230C2#1681113312yes—
231C2#1681120011yescharacteristic
232C2#1681130411yescharacteristic
233C2#1681131012yes—
234C2#1681133511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2609
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#13133173C2semidirect
2. #16810553C2semidirect
3. #16810546C2semidirect
4. #16810516C2semidirect
5. #16810505C2semidirect
7. #16810495C2semidirect
8. #16810488C2semidirect
58. C4×Dic53D4semidirect
127. C53#2609semidirect