Finite Group Databasebeta

← #20281643

Subgroups — #20281643

Order 6784.

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

normal subgroups only

This group has 504 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×#13133198C211yesmaximal, characteristic
2#16810556C212yesmaximal, complemented
3#16810551C212yesmaximal, complemented
4#16810551C212yesmaximal, complemented
5#16810526C211yesmaximal, characteristic
6#16810518C212yesmaximal, complemented
7#16810496C212yesmaximal, complemented
8#16810495C212yesmaximal, complemented
9C2×#123×C53C211yesmaximal, characteristic
10C23×Dic53C2211yescharacteristic
11#13133198—24no—
12C2×C4⋊Dic53C2211yescharacteristic
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2212yes—
20C2×C22⋊Dic53C2211yescharacteristic
21C2×C4×Dic53C2211yescharacteristic
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27C22×D4×C53C2211yescharacteristic
28C23×C4×C53C2211yescharacteristic
29#123×C53—24no—
30C2×C4⋊C4×C53C2212yes—
31C2×C22⋊C4×C53C2211yescharacteristic
32C2×C22⋊C4×C53C2212yes—
33C24×C53C2311yescharacteristic
34C24×C53C2311yescharacteristic
35C2×D4×C53—41no—
36C2×D4×C53—41no—
37C2×D4×C53D414yes—
38C22×C4×C53—22no—
39C22×C4×C53—41no—
40C22×C4×C53—41no—
41C22×C4×C53C2312yes—
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×Dic53—22no—
45C22×Dic53—41no—
46C22×Dic53—41no—
47C22×Dic53C2312yes—
48C22×Dic53C2312yes—
49C22×Dic53C2312yes—
50C22×Dic53C2312yes—
51C4⋊C4×C53—24no—
52C22⋊C4×C53—24no—
53C22⋊C4×C53—22no—
54C22⋊Dic53—24no—
55C22⋊Dic53—22no—
56C22⋊Dic53—24no—
57C22⋊Dic53—24no—
58C4⋊Dic53—41no—
59#11719919—24no—
60#11719919—24no—
61#11719919—42no—
62#11719919—42no—
63C4×Dic53—41no—
64C23×C53—22no—
65C23×C53—22no—
66C23×C53—21no—
67C23×C53—21no—
68C23×C53—21no—
69C23×C53—21no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53C2×D412yes—
73C23×C53C2×D412yes—
74C23×C53C2411yescharacteristic
75D4×C53—44no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—44no—
85C2×C4×C53C2×D412yes—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—44no—
98C2×Dic53#4114yes—
99C22×C53—22no—
100C22×C53—22no—
101C22×C53—24no—
102C22×C53—24no—
103C22×C53—42no—
104C22×C53—42no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—41no—
108C22×C53—41no—
109C22×C53C22×D411yescharacteristic
110C22×C53C2×#4112yes—
111C22×C53H(5,𝔽2)11yescharacteristic
112C22×C53H(5,𝔽2)11yescharacteristic
113C22×C532−1+412yes—
114C4×C53—22no—
115C4×C53—42no—
116C4×C53—42no—
117Dic53—24no—
118Dic53—42no—
119Dic53—42no—
120Dic53—42no—
121#2724—531nomaximal, Sylow
122C2×C53—42no—
123C2×C53—42no—
124C2×C53#53411yescharacteristic
125C2×C53#53912yes—
126C2×C53#54712yes—
127C2×C53#55911yescharacteristic
128C2×C53#56311yescharacteristic
129C2×#123—531no—
130C2×#123D5311yescharacteristic
131#398—532no—
132#397—532no—
133#397—532no—
134#390—531no—
135#389—532no—
136#385—532no—
137#385—532no—
138C53#272411yescharacteristic, complemented, Sylow
139C22×D4C2×D5311yescharacteristic
140C23×C4—531no—
141C23×C4C2×D5311yescharacteristic
142#123—24no—
143#123—1064no—
144C2×C4⋊C4—532no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4—531no—
149C2×C4⋊C4C2×D5312yes—
150C2×C22⋊C4—532no—
151C2×C22⋊C4—531no—
152C2×C22⋊C4—532no—
153C2×C22⋊C4—532no—
154C2×C22⋊C4C2×D5312yes—
155C2×C22⋊C4C2×D5311yescharacteristic
156C2×C42—531no—
157#95—532no—
158#95—532no—
159#95—532no—
160#95—532no—
161#95—532no—
162C24C22×D5311yescharacteristic
163C24C22×D5311yescharacteristic
164C2×D4—41no—
165C2×D4—41no—
166C2×D4C22⋊D5314yes—
167C22×C4—22no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—532no—
171C22×C4—532no—
172C22×C4—532no—
173C22×C4—532no—
174C22×C4—1062no—
175C22×C4—2121no—
176C22×C4—2121no—
177C22×C4C22×D5312yes—
178C22×C4C22×D5312yes—
179C22×C4C22×D5311yescharacteristic
180C4⋊C4—24no—
181C4⋊C4—1064no—
182C4⋊C4—1064no—
183C4⋊C4—2122no—
184C4⋊C4—2122no—
185C4⋊C4—2121no—
186C22⋊C4—22no—
187C22⋊C4—24no—
188C22⋊C4—1064no—
189C22⋊C4—1064no—
190C22⋊C4—1062no—
191C22⋊C4—1064no—
192C42—2121no—
193C23—22no—
194C23—22no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23—22no—
201C23C2×C22⋊D5312yes—
202C23C2×C22⋊D5312yes—
203C23C23×D5311yescharacteristic
204D4—44no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—44no—
214C2×C4—534no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—2124no—
227C2×C4C2×C22⋊D5312yes—
228C22—24no—
229C22—24no—
230C22—22no—
231C22—22no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22—42no—
238C22C22×C22⋊D5311yescharacteristic
239C22#41×D5312yes—
240C22#1313326211yescharacteristic
241C22#1313326211yescharacteristic
242C22#1313326412yes—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—1064no—
247C4—2122no—
248C4—2122no—
249C4—2122no—
250C2—42no—
251C2—42no—
252C2#1681118211yescharacteristic
253C2#1681121612yes—
254C2#1681127212yes—
255C2#1681135711yescharacteristic
256C2#1681137611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#123×C53C2
hypercenterC2×#123D53
hypocenterC53#2724
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. #16810556C2semidirect
3. #16810551C2semidirect
4. #16810551C2semidirect
6. #16810518C2semidirect
7. #16810496C2semidirect
8. #16810495C2semidirect
138. C53#2724semidirect