Finite Group Databasebeta

← #20281444

Subgroups — #20281444

Order 6784.

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

normal subgroups only

This group has 612 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×#13133213C211yesmaximal, characteristic
2#16810540C212yesmaximal, complemented
3#16810538C212yesmaximal, complemented
4#16810538C212yesmaximal, complemented
5#16810504C212yesmaximal, complemented
6#16810503C211yesmaximal, characteristic, complemented
7#16810502C212yesmaximal, complemented
8#16810500C212yesmaximal, complemented
9C2×#123×C53C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C22×C22⋊D53C2212yes—
13C23×Dic53C2211yescharacteristic
14#13133213—24no—
15C2×#11719919C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2211yescharacteristic
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×C4×Dic53C2211yescharacteristic
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×C53C2212yes—
32C2×C22⋊C4×C53C2211yescharacteristic
33C24×C53C2311yescharacteristic
34C24×C53C2311yescharacteristic
35C23×D53C2312yes—
36C2×D4×C53—41no—
37C2×D4×C53—41no—
38C2×D4×C53D414yes—
39C22×C4×C53—22no—
40C22×C4×C53—41no—
41C22×C4×C53—41no—
42C22×C4×C53C2312yes—
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C2×C22⋊D53—24no—
46C2×C22⋊D53—24no—
47C2×C22⋊D53—24no—
48C2×C22⋊D53—24no—
49C2×C22⋊D53—42no—
50C2×C22⋊D53—42no—
51C22×Dic53—41no—
52C22×Dic53—42no—
53C22×Dic53C2312yes—
54C22×Dic53C2312yes—
55C22×Dic53C2312yes—
56C2×D212—22no—
57C2×D212—41no—
58C2×D212—41no—
59C4⋊C4×C53—24no—
60C22⋊C4×C53—24no—
61C22⋊C4×C53—22no—
62C22⋊Dic53—42no—
63C22⋊Dic53—41no—
64#11719921—42no—
65#11719921—42no—
66#11719921—42no—
67#11719921—42no—
68#11719919—24no—
69C4×Dic53—22no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53—21no—
74C23×C53—22no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—21no—
78C23×C53C2×D412yes—
79C23×C53C2×D412yes—
80C23×C53C2411yescharacteristic
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—22no—
88D4×C53—44no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—44no—
98C2×C4×C53C2×D412yes—
99C22⋊D53—48no—
100C22⋊D53—48no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—44no—
110C2×Dic53#4114yes—
111D212—44no—
112C22×C53—22no—
113C22×C53—22no—
114C22×C53—24no—
115C22×C53—24no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53—41no—
119C22×C53—41no—
120C22×C53—41no—
121C22×C53—41no—
122C22×C53C22×D411yescharacteristic
123C22×C53C2×#4112yes—
124C22×C53H(5,𝔽2)11yescharacteristic
125C22×C53H(5,𝔽2)11yescharacteristic
126C22×C53H(5,𝔽2)12yes—
127C2×D53—42no—
128C2×D53—42no—
129C2×D53—42no—
130C2×D53—42no—
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—42no—
134C4×C53—22no—
135C4×C53—42no—
136C4×C53—42no—
137Dic53—24no—
138Dic53—42no—
139Dic53—42no—
140#2709—531nomaximal, Sylow
141C2×C53—42no—
142C2×C53—42no—
143C2×C53#53411yescharacteristic
144C2×C53#54012yes—
145C2×C53#54612yes—
146C2×C53#55911yescharacteristic
147C2×C53#56011yescharacteristic
148D53—82no—
149C2×#127—531no—
150C2×#123D5311yescharacteristic, complemented
151#394—532no—
152#394—532no—
153#394—532no—
154#387—532no—
155#386—531no—
156#386—532no—
157#386—532no—
158C53#270911yescharacteristic, complemented, Sylow
159C22×D4—532no—
160C22×D4—532no—
161C22×D4—531no—
162C22×D4C2×D5311yescharacteristic
163C23×C4—531no—
164C23×C4C2×D5311yescharacteristic
165#127—1064no—
166#123—24no—
167C2×C4⋊C4—532no—
168C2×C4⋊C4C2×D5312yes—
169C2×C22⋊C4—532no—
170C2×C22⋊C4—531no—
171C2×C22⋊C4—532no—
172C2×C22⋊C4—532no—
173C2×C22⋊C4—532no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4C2×D5312yes—
176C2×C22⋊C4C2×D5311yescharacteristic
177C2×C42—531no—
178#95—532no—
179#95—532no—
180#95—532no—
181#95—532no—
182C24—532no—
183C24C22×D5311yescharacteristic
184C24C22×D5311yescharacteristic
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—1064no—
188C2×D4—1064no—
189C2×D4—1064no—
190C2×D4—1064no—
191C2×D4—1062no—
192C2×D4—2121no—
193C2×D4—2121no—
194C2×D4—2122no—
195C2×D4—2122no—
196C2×D4C22⋊D5314yes—
197C22×C4—22no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—532no—
201C22×C4—532no—
202C22×C4—532no—
203C22×C4—2121no—
204C22×C4—2122no—
205C22×C4C22×D5312yes—
206C22×C4C22×D5312yes—
207C22×C4C22×D5311yescharacteristic
208C4⋊C4—24no—
209C4⋊C4—1064no—
210C22⋊C4—22no—
211C22⋊C4—24no—
212C22⋊C4—2122no—
213C22⋊C4—2122no—
214C22⋊C4—2122no—
215C22⋊C4—2122no—
216C22⋊C4—2121no—
217C22⋊C4—2122no—
218C42—1062no—
219C23—22no—
220C23—22no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—1062no—
228C23—1062no—
229C23—1062no—
230C23—1062no—
231C23—1062no—
232C23—1062no—
233C23—1062no—
234C23C2×C22⋊D5312yes—
235C23C2×C22⋊D5312yes—
236C23C23×D5311yescharacteristic
237D4—44no—
238D4—2124no—
239D4—2128no—
240D4—2128no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—44no—
250C2×C4—534no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—2124no—
260C2×C4C2×C22⋊D5312yes—
261C22—22no—
262C22—22no—
263C22—24no—
264C22—24no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—42no—
270C22—42no—
271C22—2122no—
272C22—2122no—
273C22—2122no—
274C22—2122no—
275C22—2122no—
276C22—2122no—
277C22—2122no—
278C22C22×C22⋊D5311yescharacteristic
279C22#41×D5312yes—
280C22#1313326211yescharacteristic
281C22#1313326211yescharacteristic
282C22#1313326312yes—
283C4—22no—
284C4—42no—
285C4—42no—
286C4—1064no—
287C4—2122no—
288C4—2122no—
289C2—42no—
290C2—42no—
291C2—4242no—
292C2#1681118211yescharacteristic
293C2#1681122512yes—
294C2#1681126112yes—
295C2#1681135711yescharacteristic
296C2#1681136311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#123×C53C2
hypercenterC2×#123D53
hypocenterC53#2709
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. #16810540C2semidirect
3. #16810538C2semidirect
4. #16810538C2semidirect
5. #16810504C2semidirect
6. #16810503C2semidirect
7. #16810502C2semidirect
8. #16810500C2semidirect
9. C2×#123×C53C2semidirect
150. C2×#123D53semidirect
158. C53#2709semidirect