Finite Group Databasebeta

← #20280657

Subgroups — #20280657

Order 6784.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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
2C2×#13133207C211yesmaximal, characteristic, complemented
3#16810541C211yesmaximal, characteristic, complemented
4#16810540C211yesmaximal, characteristic, complemented
5#16810539C212yesmaximal, complemented
6#16810538C212yesmaximal, complemented
7#16810537C211yesmaximal, characteristic, complemented
8#16810523C212yesmaximal, complemented
9#16810523C212yesmaximal, complemented
10#16810468C211yesmaximal, characteristic, complemented
11C53×#394C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53C2212yes—
14C22×C22⋊D53C2212yes—
15#13133213—24no—
16#13133207—24no—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×C22⋊Dic53C2212yes—
23C2×C22⋊Dic53C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×C4×Dic53C2212yes—
29C2×C4×Dic53C2211yescharacteristic
30#13133124C2211yescharacteristic
31#13133124C2212yes—
32C22×D4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2212yes—
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2312yes—
38C23×D53C2312yes—
39C2×D4×C53—22no—
40C2×D4×C53—42no—
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
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×Dic53C2312yes—
52C22×Dic53C2311yescharacteristic
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2311yescharacteristic
55C22×Dic53C2311yescharacteristic
56C2×D212—22no—
57C2×D212—42no—
58C4⋊C4×C53—22no—
59C22⋊C4×C53—42no—
60C22⋊C4×C53—42no—
61C22⋊Dic53—42no—
62C22⋊Dic53—42no—
63#11719921—42no—
64#11719921—42no—
65#11719921—42no—
66#11719921—42no—
67C4⋊Dic53—22no—
68#11719919—22no—
69#11719919—22no—
70#11719919—22no—
71#11719919—22no—
72C4×Dic53—24no—
73C4×Dic53D414yes—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53C2411yescharacteristic
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89D4×C53—44no—
90C2×C4×C53—22no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53C2×D412yes—
100C22⋊D53—48no—
101C22⋊D53—48no—
102C2×Dic53—21no—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—22no—
107C2×Dic53—21no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—21no—
112C2×Dic53C2×D412yes—
113C2×Dic53C2×D412yes—
114C2×Dic53#4114yes—
115D212—44no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53—42no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53—42no—
123C22×C53C22×D411yescharacteristic
124C22×C53C2×#4112yes—
125C22×C53H(5,𝔽2)12yes—
126C22×C53H(5,𝔽2)11yescharacteristic
127C22×C53H(5,𝔽2)11yescharacteristic
128C2×D53—42no—
129C2×D53—42no—
130C2×D53—42no—
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—42no—
135C4×C53—22no—
136C4×C53—41no—
137C4×C53—41no—
138C4×C53—42no—
139Dic53—22no—
140Dic53—41no—
141Dic53—41no—
142Dic53—42no—
143Dic53#12414yes—
144#2641—531nomaximal, Sylow
145C2×C53—82no—
146C2×C53C2×#12411yescharacteristic
147C2×C53#53511yescharacteristic
148C2×C53#53511yescharacteristic
149C2×C53#54012yes—
150C2×C53#54012yes—
151D53—82no—
152C2×#127—531no—
153C2×#125—531no—
154#394—531no—
155#394—531no—
156#394—532no—
157#394—532no—
158#394—531no—
159#394D5311yescharacteristic, complemented
160#390—532no—
161#390—532no—
162#376—531no—
163C53#264111yescharacteristic, complemented, Sylow
164C22×D4—532no—
165C22×D4—532no—
166C22×D4—531no—
167C22×D4C2×D5311yescharacteristic
168#127—1064no—
169#125—1064no—
170C2×C4⋊C4—531no—
171C2×C4⋊C4—531no—
172C2×C4⋊C4—531no—
173C2×C4⋊C4—531no—
174C2×C4⋊C4—531no—
175C2×C4⋊C4C2×D5311yescharacteristic
176C2×C22⋊C4—532no—
177C2×C22⋊C4—532no—
178C2×C22⋊C4—532no—
179C2×C22⋊C4—532no—
180C2×C22⋊C4—532no—
181C2×C22⋊C4—532no—
182C2×C22⋊C4C2×D5312yes—
183C2×C22⋊C4C2×D5312yes—
184C2×C42—531no—
185C2×C42—532no—
186#95—532no—
187#95—531no—
188#95C2×D5311yescharacteristic
189C24—532no—
190C24C22×D5312yes—
191C2×D4—22no—
192C2×D4—42no—
193C2×D4—1064no—
194C2×D4—1064no—
195C2×D4—1064no—
196C2×D4—1064no—
197C2×D4—1062no—
198C2×D4—2122no—
199C2×D4—2122no—
200C2×D4—2122no—
201C22×C4—531no—
202C22×C4—531no—
203C22×C4—532no—
204C22×C4—531no—
205C22×C4—531no—
206C22×C4C22×D5312yes—
207C22×C4C22×D5311yescharacteristic
208C22×C4C22×D5311yescharacteristic
209C22×C4C22×D5311yescharacteristic
210C4⋊C4—22no—
211C4⋊C4—1062no—
212C4⋊C4—1062no—
213C4⋊C4—1062no—
214C4⋊C4—1062no—
215C4⋊C4—1062no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—2122no—
219C22⋊C4—2122no—
220C22⋊C4—2122no—
221C22⋊C4—2122no—
222C22⋊C4—2122no—
223C22⋊C4—2122no—
224C42—534no—
225C42—1064no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—1062no—
234C23—1062no—
235C23—1062no—
236C23—1062no—
237C23—1062no—
238C23—1062no—
239C23—1062no—
240C23C23×D5311yescharacteristic
241D4—44no—
242D4—2124no—
243D4—2128no—
244D4—2128no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—532no—
255C2×C4—534no—
256C2×C4—532no—
257C2×C4—1061no—
258C2×C4—1061no—
259C2×C4—1061no—
260C2×C4—1061no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—1061no—
264C2×C4—1062no—
265C2×C4—1061no—
266C2×C4—1062no—
267C2×C4D4×D5312yes—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—2122no—
276C22—2122no—
277C22—2122no—
278C22—2122no—
279C22—2122no—
280C22—2122no—
281C22—2122no—
282C22C2×D4×D5311yescharacteristic
283C22#41×D5312yes—
284C22#1313326212yes—
285C22#1313326211yescharacteristic
286C22#1313326311yescharacteristic
287C4—22no—
288C4—41no—
289C4—41no—
290C4—42no—
291C4—534no—
292C4—1062no—
293C4—2121no—
294C4—2121no—
295C4—2122no—
296C2—82no—
297C2—4242no—
298C2#124×D5311yescharacteristic
299C2#1681118811yescharacteristic
300C2#1681118911yescharacteristic
301C2#1681122312yes—
302C2#1681122512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2641
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#13133207C2semidirect
3. #16810541C2semidirect
4. #16810540C2semidirect
5. #16810539C2semidirect
6. #16810538C2semidirect
7. #16810537C2semidirect
8. #16810523C2semidirect
9. #16810523C2semidirect
10. #16810468C2semidirect
11. C53×#394C2semidirect
159. #394D53semidirect
163. C53#2641semidirect