Finite Group Databasebeta

← #20278073

Subgroups — #20278073

Order 6784.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×Q8×D53C211yesmaximal, characteristic, complemented
2C2×#13133194C212yesmaximal, complemented
3C2×#13133177C211yesmaximal, characteristic
4#16810544C211yesmaximal, characteristic, complemented
5#16810542C211yesmaximal, characteristic, complemented
6#16810535C211yesmaximal, characteristic, complemented
7#16810527C212yesmaximal, complemented
8#16810499C212yesmaximal
9#16810497C212yesmaximal, complemented
10#16810482C211yesmaximal, characteristic, complemented
11C53×#393C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×Dic106C2212yes—
14C2×Q8×D53—24no—
15C2×Q8×D53—28no—
16C22×C22⋊D53C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18C22×C4×D53C2212yes—
19C22×C4×D53C2211yescharacteristic
20#13133194—28no—
21#13133177—24no—
22C2×C4⋊Dic53C2212yes—
23C2×#11719919C2212yes—
24C2×#11719919C2212yes—
25C2×#11719919C2211yescharacteristic
26C2×C22⋊Dic53C2212yes—
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2212yes—
29C2×C4×Dic53C2211yescharacteristic
30#13133124C2212yes—
31#13133124C2212yes—
32#13133124C2212yes—
33#13133124C2211yescharacteristic
34C22×Q8×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36C2×C22⋊C4×C53C2211yescharacteristic
37#95×C53C2212yes—
38#95×C53C2211yescharacteristic
39C24×C53C2311yescharacteristic
40C23×D53C2311yescharacteristic
41C2×Q8×C53—22no—
42C2×Q8×C53—24no—
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2312yes—
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2312yes—
47C2×C22⋊D53—22no—
48C2×C22⋊D53—22no—
49C2×C22⋊D53Q814yescomplemented
50C22×Dic53—22no—
51C22×Dic53—24no—
52C22×Dic53C2312yes—
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2312yes—
55C22×Dic53C2311yescharacteristic
56C22×Dic53C2311yescharacteristic
57Q8×D53—232no—
58C2×C4×D53—24no—
59C2×C4×D53—24no—
60C2×C4×D53—24no—
61C2×C4×D53—24no—
62C2×C4×D53D414yes—
63C2×Dic106—22no—
64C2×Dic106—24no—
65C2×Dic106—24no—
66C2×Dic106—24no—
67C2×Dic106D418yescomplemented
68C22⋊C4×C53—22no—
69C22⋊C4×C53—24no—
70C22⋊Dic53—24no—
71#11719921—22no—
72#11719921—24no—
73C4⋊Dic53—24no—
74#11719919—22no—
75#11719919—24no—
76#11719919—24no—
77C4×Dic53—22no—
78C23×C53—21no—
79C23×C53—22no—
80C23×C53—21no—
81C23×C53—22no—
82C23×C53C2×Q812yescomplemented
83C23×C53C2411yescharacteristic
84C22×D53—22no—
85C22×D53—22no—
86C22×D53C2×D412yes—
87C22×D53C2×Q812yes—
88C22×D53#4112yes—
89Q8×C53—28no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53C2×D412yes—
99C2×C4×C53C2×D414yes—
100C22⋊D53—28no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—22no—
106C2×Dic53—21no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—44no—
111C2×Dic53C2×D414yes—
112C2×Dic53C2×D412yes—
113C2×Dic53C2×D414yes—
114C2×Dic53C2×Q812yescomplemented
115C4×D53—216no—
116C4×D53—28no—
117Dic106—28no—
118Dic106—216no—
119C22×C53—22no—
120C22×C53—24no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53—42no—
124C22×C53C22×D411yescharacteristic
125C22×C53C22×D412yes—
126C22×C53C22×Q811yescharacteristic
127C22×C53C2×#4111yescharacteristic
128C22×C532−1+412yes—
129C2×D53—22no—
130C2×D53—22no—
131C2×D53—24no—
132C2×D53—24no—
133C2×D53C22⋊Q814yescomplemented
134C4×C53—22no—
135C4×C53—24no—
136C4×C53—41no—
137C4×C53—42no—
138Dic53—24no—
139Dic53—22no—
140Dic53—41no—
141Dic53—42no—
142Dic53#12014yes—
143#2409—531nomaximal, Sylow
144C2×C53—42no—
145C2×C53C2×#12011yescharacteristic
146C2×C53C2×C22⋊Q811yescharacteristic
147C2×C53#53611yescharacteristic
148C2×C53#54812yes—
149C2×C53D4×Q812yes—
150D53—24no—
151C23×Q8—531no—
152C2×C22⋊Q8—532no—
153C2×C4×D4—531no—
154#395—531no—
155#395—531no—
156#393—531no—
157#393D5311yescharacteristic, complemented
158#391—532no—
159#386—532no—
160#385—532no—
161#380—531no—
162C53#240911yescharacteristic, complemented, Sylow
163C22×Q8—532no—
164C22×Q8—531no—
165C22×Q8—1064no—
166C22×Q8—1068no—
167C22×Q8C2×D5311yescharacteristic
168C22×D4—531no—
169C23×C4—531no—
170C23×C4—531no—
171C23×C4—532no—
172C22⋊Q8—1068no—
173C4×D4—1064no—
174C2×C4⋊C4—531no—
175C2×C4⋊C4—532no—
176C2×C4⋊C4—532no—
177C2×C4⋊C4—532no—
178C2×C22⋊C4—532no—
179C2×C22⋊C4—531no—
180C2×C22⋊C4—532no—
181C2×C22⋊C4C2×D5311yescharacteristic
182C2×C22⋊C4C2×D5312yes—
183C2×C42—531no—
184#95—531no—
185#95—532no—
186#95—532no—
187#95—532no—
188#95C2×D5311yescharacteristic
189#95C2×D5312yes—
190C24—531no—
191C24C22×D5311yescharacteristic
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—538no—
195C2×Q8—1064no—
196C2×Q8—1064no—
197C2×Q8—1064no—
198C2×Q8—10632no—
199C2×Q8—1062no—
200C2×D4—534no—
201C2×D4—1062no—
202C2×D4—1062no—
203C22×C4—531no—
204C22×C4—531no—
205C22×C4—532no—
206C22×C4—534no—
207C22×C4—532no—
208C22×C4—531no—
209C22×C4—1064no—
210C22×C4—1064no—
211C22×C4—1064no—
212C22×C4—1062no—
213C22×C4—1064no—
214C22×C4—1064no—
215C22×C4C22×D5311yescharacteristic
216C22×C4C22×D5312yes—
217C22×C4C22×D5311yescharacteristic
218C22×C4C22×D5312yes—
219C4⋊C4—1064no—
220C4⋊C4—1064no—
221C4⋊C4—1062no—
222C4⋊C4—1064no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—1062no—
226C22⋊C4—1064no—
227C22⋊C4—1064no—
228C42—1062no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—532no—
234C23—532no—
235C23—532no—
236C23—1062no—
237C23—1062no—
238C23Q8×D5312yescomplemented
239C23C23×D5311yescharacteristic
240Q8—28no—
241Q8—10616no—
242Q8—1068no—
243D4—1068no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—532no—
253C2×C4—534no—
254C2×C4—534no—
255C2×C4—532no—
256C2×C4—1068no—
257C2×C4—10616no—
258C2×C4—1061no—
259C2×C4—1062no—
260C2×C4—1062no—
261C2×C4—1061no—
262C2×C4—1062no—
263C2×C4—1061no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—1062no—
267C2×C4—2124no—
268C2×C4D4×D5312yes—
269C2×C4D4×D5314yes—
270C22—24no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—534no—
276C22—1064no—
277C22—1064no—
278C22—1062no—
279C22—1062no—
280C22C2×D4×D5312yes—
281C22C2×D4×D5311yescharacteristic
282C22C2×Q8×D5311yescharacteristic
283C22#41×D5311yescharacteristic
284C22#1313326412yes—
285C4—22no—
286C4—24no—
287C4—42no—
288C4—41no—
289C4—534no—
290C4—1064no—
291C4—1062no—
292C4—2122no—
293C4—2121no—
294C2—42no—
295C2—1064no—
296C2#120×D5311yescharacteristic
297C2C22⋊Q8×D5311yescharacteristic
298C2#1681119311yescharacteristic
299C2#1681128112yes—
300C2#1681128612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53#2409
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D53C2semidirect
2. C2×#13133194C2semidirect
4. #16810544C2semidirect
5. #16810542C2semidirect
6. #16810535C2semidirect
7. #16810527C2semidirect
9. #16810497C2semidirect
10. #16810482C2semidirect
11. C53×#393C2semidirect
49. C2×C22⋊D53Q8semidirect
67. C2×Dic106D4semidirect
82. C23×C53C2×Q8semidirect
114. C2×Dic53C2×Q8semidirect
133. C2×D53C22⋊Q8semidirect
157. #393D53semidirect
162. C53#2409semidirect
238. C23Q8×D53semidirect