Finite Group Databasebeta

← #20278075

Subgroups — #20278075

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×#13133195C212yesmaximal, complemented
3C2×#13133179C211yesmaximal, characteristic
4#16810543C211yesmaximal, characteristic, complemented
5#16810536C211yesmaximal, characteristic, complemented
6#16810534C211yesmaximal, characteristic, complemented
7#16810527C212yesmaximal, complemented
8#16810501C212yesmaximal
9#16810494C212yesmaximal, complemented
10#16810482C211yesmaximal, characteristic, complemented
11C53×#395C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×Dic106C2212yes—
14C2×Q8×D53—24no—
15C2×Q8×D53—28no—
16C22×D212C2211yescharacteristic
17C22×C4×D53C2212yes—
18C22×C4×D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20#13133195—28no—
21#13133179—24no—
22C2×#11719919C2212yes—
23C2×#11719919C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×C4×Dic53C2211yescharacteristic
30#13133124C2212yes—
31#13133124C2212yes—
32#13133124C2212yes—
33#13133124C2211yescharacteristic
34C22×Q8×C53C2211yescharacteristic
35C2×C4⋊C4×C53C2211yescharacteristic
36C2×C4⋊C4×C53C2212yes—
37#95×C53C2212yes—
38#95×C53C2211yescharacteristic
39C23×D53C2311yescharacteristic
40C23×D53C2311yescharacteristic
41C2×Q8×C53—22no—
42C2×Q8×C53—24no—
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2312yes—
47C22×C4×C53C2311yescharacteristic
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2312yes—
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52Q8×D53—232no—
53C2×D212—22no—
54C2×D212—22no—
55C2×D212Q814yescomplemented
56C2×C4×D53—24no—
57C2×C4×D53—24no—
58C2×C4×D53—24no—
59C2×C4×D53—24no—
60C2×C4×D53—22no—
61C2×C4×D53—24no—
62C2×C4×D53D414yes—
63C2×Dic106—22no—
64C2×Dic106—24no—
65C2×Dic106—24no—
66C2×Dic106—24no—
67C2×Dic106D418yescomplemented
68C4⋊C4×C53—22no—
69C4⋊C4×C53—24no—
70#11719921—22no—
71#11719921—24no—
72#11719921—22no—
73#11719921—24no—
74#11719921—24no—
75#11719919—24no—
76#11719919—24no—
77C4×Dic53—22no—
78C23×C53C2411yescharacteristic
79C22×D53—22no—
80C22×D53—22no—
81C22×D53—21no—
82C22×D53—22no—
83C22×D53—21no—
84C22×D53—22no—
85C22×D53C2×D412yes—
86C22×D53C2×Q812yes—
87C22×D53C2×Q812yescomplemented
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—21no—
95C2×C4×C53—22no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53C2×D412yes—
101C2×C4×C53C2×D414yes—
102C2×C4×C53C2×Q812yescomplemented
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—21no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53C2×D414yes—
111C2×Dic53C2×D412yes—
112C2×Dic53C2×D414yes—
113D212—28no—
114C4×D53—216no—
115C4×D53—28no—
116C4×D53—44no—
117Dic106—28no—
118Dic106—216no—
119C22×C53C22×D411yescharacteristic
120C22×C53C22×D412yes—
121C22×C53C22×Q811yescharacteristic
122C22×C53C2×#4111yescharacteristic
123C22×C532−1+412yes—
124C2×D53—22no—
125C2×D53—22no—
126C2×D53—24no—
127C2×D53—24no—
128C2×D53—22no—
129C2×D53—24no—
130C2×D53—41no—
131C2×D53—41no—
132C2×D53—42no—
133C2×D53C22⋊Q814yescomplemented
134C4×C53—22no—
135C4×C53—24no—
136C4×C53—22no—
137C4×C53—41no—
138C4×C53—42no—
139Dic53—24no—
140Dic53—41no—
141Dic53—42no—
142Dic53#12014yes—
143#2409—531nomaximal, Sylow
144C2×C53C2×#12011yescharacteristic
145C2×C53C2×C22⋊Q811yescharacteristic
146C2×C53#53611yescharacteristic
147C2×C53#54812yes—
148C2×C53D4×Q812yes—
149D53—24no—
150D53—42no—
151C23×Q8—531no—
152C2×C22⋊Q8—532no—
153C2×C4×D4—531no—
154#395—531no—
155#395D5311yescharacteristic, complemented
156#393—531no—
157#393—531no—
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—532no—
175C2×C4⋊C4—532no—
176C2×C4⋊C4C2×D5311yescharacteristic
177C2×C4⋊C4C2×D5312yes—
178C2×C22⋊C4—531no—
179C2×C22⋊C4—531no—
180C2×C22⋊C4—532no—
181C2×C22⋊C4—532no—
182C2×C22⋊C4—532no—
183C2×C42—531no—
184#95—531no—
185#95—532no—
186#95—532no—
187#95—532no—
188#95C2×D5311yescharacteristic
189#95C2×D5312yes—
190C24—531no—
191C24—531no—
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—532no—
205C22×C4—534no—
206C22×C4—531no—
207C22×C4—532no—
208C22×C4—1064no—
209C22×C4—1064no—
210C22×C4—1062no—
211C22×C4—1064no—
212C22×C4—1064no—
213C22×C4—1064no—
214C22×C4C22×D5312yes—
215C22×C4C22×D5311yescharacteristic
216C22×C4C22×D5311yescharacteristic
217C22×C4C22×D5312yes—
218C22×C4C22×D5311yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—1064no—
222C4⋊C4—1064no—
223C22⋊C4—1064no—
224C22⋊C4—1062no—
225C22⋊C4—1064no—
226C22⋊C4—1062no—
227C22⋊C4—1064no—
228C42—1062no—
229C23—532no—
230C23—532no—
231C23—532no—
232C23—532no—
233C23—1062no—
234C23—1062no—
235C23—1062no—
236C23—1061no—
237C23—1061no—
238C23—1062no—
239C23C23×D5311yescharacteristic
240Q8—28no—
241Q8—10616no—
242Q8—1068no—
243D4—1068no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—532no—
255C2×C4—534no—
256C2×C4—534no—
257C2×C4—1068no—
258C2×C4—10616no—
259C2×C4—1062no—
260C2×C4—1062no—
261C2×C4—1061no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—2124no—
267C2×C4D4×D5312yes—
268C2×C4D4×D5314yes—
269C2×C4Q8×D5312yescomplemented
270C22—534no—
271C22—1064no—
272C22—1064no—
273C22—1062no—
274C22—1062no—
275C22—1064no—
276C22—1062no—
277C22—2122no—
278C22—2121no—
279C22—2121no—
280C22C2×D4×D5312yes—
281C22C2×D4×D5311yescharacteristic
282C22C2×Q8×D5311yescharacteristic
283C22#41×D5311yescharacteristic
284C22#1313326512yes—
285C4—22no—
286C4—24no—
287C4—22no—
288C4—42no—
289C4—41no—
290C4—534no—
291C4—1064no—
292C4—2122no—
293C4—2121no—
294C2—1064no—
295C2—2122no—
296C2#120×D5311yescharacteristic
297C2C22⋊Q8×D5311yescharacteristic
298C2#1681119811yescharacteristic
299C2#1681128312yes—
300C2#1681128812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2409
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D53C2semidirect
2. C2×#13133195C2semidirect
4. #16810543C2semidirect
5. #16810536C2semidirect
6. #16810534C2semidirect
7. #16810527C2semidirect
9. #16810494C2semidirect
10. #16810482C2semidirect
11. C53×#395C2semidirect
55. C2×D212Q8semidirect
67. C2×Dic106D4semidirect
87. C22×D53C2×Q8semidirect
102. C2×C4×C53C2×Q8semidirect
133. C2×D53C22⋊Q8semidirect
155. #395D53semidirect
162. C53#2409semidirect
269. C2×C4Q8×D53semidirect