Finite Group Databasebeta

← #20281048

Subgroups — #20281048

Order 6784.

1000 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133187C212yesmaximal, complemented
2C2×#13133187C212yesmaximal, complemented
3C2×#13133185C212yesmaximal, complemented
4#16810541C212yesmaximal, complemented
5#16810540C211yesmaximal, characteristic, complemented
6#16810538C212yesmaximal, complemented
7#16810524C211yesmaximal, characteristic, complemented
8#16810523C212yesmaximal, complemented
9C53×#394C211yesmaximal, characteristic, complemented
10C24×D53C2212yescomplemented
11C22×D212C2211yescharacteristic
12C22×D212C2212yes—
13C22×D212C2211yescharacteristic
14C22×C22⋊D53C2212yes—
15C22×C22⋊D53C2212yes—
16#13133187—28no—
17#13133187—28no—
18#13133185—28no—
19C2×#11719919C2212yescomplemented
20C2×C22⋊Dic53C2212yescomplemented
21C2×#11719921C2212yescomplemented
22C2×#11719921C2212yescomplemented
23C2×#11719921C2212yescomplemented
24C2×#11719921C2212yescomplemented
25C2×#11719921C2212yescomplemented
26C2×#11719921C2212yescomplemented
27C2×C4×Dic53C2211yescharacteristic, complemented
28#13133124C2211yescharacteristic, complemented
29C22×D4×C53C2211yescharacteristic
30C2×C4⋊C4×C53C2211yescharacteristic, complemented
31C2×C22⋊C4×C53C2212yescomplemented
32C2×C22⋊C4×C53C2212yescomplemented
33#95×C53C2211yescharacteristic, complemented
34C24×C53C2312yes—
35C23×D53—24no—
36C23×D53—24no—
37C23×D53—24no—
38C23×D53—42no—
39C23×D53—42no—
40C23×D53—42no—
41C23×D53—42no—
42C23×D53C2312yes—
43C23×D53C2312yes—
44C2×D4×C53—24no—
45C2×D4×C53—41no—
46C22×C4×C53C2312yes—
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2311yescharacteristic
49C22×C4×C53C2311yescharacteristic
50C2×C22⋊D53—24no—
51C2×C22⋊D53—24no—
52C2×C22⋊D53—42no—
53C2×C22⋊D53—42no—
54C2×C22⋊D53—42no—
55C2×C22⋊D53D418yescomplemented
56C22×Dic53C2312yes—
57C22×Dic53C2312yes—
58C2×D212—24no—
59C2×D212—24no—
60C2×D212—24no—
61C2×D212—41no—
62C2×D212—41no—
63C2×D212—42no—
64C2×D212—41no—
65C2×D212D414yescomplemented
66C4⋊C4×C53—41no—
67C22⋊C4×C53—24no—
68C22⋊C4×C53—24no—
69C22⋊Dic53—24no—
70#11719921—24no—
71#11719921—24no—
72#11719921—24no—
73#11719921—24no—
74#11719921—24no—
75#11719921—24no—
76#11719919—42no—
77C4×Dic53—41no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53C2×D414yes—
85C23×C53C2411yescharacteristic
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—22no—
90C22×D53—22no—
91C22×D53—22no—
92C22×D53—22no—
93C22×D53—22no—
94C22×D53—22no—
95C22×D53—22no—
96C22×D53—22no—
97C22×D53—22no—
98C22×D53—48no—
99C22×D53—48no—
100C22×D53—48no—
101C22×D53—48no—
102C22×D53—48no—
103C22×D53—48no—
104C22×D53—48no—
105C22×D53C2×D414yes—
106C22×D53C2×D414yes—
107D4×C53—44no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53C2×D412yes—
118C22⋊D53—48no—
119C22⋊D53—48no—
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53—22no—
123C2×Dic53—22no—
124C2×Dic53—22no—
125C2×Dic53C2×D414yes—
126D212—44no—
127D212—44no—
128D212—48no—
129C22×C53—24no—
130C22×C53—24no—
131C22×C53—24no—
132C22×C53—42no—
133C22×C53—42no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53C22×D411yescharacteristic
137C22×C53C22×D412yes—
138C22×C53H(5,𝔽2)11yescharacteristic
139C22×C53H(5,𝔽2)11yescharacteristic
140C22×C53H(5,𝔽2)12yes—
141C2×D53—24no—
142C2×D53—24no—
143C2×D53—24no—
144C2×D53—24no—
145C2×D53—24no—
146C2×D53—24no—
147C2×D53—432no—
148C2×D53—42no—
149C2×D53—42no—
150C2×D53—42no—
151C2×D53—42no—
152C2×D53—42no—
153C2×D53—42no—
154C2×D53—42no—
155C2×D53—42no—
156C4×C53—22no—
157C4×C53—41no—
158C4×C53—42no—
159C4×C53—41no—
160Dic53—24no—
161Dic53—42no—
162#2679—531nomaximal, Sylow
163C2×C53—44no—
164C2×C53#53411yescharacteristic
165C2×C53#53412yes—
166C2×C53D4211yescharacteristic
167C2×C53D4212yes—
168C2×C53H(3,𝔽4)11yescharacteristic
169D53—44no—
170D53—44no—
171C2×#120—532no—
172C2×#120—532no—
173C2×#120—532no—
174#394—531no—
175#394—532no—
176#394—532no—
177#394D5311yescharacteristic, complemented
178#390—531no—
179#390—532no—
180C53#267911yescharacteristic, complemented, Sylow
181C25—532no—
182C22×D4—532no—
183C22×D4—532no—
184C22×D4—531no—
185C22×D4—531no—
186C22×D4—532no—
187C22×D4C2×D5311yescharacteristic
188#120—1068no—
189#120—1068no—
190#120—1068no—
191C2×C4⋊C4—532no—
192C2×C4⋊C4C2×D5311yescharacteristic, complemented
193C2×C22⋊C4—532no—
194C2×C22⋊C4—532no—
195C2×C22⋊C4—532no—
196C2×C22⋊C4—532no—
197C2×C22⋊C4—532no—
198C2×C22⋊C4—532no—
199C2×C22⋊C4—532no—
200C2×C22⋊C4C2×D5312yescomplemented
201C2×C22⋊C4C2×D5312yescomplemented
202C2×C42—531no—
203#95—531no—
204#95C2×D5311yescharacteristic, complemented
205C24—532no—
206C24—532no—
207C24—1064no—
208C24—1064no—
209C24—1064no—
210C24—2122no—
211C24—2122no—
212C24—2122no—
213C24—2122no—
214C24C22×D5312yes—
215C2×D4—24no—
216C2×D4—41no—
217C2×D4—538no—
218C2×D4—534no—
219C2×D4—1064no—
220C2×D4—1064no—
221C2×D4—1064no—
222C2×D4—1064no—
223C2×D4—1064no—
224C2×D4—2121no—
225C2×D4—2121no—
226C2×D4—2121no—
227C2×D4—2122no—
228C2×D4—2122no—
229C2×D4—2122no—
230C2×D4—2122no—
231C22×C4—532no—
232C22×C4—532no—
233C22×C4C22×D5311yescharacteristic
234C22×C4C22×D5311yescharacteristic
235C22×C4C22×D5312yes—
236C22×C4C22×D5311yescharacteristic
237C4⋊C4—41no—
238C4⋊C4—2122no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—1064no—
242C22⋊C4—1064no—
243C22⋊C4—1064no—
244C22⋊C4—1064no—
245C22⋊C4—1064no—
246C22⋊C4—1064no—
247C22⋊C4—1064no—
248C42—2121no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—534no—
256C23—534no—
257C23—1062no—
258C23—1062no—
259C23—1062no—
260C23—1062no—
261C23—1062no—
262C23—1062no—
263C23—1062no—
264C23—1062no—
265C23—1062no—
266C23—1062no—
267C23—1062no—
268C23—1062no—
269C23—2128no—
270C23—2128no—
271C23—2128no—
272C23—2128no—
273C23—2128no—
274C23—2128no—
275C23—2128no—
276C23D4×D5314yes—
277C23C23×D5311yescharacteristic
278D4—44no—
279D4—2124no—
280D4—2124no—
281D4—2128no—
282D4—2128no—
283D4—2128no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—534no—
294C2×C4—1062no—
295C2×C4—1062no—
296C2×C4—1062no—
297C2×C4—1062no—
298C2×C4—1062no—
299C2×C4D4×D5312yes—
300C22—24no—
301C22—24no—
302C22—24no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—1064no—
308C22—1064no—
309C22—1064no—
310C22—1064no—
311C22—1064no—
312C22—1064no—
313C22—21232no—
314C22—2122no—
315C22—2122no—
316C22—2122no—
317C22—2122no—
318C22—2122no—
319C22—2122no—
320C22—2122no—
321C22—2122no—
322C22C2×D4×D5311yescharacteristic
323C22C2×D4×D5312yes—
324C22#1313326211yescharacteristic
325C22#1313326312yes—
326C22#1313326311yescharacteristic
327C4—22no—
328C4—41no—
329C4—41no—
330C4—42no—
331C4—1064no—
332C4—2122no—
333C2—44no—
334C2—2124no—
335C2—2124no—
336C2#1681118111yescharacteristic
337C2#1681118412yes—
338C2#1681125511yescharacteristic
339C2#1681125712yes—
340C2#1681136511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2679
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#13133187C2semidirect
2. C2×#13133187C2semidirect
3. C2×#13133185C2semidirect
4. #16810541C2semidirect
5. #16810540C2semidirect
6. #16810538C2semidirect
7. #16810524C2semidirect
8. #16810523C2semidirect
9. C53×#394C2semidirect
10. C24×D53C22semidirect
19. C2×#11719919C22semidirect
20. C2×C22⋊Dic53C22semidirect
21. C2×#11719921C22semidirect
22. C2×#11719921C22semidirect
23. C2×#11719921C22semidirect
24. C2×#11719921C22semidirect
25. C2×#11719921C22semidirect
26. C2×#11719921C22semidirect
27. C2×C4×Dic53C22semidirect
28. #13133124C22semidirect
30. C2×C4⋊C4×C53C22semidirect
31. C2×C22⋊C4×C53C22semidirect
32. C2×C22⋊C4×C53C22semidirect
33. #95×C53C22semidirect
55. C2×C22⋊D53D4semidirect
65. C2×D212D4semidirect
177. #394D53semidirect
180. C53#2679semidirect
192. C2×C4⋊C4C2×D53semidirect
200. C2×C22⋊C4C2×D53semidirect
201. C2×C22⋊C4C2×D53semidirect
204. #95C2×D53semidirect