Finite Group Databasebeta

← #20281049

Subgroups — #20281049

Order 6784.

1000 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13133186C212yesmaximal, complemented
2C2×#13133185C212yesmaximal, complemented
3C2×#13133185C211yesmaximal, characteristic, complemented
4#16810539C212yesmaximal, complemented
5#16810538C212yesmaximal, complemented
6#16810537C212yesmaximal, complemented
7#16810526C211yesmaximal, characteristic, complemented
8#16810523C212yesmaximal, complemented
9C2×#120×C53C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic, complemented
11C22×C22⋊D53C2212yes—
12C22×C22⋊D53C2212yes—
13C22×C22⋊D53C2212yes—
14#13133186—28no—
15#13133185—24no—
16#13133185—28no—
17C2×C4⋊Dic53C2211yescharacteristic, complemented
18C2×#11719919C2212yescomplemented
19C2×C22⋊Dic53C2212yescomplemented
20C2×C22⋊Dic53C2212yescomplemented
21C2×C22⋊Dic53C2212yescomplemented
22C2×C22⋊Dic53C2211yescharacteristic, complemented
23C2×C22⋊Dic53C2212yescomplemented
24C2×#11719921C2212yescomplemented
25C2×#11719921C2212yescomplemented
26C2×#11719921C2212yescomplemented
27C2×C4×Dic53C2211yescharacteristic, complemented
28#13133124C2212yescomplemented
29C25×C53C2211yescharacteristic, complemented
30C22×D4×C53C2212yes—
31C22×D4×C53C2211yescharacteristic
32#120×C53—24no—
33C2×C22⋊C4×C53C2212yescomplemented
34C2×C22⋊C4×C53C2211yescharacteristic, complemented
35C24×C53—22no—
36C24×C53—24no—
37C24×C53—42no—
38C24×C53—41no—
39C24×C53—41no—
40C24×C53C2311yescharacteristic
41C24×C53C2312yes—
42C24×C53C2311yescharacteristic
43C23×D53—22no—
44C23×D53—24no—
45C23×D53—41no—
46C23×D53—42no—
47C23×D53—41no—
48C23×D53C2312yes—
49C2×D4×C53—24no—
50C2×D4×C53—24no—
51C2×D4×C53—41no—
52C2×D4×C53—41no—
53C2×D4×C53—42no—
54C2×D4×C53D414yescomplemented
55C22×C4×C53C2312yes—
56C22×C4×C53C2311yescharacteristic
57C2×C22⋊D53—24no—
58C2×C22⋊D53—24no—
59C2×C22⋊D53—24no—
60C2×C22⋊D53—24no—
61C2×C22⋊D53—42no—
62C2×C22⋊D53—42no—
63C2×C22⋊D53—42no—
64C2×C22⋊D53—42no—
65C2×C22⋊D53D418yescomplemented
66C22×Dic53C2312yes—
67C22×Dic53C2312yes—
68C22×Dic53C2312yes—
69C22⋊C4×C53—22no—
70C22⋊C4×C53—24no—
71C22⋊Dic53—22no—
72C22⋊Dic53—24no—
73C22⋊Dic53—24no—
74C22⋊Dic53—24no—
75C22⋊Dic53—24no—
76#11719921—24no—
77#11719921—24no—
78#11719921—24no—
79C4⋊Dic53—41no—
80#11719919—42no—
81C4×Dic53—41no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—22no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—22no—
96C23×C53—44no—
97C23×C53—48no—
98C23×C53—48no—
99C23×C53—44no—
100C23×C53—44no—
101C23×C53C2×D414yes—
102C23×C53C2×D412yes—
103C23×C53C2×D412yes—
104C23×C53C2411yescharacteristic
105C22×D53—22no—
106C22×D53—22no—
107C22×D53—22no—
108C22×D53—22no—
109C22×D53—22no—
110C22×D53—22no—
111C22×D53—44no—
112C22×D53—48no—
113C22×D53—48no—
114C22×D53—44no—
115C22×D53—44no—
116C22×D53C2×D414yes—
117D4×C53—44no—
118D4×C53—48no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53C2×D412yes—
125C22⋊D53—48no—
126C22⋊D53—48no—
127C22⋊D53—48no—
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—22no—
132C2×Dic53—22no—
133C2×Dic53—22no—
134C2×Dic53—22no—
135C2×Dic53—22no—
136C2×Dic53C2×D414yes—
137C22×C53—24no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—24no—
141C22×C53—24no—
142C22×C53—24no—
143C22×C53—24no—
144C22×C53—416no—
145C22×C53—42no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—42no—
152C22×C53—41no—
153C22×C53—42no—
154C22×C53—41no—
155C22×C53C22×D412yes—
156C22×C53C22×D411yescharacteristic
157C22×C53H(5,𝔽2)11yescharacteristic
158C22×C53H(5,𝔽2)11yescharacteristic
159C22×C53H(5,𝔽2)12yes—
160C2×D53—24no—
161C2×D53—24no—
162C2×D53—24no—
163C2×D53—416no—
164C2×D53—42no—
165C2×D53—42no—
166C2×D53—42no—
167C2×D53—42no—
168C4×C53—22no—
169C4×C53—42no—
170Dic53—24no—
171Dic53—42no—
172Dic53—42no—
173#2679—531nomaximal, Sylow
174C2×C53—42no—
175C2×C53—44no—
176C2×C53—42no—
177C2×C53#53411yescharacteristic
178C2×C53#53412yes—
179C2×C53D4211yescharacteristic
180C2×C53D4212yes—
181C2×C53H(3,𝔽4)11yescharacteristic
182D53—44no—
183C2×#120—532no—
184C2×#120—532no—
185C2×#120—531no—
186C2×#120D5311yescharacteristic, complemented
187#394—532no—
188#394—532no—
189#394—532no—
190#390—531no—
191#390—532no—
192C53#267911yescharacteristic, complemented, Sylow
193C25—531no—
194C25C2×D5311yescharacteristic, complemented
195C22×D4—532no—
196C22×D4—532no—
197C22×D4—532no—
198C22×D4C2×D5312yes—
199C22×D4C2×D5311yescharacteristic
200#120—24no—
201#120—1068no—
202#120—1068no—
203#120—1064no—
204C2×C4⋊C4—532no—
205C2×C4⋊C4—531no—
206C2×C22⋊C4—532no—
207C2×C22⋊C4—532no—
208C2×C22⋊C4—532no—
209C2×C22⋊C4—532no—
210C2×C22⋊C4—531no—
211C2×C22⋊C4—532no—
212C2×C22⋊C4—532no—
213C2×C22⋊C4—532no—
214C2×C22⋊C4C2×D5312yescomplemented
215C2×C22⋊C4C2×D5311yescharacteristic, complemented
216C2×C42—531no—
217#95—532no—
218C24—24no—
219C24—22no—
220C24—41no—
221C24—42no—
222C24—41no—
223C24—532no—
224C24—1064no—
225C24—1062no—
226C24—2122no—
227C24—2121no—
228C24—2121no—
229C24C22×D5312yes—
230C24C22×D5311yescharacteristic
231C24C22×D5311yescharacteristic
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—42no—
237C2×D4—538no—
238C2×D4—1064no—
239C2×D4—1064no—
240C2×D4—1064no—
241C2×D4—1064no—
242C2×D4—2122no—
243C2×D4—2122no—
244C2×D4—2122no—
245C2×D4—2122no—
246C2×D4C22⋊D5314yescomplemented
247C22×C4—532no—
248C22×C4—532no—
249C22×C4—532no—
250C22×C4C22×D5312yes—
251C22×C4C22×D5311yescharacteristic
252C4⋊C4—2122no—
253C4⋊C4—2121no—
254C22⋊C4—22no—
255C22⋊C4—24no—
256C22⋊C4—1064no—
257C22⋊C4—1064no—
258C22⋊C4—1064no—
259C22⋊C4—1064no—
260C22⋊C4—1064no—
261C22⋊C4—1064no—
262C22⋊C4—1062no—
263C22⋊C4—1064no—
264C42—2121no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—21no—
273C23—21no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—44no—
280C23—48no—
281C23—44no—
282C23—48no—
283C23—44no—
284C23—534no—
285C23—1062no—
286C23—1062no—
287C23—1062no—
288C23—1062no—
289C23—1062no—
290C23—1062no—
291C23—2128no—
292C23—2124no—
293C23—2124no—
294C23—2128no—
295C23—2124no—
296C23D4×D5314yes—
297C23C2×C22⋊D5312yes—
298C23C2×C22⋊D5312yes—
299C23C23×D5311yescharacteristic
300D4—44no—
301D4—48no—
302D4—2128no—
303D4—2128no—
304D4—2128no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—534no—
311C2×C4—1062no—
312C2×C4—1062no—
313C2×C4—1062no—
314C2×C4—1062no—
315C2×C4—1062no—
316C2×C4—1062no—
317C2×C4—1062no—
318C2×C4—1062no—
319C2×C4C2×C22⋊D5312yes—
320C22—22no—
321C22—22no—
322C22—24no—
323C22—24no—
324C22—24no—
325C22—24no—
326C22—24no—
327C22—416no—
328C22—41no—
329C22—42no—
330C22—41no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—42no—
338C22—1064no—
339C22—1064no—
340C22—1064no—
341C22—21216no—
342C22—2122no—
343C22—2122no—
344C22—2122no—
345C22—2122no—
346C22C2×D4×D5312yes—
347C22C22×C22⋊D5311yescharacteristic
348C22#1313326211yescharacteristic
349C22#1313326212yes—
350C22#1313326211yescharacteristic
351C4—22no—
352C4—42no—
353C4—1064no—
354C4—2122no—
355C4—2122no—
356C2—42no—
357C2—44no—
358C2—42no—
359C2—2124no—
360C2#1681118211yescharacteristic
361C2#1681118312yes—
362C2#1681125511yescharacteristic
363C2D4×C22⋊D5312yes—
364C2#1681136611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#120×C53C2
hypercenterC2×#120D53
hypocenterC53#2679
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#13133186C2semidirect
2. C2×#13133185C2semidirect
3. C2×#13133185C2semidirect
4. #16810539C2semidirect
5. #16810538C2semidirect
6. #16810537C2semidirect
7. #16810526C2semidirect
8. #16810523C2semidirect
9. C2×#120×C53C2semidirect
10. C24×D53C22semidirect
17. C2×C4⋊Dic53C22semidirect
18. C2×#11719919C22semidirect
19. C2×C22⋊Dic53C22semidirect
20. C2×C22⋊Dic53C22semidirect
21. C2×C22⋊Dic53C22semidirect
22. C2×C22⋊Dic53C22semidirect
23. C2×C22⋊Dic53C22semidirect
24. C2×#11719921C22semidirect
25. C2×#11719921C22semidirect
26. C2×#11719921C22semidirect
27. C2×C4×Dic53C22semidirect
28. #13133124C22semidirect
29. C25×C53C22semidirect
33. C2×C22⋊C4×C53C22semidirect
34. C2×C22⋊C4×C53C22semidirect
54. C2×D4×C53D4semidirect
65. C2×C22⋊D53D4semidirect
186. C2×#120D53semidirect
192. C53#2679semidirect
194. C25C2×D53semidirect
214. C2×C22⋊C4C2×D53semidirect
215. C2×C22⋊C4C2×D53semidirect
246. C2×D4C22⋊D53semidirect