Finite Group Databasebeta

← #20278044

Subgroups — #20278044

Order 6784.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×D212C211yesmaximal, characteristic
2C2×#13133202C212yesmaximal, complemented
3C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
4#16810541C212yesmaximal, complemented
5#16810540C212yesmaximal, complemented
6#16810501C212yesmaximal, complemented
7#16810496C211yesmaximal, characteristic, complemented
8#16810478C211yesmaximal, characteristic, complemented
9#16810478C211yesmaximal, characteristic, complemented
10#16810478C211yesmaximal, characteristic, complemented
11C53×#385C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C24×D53C2211yescharacteristic
14C22×D212—24no—
15C22×D212—28no—
16C22×D212C2212yes—
17C22×D212C2212yes—
18C22×C4×D53C2211yescharacteristic, complemented
19C22×C4×D53C2211yescharacteristic, complemented
20#13133202—28no—
21C2×#11719919C2212yescomplemented
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2211yescharacteristic
24C2×#11719921C2212yescomplemented
25C2×#11719921C2211yescharacteristic, complemented
26C2×#11719921C2212yescomplemented
27C2×#11719921C2211yescharacteristic, complemented
28C2×#11719921C2212yescomplemented
29C2×#11719921C2212yescomplemented
30C2×#11719921C2212yescomplemented
31C2×#11719921C2212yescomplemented
32C22⋊C4×D53—24no—
33#13133124C2212yescomplemented
34C23×C4×C53C2211yescharacteristic
35C2×C4⋊C4×C53C2212yescomplemented
36C2×C22⋊C4×C53C2211yescharacteristic
37C2×C22⋊C4×C53C2211yescharacteristic
38#95×C53C2212yescomplemented
39C24×C53C2311yescharacteristic
40C23×D53—22no—
41C23×D53—22no—
42C23×D53—22no—
43C23×D53—24no—
44C23×D53—24no—
45C23×D53—22no—
46C23×D53—22no—
47C23×D53—42no—
48C23×D53—42no—
49C23×D53D414yes—
50C23×D53C2311yescharacteristic
51C23×D53C2312yes—
52C23×D53C2311yescharacteristic
53C22×C4×C53—22no—
54C22×C4×C53—24no—
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2312yes—
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2312yes—
59C22×Dic53C2311yescharacteristic
60C22×Dic53C2312yes—
61C22×Dic53C2311yescharacteristic
62C2×D212—232no—
63C2×D212—24no—
64C2×D212—24no—
65C2×D212—24no—
66C2×D212—24no—
67C2×D212—24no—
68C2×D212D418yes—
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53—24no—
72C2×C4×D53—42no—
73C4⋊C4×C53—24no—
74C22⋊C4×C53—22no—
75C22⋊C4×C53—22no—
76C22⋊Dic53—22no—
77C22⋊Dic53—22no—
78#11719921—22no—
79#11719921—22no—
80#11719921—24no—
81#11719921—24no—
82#11719921—24no—
83#11719921—42no—
84#11719921—42no—
85#11719921—42no—
86#11719919—24no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53C2×D412yes—
90C23×C53#4112yes—
91C23×C53#4112yes—
92C23×C53C2411yescharacteristic
93C22×D53—21no—
94C22×D53—21no—
95C22×D53—22no—
96C22×D53—28no—
97C22×D53—28no—
98C22×D53—22no—
99C22×D53—22no—
100C22×D53—22no—
101C22×D53—21no—
102C22×D53—22no—
103C22×D53—21no—
104C22×D53—22no—
105C22×D53—22no—
106C22×D53—22no—
107C22×D53—22no—
108C22×D53—28no—
109C22×D53—44no—
110C22×D53—48no—
111C22×D53—48no—
112C22×D53—48no—
113C22×D53—48no—
114C22×D53—44no—
115C22×D53—44no—
116C22×D53C2×D412yes—
117C22×D53C2×D414yes—
118C22×D53C2×D412yes—
119C22×D53C2×D414yes—
120C2×C4×C53—28no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53C2×D414yes—
131C2×Dic53—22no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—22no—
135C2×Dic53—22no—
136C2×Dic53—22no—
137C2×Dic53—22no—
138D212—216no—
139D212—216no—
140C4×D53—44no—
141C4×D53—44no—
142C22×C53—22no—
143C22×C53—22no—
144C22×C53—24no—
145C22×C53—24no—
146C22×C53#12314yescomplemented
147C22×C53C22×D411yescharacteristic
148C22×C53C22×D412yes—
149C22×C53C2×#4111yescharacteristic
150C22×C53C2×#4111yescharacteristic
151C22×C53H(5,𝔽2)12yes—
152C2×D53—22no—
153C2×D53—22no—
154C2×D53—24no—
155C2×D53—24no—
156C2×D53—22no—
157C2×D53—24no—
158C2×D53—22no—
159C2×D53—22no—
160C2×D53—24no—
161C2×D53—24no—
162C2×D53—416no—
163C2×D53—416no—
164C2×D53—42no—
165C2×D53—42no—
166C2×D53—42no—
167C2×D53—42no—
168C2×D53—42no—
169C2×D53—42no—
170C2×D53#12014yes—
171C4×C53—24no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—42no—
175Dic53—41no—
176Dic53—41no—
177Dic53—42no—
178#2405—531nomaximal, Sylow
179C2×C53—24no—
180C2×C53C2×#12011yescharacteristic
181C2×C53C2×#12311yescharacteristic
182C2×C53#53411yescharacteristic
183C2×C53#54612yes—
184C2×C53#54612yes—
185D53—24no—
186D53—42no—
187D53—44no—
188C23×D4—531no—
189C2×#123—532no—
190C22×C22⋊C4—531no—
191#394—532no—
192#394—532no—
193#386—532no—
194#385—531no—
195#385D5311yescharacteristic, complemented
196#379—531no—
197#379—531no—
198#379—531no—
199C53#240511yescharacteristic, complemented, Sylow
200C25—531no—
201C25—531no—
202C22×D4—532no—
203C22×D4—532no—
204C22×D4—1068no—
205C22×D4—1064no—
206C23×C4—531no—
207C23×C4—531no—
208C23×C4C2×D5311yescharacteristic
209#123—1068no—
210C2×C4⋊C4—532no—
211C2×C4⋊C4C2×D5312yescomplemented
212C2×C22⋊C4—531no—
213C2×C22⋊C4—531no—
214C2×C22⋊C4—531no—
215C2×C22⋊C4—531no—
216C2×C22⋊C4—532no—
217C2×C22⋊C4—532no—
218C2×C22⋊C4—532no—
219C2×C22⋊C4—532no—
220C2×C22⋊C4—532no—
221C2×C22⋊C4—532no—
222C2×C22⋊C4—1064no—
223C2×C22⋊C4C2×D5311yescharacteristic
224C2×C22⋊C4C2×D5311yescharacteristic
225#95—532no—
226#95C2×D5312yescomplemented
227C24—531no—
228C24—532no—
229C24—531no—
230C24—534no—
231C24—1064no—
232C24—1064no—
233C24—1062no—
234C24—1062no—
235C24—1062no—
236C24—1062no—
237C24—1062no—
238C24—2122no—
239C24—2122no—
240C24C22×D5311yescharacteristic
241C2×D4—538no—
242C2×D4—1064no—
243C2×D4—1064no—
244C2×D4—1064no—
245C2×D4—1064no—
246C2×D4—10632no—
247C2×D4—1064no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—531no—
251C22×C4—532no—
252C22×C4—531no—
253C22×C4—1062no—
254C22×C4—1062no—
255C22×C4—1064no—
256C22×C4—2122no—
257C22×C4C22×D5311yescharacteristic
258C22×C4C22×D5312yes—
259C22×C4C22×D5311yescharacteristic
260C22×C4C22×D5312yes—
261C4⋊C4—24no—
262C4⋊C4—1064no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—1062no—
266C22⋊C4—1062no—
267C22⋊C4—1064no—
268C22⋊C4—1064no—
269C22⋊C4—1064no—
270C22⋊C4—1062no—
271C22⋊C4—1062no—
272C22⋊C4—2122no—
273C22⋊C4—2122no—
274C22⋊C4—2122no—
275C23—22no—
276C23—22no—
277C23—534no—
278C23—532no—
279C23—534no—
280C23—532no—
281C23—1068no—
282C23—1068no—
283C23—1061no—
284C23—1061no—
285C23—1062no—
286C23—1068no—
287C23—1062no—
288C23—1062no—
289C23—1062no—
290C23—1062no—
291C23—1061no—
292C23—1062no—
293C23—1061no—
294C23—1062no—
295C23—1062no—
296C23—1062no—
297C23—2124no—
298C23—2128no—
299C23—2128no—
300C23—2128no—
301C23—2128no—
302C23—2124no—
303C23—2124no—
304C23#1171994512yes—
305C23D4×D5312yes—
306C23#1171994912yes—
307C23C23×D5311yescharacteristic
308D4—10616no—
309D4—10616no—
310C2×C4—28no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—1062no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1062no—
324C2×C4—1062no—
325C2×C4—1062no—
326C2×C4—1062no—
327C2×C4—2124no—
328C2×C4—2124no—
329C2×C4D4×D5314yes—
330C22—24no—
331C22—24no—
332C22—22no—
333C22—22no—
334C22—534no—
335C22—1064no—
336C22—1064no—
337C22—1062no—
338C22—1062no—
339C22—1064no—
340C22—1064no—
341C22—1064no—
342C22—1062no—
343C22—1062no—
344C22—1062no—
345C22—21216no—
346C22—21216no—
347C22—2122no—
348C22—2122no—
349C22—2122no—
350C22—2122no—
351C22—2122no—
352C22—2122no—
353C22#1313320214yescomplemented
354C22C2×D4×D5312yes—
355C22C2×D4×D5311yescharacteristic
356C22C2×#1171994511yescharacteristic
357C22C2×#1171994911yescharacteristic
358C22#1313326312yes—
359C4—24no—
360C4—42no—
361C4—41no—
362C4—41no—
363C4—2122no—
364C4—2121no—
365C4—2121no—
366C2—24no—
367C2—1064no—
368C2—2124no—
369C2—2122no—
370C2#120×D5311yescharacteristic
371C2C2×#1313320211yescharacteristic
372C2#1681118411yescharacteristic
373C2#1681126412yes—
374C2#1681126912yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#385C2
hypercenter#385D53
hypocenterC53#2405
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#13133202C2semidirect
3. C2×C22⋊C4×D53C2semidirect
4. #16810541C2semidirect
5. #16810540C2semidirect
6. #16810501C2semidirect
7. #16810496C2semidirect
8. #16810478C2semidirect
9. #16810478C2semidirect
10. #16810478C2semidirect
11. C53×#385C2semidirect
18. C22×C4×D53C22semidirect
19. C22×C4×D53C22semidirect
21. C2×#11719919C22semidirect
24. C2×#11719921C22semidirect
25. C2×#11719921C22semidirect
26. C2×#11719921C22semidirect
27. C2×#11719921C22semidirect
28. C2×#11719921C22semidirect
29. C2×#11719921C22semidirect
30. C2×#11719921C22semidirect
31. C2×#11719921C22semidirect
33. #13133124C22semidirect
35. C2×C4⋊C4×C53C22semidirect
38. #95×C53C22semidirect
146. C22×C53#123semidirect
195. #385D53semidirect
199. C53#2405semidirect
211. C2×C4⋊C4C2×D53semidirect
226. #95C2×D53semidirect
353. C22#13133202semidirect