Finite Group Databasebeta

← #20278347

Subgroups — #20278347

Order 6784.

1348 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 181 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×D4×D53C211yesmaximal, characteristic, complemented
2C2×#13133213C212yesmaximal, complemented
3C2×#13133190C211yesmaximal, characteristic, complemented
4C2×#13133187C212yesmaximal, complemented
5C2×#13133185C212yesmaximal, complemented
6#16810533C211yesmaximal, characteristic, complemented
7#16810532C211yesmaximal, characteristic, complemented
8#16810531C211yesmaximal, characteristic, complemented
9#16810523C212yesmaximal, complemented
10#16810470C211yesmaximal, characteristic, complemented
11C53×#392C211yesmaximal, characteristic, complemented
12C24×D53C2212yes—
13C22×D212C2211yescharacteristic
14C22×D212C2211yescharacteristic
15C22×D212C2212yescomplemented
16C22×C22⋊D53C2211yescharacteristic
17C22×C22⋊D53C2212yes—
18C22×C22⋊D53C2211yescharacteristic, complemented
19C22×C22⋊D53C2212yescomplemented
20C22×C22⋊D53C2212yescomplemented
21C2×D4×D53—28no—
22C2×D4×D53—24no—
23C22×C4×D53C2211yescharacteristic
24#13133213—28no—
25#13133190—24no—
26#13133187—28no—
27#13133185—28no—
28C2×#11719919C2211yescharacteristic, complemented
29C2×C22⋊Dic53C2212yescomplemented
30C2×#11719921C2212yes—
31C2×#11719921C2212yescomplemented
32C2×#11719921C2212yescomplemented
33C2×#11719921C2211yescharacteristic, complemented
34C2×C4×Dic53C2212yescomplemented
35#13133124C2211yescharacteristic, complemented
36C22×D4×C53C2211yescharacteristic
37C22×D4×C53C2212yescomplemented
38C2×C22⋊C4×C53C2212yescomplemented
39C2×C22⋊C4×C53C2211yescharacteristic, complemented
40#95×C53C2211yescharacteristic, complemented
41C24×C53C2311yescharacteristic
42C24×C53C2312yes—
43C23×D53—24no—
44C23×D53—24no—
45C23×D53—24no—
46C23×D53—24no—
47C23×D53—24no—
48C23×D53—42no—
49C23×D53—42no—
50C23×D53C2311yescharacteristic
51C23×D53C2312yes—
52C23×D53C2311yescharacteristic
53C2×D4×C53—24no—
54C2×D4×C53—24no—
55C2×D4×C53—22no—
56C2×D4×C53—24no—
57C2×D4×C53—42no—
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2312yes—
61C2×C22⋊D53—24no—
62C2×C22⋊D53—24no—
63C2×C22⋊D53—22no—
64C2×C22⋊D53—24no—
65C2×C22⋊D53—24no—
66C2×C22⋊D53—24no—
67C2×C22⋊D53—22no—
68C2×C22⋊D53—24no—
69C2×C22⋊D53—24no—
70C2×C22⋊D53—24no—
71C2×C22⋊D53—42no—
72C2×C22⋊D53—42no—
73C2×C22⋊D53D418yescomplemented
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2312yes—
76C22×Dic53C2311yescharacteristic
77D4×D53—232no—
78C2×D212—24no—
79C2×D212—22no—
80C2×D212—24no—
81C2×D212—22no—
82C2×D212—24no—
83C2×D212—24no—
84C2×D212—42no—
85C2×C4×D53—24no—
86C2×C4×D53D414yescomplemented
87C22⋊C4×C53—22no—
88C22⋊C4×C53—24no—
89C22⋊Dic53—24no—
90#11719921—22no—
91#11719921—24no—
92#11719921—24no—
93#11719921D418yescomplemented
94#11719919—22no—
95C4×Dic53—24no—
96C23×C53—22no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53—22no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—22no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53C2×D414yes—
108C23×C53C2411yescharacteristic
109C22×D53—22no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53—216no—
113C22×D53—22no—
114C22×D53—22no—
115C22×D53—22no—
116C22×D53—22no—
117C22×D53—22no—
118C22×D53—22no—
119C22×D53—22no—
120C22×D53—22no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—48no—
125C22×D53—48no—
126C22×D53—48no—
127C22×D53—48no—
128C22×D53—48no—
129C22×D53—48no—
130C22×D53C2×D414yes—
131C22×D53C2×D412yes—
132C22×D53C2×D414yes—
133D4×C53—28no—
134D4×C53—48no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—21no—
139C2×C4×C53—22no—
140C2×C4×C53C2×D412yes—
141C2×C4×C53C2×D414yescomplemented
142C22⋊D53—216no—
143C22⋊D53—48no—
144C22⋊D53—48no—
145C22⋊D53—44no—
146C22⋊D53—44no—
147C2×Dic53—22no—
148C2×Dic53—22no—
149C2×Dic53—21no—
150C2×Dic53—22no—
151C2×Dic53C2×D414yes—
152C2×Dic53C2×D412yes—
153C2×Dic53C2×D414yescomplemented
154D212—28no—
155D212—48no—
156D212—44no—
157C4×D53—28no—
158C22×C53—24no—
159C22×C53—24no—
160C22×C53—24no—
161C22×C53—42no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53—42no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—42no—
169C22×C53—42no—
170C22×C53C22×D411yescharacteristic
171C22×C53C22×D412yes—
172C22×C53C22×D412yes—
173C22×C53H(5,𝔽2)11yescharacteristic
174C22×C53H(5,𝔽2)11yescharacteristic
175C2×D53—22no—
176C2×D53—22no—
177C2×D53—24no—
178C2×D53—24no—
179C2×D53—24no—
180C2×D53—24no—
181C2×D53—24no—
182C2×D53—432no—
183C2×D53—42no—
184C2×D53—42no—
185C2×D53—42no—
186C2×D53—42no—
187C2×D53—41no—
188C2×D53—41no—
189C2×D53—41no—
190C2×D53—42no—
191C2×D53—42no—
192C2×D53#12714yescomplemented
193C4×C53—22no—
194C4×C53—24no—
195C4×C53—41no—
196Dic53—24no—
197Dic53—41no—
198Dic53#12014yescomplemented
199#2433—531nomaximal, Sylow
200C2×C53—44no—
201C2×C53—81no—
202C2×C53C2×#12011yescharacteristic
203C2×C53C2×#12711yescharacteristic
204C2×C53#53511yescharacteristic
205C2×C53D4212yes—
206C2×C53D4212yes—
207D53—24no—
208D53—44no—
209D53—81no—
210C23×D4—531no—
211C2×#127—532no—
212C2×#121—531no—
213C2×#120—532no—
214C2×#120—532no—
215#392—531no—
216#392—531no—
217#392—531no—
218#392D5311yescharacteristic, complemented
219#390—532no—
220C4×C22⋊C4—531no—
221C53#243311yescharacteristic, complemented, Sylow
222C25—532no—
223C22×D4—532no—
224C22×D4—531no—
225C22×D4—531no—
226C22×D4—532no—
227C22×D4—532no—
228C22×D4—531no—
229C22×D4—531no—
230C22×D4—532no—
231C22×D4—1064no—
232C22×D4—1068no—
233C22×D4C2×D5311yescharacteristic
234C22×D4C2×D5312yescomplemented
235C23×C4—531no—
236#127—1068no—
237#121—1064no—
238#120—1068no—
239#120—1068no—
240C2×C4⋊C4—531no—
241C2×C22⋊C4—532no—
242C2×C22⋊C4—532no—
243C2×C22⋊C4—532no—
244C2×C22⋊C4—532no—
245C2×C22⋊C4—531no—
246C2×C22⋊C4C2×D5312yescomplemented
247C2×C22⋊C4C2×D5311yescharacteristic, complemented
248C2×C42—532no—
249#95—531no—
250#95C2×D5311yescharacteristic, complemented
251C24—531no—
252C24—532no—
253C24—531no—
254C24—1064no—
255C24—1064no—
256C24—1064no—
257C24—1064no—
258C24—1064no—
259C24—2122no—
260C24—2122no—
261C24C22×D5311yescharacteristic
262C24C22×D5312yes—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—42no—
268C2×D4—538no—
269C2×D4—1064no—
270C2×D4—1064no—
271C2×D4—1062no—
272C2×D4—1064no—
273C2×D4—1064no—
274C2×D4—1064no—
275C2×D4—1064no—
276C2×D4—1064no—
277C2×D4—1062no—
278C2×D4—1064no—
279C2×D4—10632no—
280C2×D4—1064no—
281C2×D4—1062no—
282C2×D4—1064no—
283C2×D4—1064no—
284C2×D4—1062no—
285C2×D4—1064no—
286C2×D4—2122no—
287C2×D4—2122no—
288C2×D4—2122no—
289C22×C4—531no—
290C22×C4—532no—
291C22×C4—534no—
292C22×C4—531no—
293C22×C4—1064no—
294C22×C4C22×D5311yescharacteristic
295C22×C4C22×D5311yescharacteristic
296C22×C4C22×D5312yes—
297C4⋊C4—1062no—
298C22⋊C4—22no—
299C22⋊C4—24no—
300C22⋊C4—538no—
301C22⋊C4—1064no—
302C22⋊C4—1064no—
303C22⋊C4—1062no—
304C22⋊C4—1064no—
305C42—1064no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—21no—
313C23—22no—
314C23—22no—
315C23—21no—
316C23—21no—
317C23—534no—
318C23—532no—
319C23—534no—
320C23—10616no—
321C23—1062no—
322C23—1061no—
323C23—1061no—
324C23—1062no—
325C23—1062no—
326C23—1062no—
327C23—1062no—
328C23—1062no—
329C23—1062no—
330C23—1061no—
331C23—1062no—
332C23—1062no—
333C23—1061no—
334C23—1061no—
335C23—2128no—
336C23—2128no—
337C23—2128no—
338C23—2128no—
339C23—2128no—
340C23—2128no—
341C23D4×D5314yes—
342C23C23×D5311yescharacteristic
343D4—28no—
344D4—48no—
345D4—1068no—
346D4—10616no—
347D4—2128no—
348D4—2124no—
349D4—2124no—
350D4—2128no—
351D4—2128no—
352D4—2124no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—534no—
359C2×C4—532no—
360C2×C4—534no—
361C2×C4—1068no—
362C2×C4—1062no—
363C2×C4—1062no—
364C2×C4—1062no—
365C2×C4—1061no—
366C2×C4D4×D5312yes—
367C2×C4D4×D5314yescomplemented
368C22—24no—
369C22—24no—
370C22—24no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—42no—
380C22—534no—
381C22—1062no—
382C22—1062no—
383C22—1064no—
384C22—1064no—
385C22—1064no—
386C22—1064no—
387C22—1064no—
388C22—21232no—
389C22—2121no—
390C22—2121no—
391C22—2121no—
392C22—2122no—
393C22—2122no—
394C22—2122no—
395C22—2122no—
396C22—2122no—
397C22—2122no—
398C22C2×D4×D5311yescharacteristic
399C22C2×D4×D5312yes—
400C22C2×D4×D5312yes—
401C22#1313326211yescharacteristic
402C22#1313326311yescharacteristic
403C4—22no—
404C4—24no—
405C4—41no—
406C4—534no—
407C4—1064no—
408C4—2121no—
409C2—44no—
410C2—81no—
411C2—1064no—
412C2—4241no—
413C2—2124no—
414C2#120×D5311yescharacteristic
415C2#127×D5311yescharacteristic
416C2#1681118911yescharacteristic
417C2#1681125512yes—
418C2#1681125712yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2433
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D53C2semidirect
2. C2×#13133213C2semidirect
3. C2×#13133190C2semidirect
4. C2×#13133187C2semidirect
5. C2×#13133185C2semidirect
6. #16810533C2semidirect
7. #16810532C2semidirect
8. #16810531C2semidirect
9. #16810523C2semidirect
10. #16810470C2semidirect
11. C53×#392C2semidirect
15. C22×D212C22semidirect
18. C22×C22⋊D53C22semidirect
19. C22×C22⋊D53C22semidirect
20. C22×C22⋊D53C22semidirect
28. C2×#11719919C22semidirect
29. C2×C22⋊Dic53C22semidirect
31. C2×#11719921C22semidirect
32. C2×#11719921C22semidirect
33. C2×#11719921C22semidirect
34. C2×C4×Dic53C22semidirect
35. #13133124C22semidirect
37. C22×D4×C53C22semidirect
38. C2×C22⋊C4×C53C22semidirect
39. C2×C22⋊C4×C53C22semidirect
40. #95×C53C22semidirect
73. C2×C22⋊D53D4semidirect
86. C2×C4×D53D4semidirect
93. #11719921D4semidirect
141. C2×C4×C53C2×D4semidirect
153. C2×Dic53C2×D4semidirect
192. C2×D53#127semidirect
198. Dic53#120semidirect
218. #392D53semidirect
221. C53#2433semidirect
234. C22×D4C2×D53semidirect
246. C2×C22⋊C4C2×D53semidirect
247. C2×C22⋊C4C2×D53semidirect
250. #95C2×D53semidirect
367. C2×C4D4×D53semidirect