Finite Group Databasebeta

← #20278024

Subgroups — #20278024

Order 6784.

1564 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1564 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×D53C212yesmaximal, complemented
2C2×#13133187C212yesmaximal, complemented
3C2×#13133185C212yesmaximal, complemented
4#16810532C211yesmaximal, characteristic, complemented
5#16810531C212yesmaximal, complemented
6#16810501C212yesmaximal, complemented
7#16810499C212yesmaximal, complemented
8#16810478C211yesmaximal, characteristic, complemented
9C53×#392C211yesmaximal, characteristic, complemented
10C24×D53C2211yescharacteristic
11C24×D53C2212yescomplemented
12C22×D212C2212yes—
13C22×D212C2211yescharacteristic, complemented
14C22×C22⋊D53C2212yes—
15C22×C22⋊D53C2212yescomplemented
16C22×C22⋊D53C2212yescomplemented
17C2×D4×D53—28no—
18C2×D4×D53—28no—
19C2×D4×D53—28no—
20C22×C4×D53C2212yescomplemented
21#13133187—28no—
22#13133185—28no—
23C2×C22⋊Dic53C2212yescomplemented
24C2×C22⋊Dic53C2211yescharacteristic, complemented
25C2×#11719921C2212yescomplemented
26C2×#11719921C2212yescomplemented
27C2×#11719921C2212yescomplemented
28C2×#11719921C2212yescomplemented
29#13133124C2211yescharacteristic, complemented
30#13133124C2212yescomplemented
31C22×D4×C53C2212yescomplemented
32C22×D4×C53C2211yescharacteristic, complemented
33C2×C22⋊C4×C53C2212yescomplemented
34C2×C22⋊C4×C53C2211yescharacteristic, complemented
35#95×C53C2211yescharacteristic, complemented
36C24×C53C2311yescharacteristic
37C24×C53C2312yes—
38C23×D53—24no—
39C23×D53—24no—
40C23×D53—24no—
41C23×D53—24no—
42C23×D53—24no—
43C23×D53—24no—
44C23×D53—24no—
45C23×D53—41no—
46C23×D53—41no—
47C23×D53—42no—
48C23×D53—42no—
49C23×D53D414yescomplemented
50C23×D53C2312yes—
51C23×D53C2312yes—
52C2×D4×C53—24no—
53C2×D4×C53—24no—
54C2×D4×C53—24no—
55C2×D4×C53—24no—
56C2×D4×C53—41no—
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2312yes—
60C2×C22⋊D53—24no—
61C2×C22⋊D53—24no—
62C2×C22⋊D53—24no—
63C2×C22⋊D53—24no—
64C2×C22⋊D53—24no—
65C2×C22⋊D53—24no—
66C2×C22⋊D53—24no—
67C2×C22⋊D53—42no—
68C2×C22⋊D53D418yescomplemented
69C22×Dic53C2312yes—
70C22×Dic53C2312yes—
71D4×D53—264no—
72C2×D212—24no—
73C2×D212—24no—
74C2×D212—24no—
75C2×D212—41no—
76C2×D212D418yescomplemented
77C2×C4×D53—24no—
78C2×C4×D53—24no—
79C2×C4×D53—24no—
80C22⋊C4×C53—24no—
81C22⋊C4×C53—41no—
82C22⋊Dic53—24no—
83C22⋊Dic53—41no—
84#11719921—24no—
85#11719921—24no—
86#11719921—24no—
87#11719921—24no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53—22no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53C2×D412yes—
99C23×C53C2×D414yescomplemented
100C23×C53C2411yescharacteristic
101C22×D53—22no—
102C22×D53—22no—
103C22×D53—22no—
104C22×D53—22no—
105C22×D53—216no—
106C22×D53—22no—
107C22×D53—22no—
108C22×D53—22no—
109C22×D53—22no—
110C22×D53—22no—
111C22×D53—22no—
112C22×D53—216no—
113C22×D53—44no—
114C22×D53—44no—
115C22×D53—48no—
116C22×D53—44no—
117C22×D53—48no—
118C22×D53—48no—
119C22×D53—48no—
120C22×D53—48no—
121C22×D53—48no—
122C22×D53—48no—
123C22×D53C2×D414yes—
124C22×D53C2×D414yes—
125C22×D53C2×D414yes—
126C22×D53C2×D414yescomplemented
127D4×C53—216no—
128D4×C53—44no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53C2×D414yescomplemented
136C22⋊D53—216no—
137C22⋊D53—216no—
138C22⋊D53—48no—
139C2×Dic53—22no—
140C2×Dic53—22no—
141C2×Dic53—22no—
142C2×Dic53—22no—
143C2×Dic53—22no—
144C2×Dic53C2×D414yescomplemented
145D212—216no—
146D212—44no—
147C4×D53—216no—
148C22×C53—24no—
149C22×C53—24no—
150C22×C53—22no—
151C22×C53—24no—
152C22×C53—24no—
153C22×C53—42no—
154C22×C53—42no—
155C22×C53—41no—
156C22×C53—42no—
157C22×C53—42no—
158C22×C53—41no—
159C22×C53—42no—
160C22×C53C22×D411yescharacteristic
161C22×C53C22×D412yes—
162C22×C53C22×D412yes—
163C22×C53H(5,𝔽2)11yescharacteristic
164C22×C53H(5,𝔽2)11yescharacteristic
165C2×D53—24no—
166C2×D53—24no—
167C2×D53—24no—
168C2×D53—24no—
169C2×D53—24no—
170C2×D53—24no—
171C2×D53—24no—
172C2×D53—24no—
173C2×D53—24no—
174C2×D53—416no—
175C2×D53—432no—
176C2×D53—42no—
177C2×D53—42no—
178C2×D53—42no—
179C2×D53—42no—
180C2×D53#12018yescomplemented
181C4×C53—24no—
182C4×C53—41no—
183C4×C53—41no—
184Dic53—24no—
185Dic53—42no—
186#2403—531nomaximal, Sylow
187C2×C53—42no—
188C2×C53—44no—
189C2×C53C2×#12012yes—
190C2×C53#53411yescharacteristic
191C2×C53D4211yescharacteristic
192C2×C53D4212yes—
193C2×C53D4211yescharacteristic
194D53—28no—
195D53—44no—
196C23×D4—532no—
197C2×#120—532no—
198C2×#120—532no—
199#392—532no—
200#392—531no—
201#392D5311yescharacteristic, complemented
202#386—532no—
203#386—532no—
204#379—531no—
205C53#240311yescharacteristic, complemented, Sylow
206C25—531no—
207C25—532no—
208C22×D4—532no—
209C22×D4—532no—
210C22×D4—532no—
211C22×D4—532no—
212C22×D4—531no—
213C22×D4—1068no—
214C22×D4—1068no—
215C22×D4—1068no—
216C22×D4C2×D5312yescomplemented
217C22×D4C2×D5311yescharacteristic, complemented
218C23×C4—532no—
219#120—1068no—
220#120—1068no—
221C2×C22⋊C4—532no—
222C2×C22⋊C4—532no—
223C2×C22⋊C4—532no—
224C2×C22⋊C4—532no—
225C2×C22⋊C4—532no—
226C2×C22⋊C4—531no—
227C2×C22⋊C4C2×D5312yescomplemented
228C2×C22⋊C4C2×D5311yescharacteristic, complemented
229#95—531no—
230#95—532no—
231#95C2×D5311yescharacteristic, complemented
232C24—532no—
233C24—532no—
234C24—534no—
235C24—1064no—
236C24—1064no—
237C24—1064no—
238C24—1064no—
239C24—1064no—
240C24—1064no—
241C24—1064no—
242C24—2121no—
243C24—2122no—
244C24—2121no—
245C24—2122no—
246C24C22×D5311yescharacteristic
247C24C22×D5312yes—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—41no—
253C2×D4—538no—
254C2×D4—538no—
255C2×D4—1064no—
256C2×D4—1064no—
257C2×D4—1064no—
258C2×D4—1064no—
259C2×D4—10664no—
260C2×D4—1064no—
261C2×D4—1064no—
262C2×D4—1064no—
263C2×D4—1064no—
264C2×D4—1064no—
265C2×D4—1064no—
266C2×D4—2121no—
267C2×D4—2122no—
268C22×C4—532no—
269C22×C4—532no—
270C22×C4—1064no—
271C22×C4—1064no—
272C22×C4—1064no—
273C22×C4C22×D5311yescharacteristic
274C22×C4C22×D5311yescharacteristic
275C22×C4C22×D5312yes—
276C22⋊C4—24no—
277C22⋊C4—41no—
278C22⋊C4—1064no—
279C22⋊C4—1064no—
280C22⋊C4—1064no—
281C22⋊C4—1064no—
282C22⋊C4—1064no—
283C22⋊C4—2121no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—21no—
291C23—22no—
292C23—21no—
293C23—22no—
294C23—534no—
295C23—534no—
296C23—534no—
297C23—534no—
298C23—10616no—
299C23—10616no—
300C23—1062no—
301C23—1062no—
302C23—1062no—
303C23—1062no—
304C23—1062no—
305C23—1062no—
306C23—1062no—
307C23—1062no—
308C23—1062no—
309C23—1062no—
310C23—2124no—
311C23—2128no—
312C23—2128no—
313C23—2128no—
314C23—2124no—
315C23—2128no—
316C23—2128no—
317C23—2124no—
318C23—2128no—
319C23—2128no—
320C23D4×D5312yes—
321C23D4×D5314yescomplemented
322C23C23×D5311yescharacteristic
323D4—216no—
324D4—44no—
325D4—10616no—
326D4—10616no—
327D4—10616no—
328D4—2124no—
329D4—2128no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—534no—
337C2×C4—10616no—
338C2×C4—1062no—
339C2×C4—1062no—
340C2×C4—1062no—
341C2×C4—1062no—
342C2×C4—1062no—
343C2×C4D4×D5314yescomplemented
344C22—24no—
345C22—24no—
346C22—22no—
347C22—24no—
348C22—24no—
349C22—42no—
350C22—42no—
351C22—41no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—41no—
356C22—538no—
357C22—1064no—
358C22—1064no—
359C22—1064no—
360C22—1064no—
361C22—1064no—
362C22—1064no—
363C22—1064no—
364C22—1064no—
365C22—1064no—
366C22—21216no—
367C22—21232no—
368C22—2122no—
369C22—2122no—
370C22—2122no—
371C22—2122no—
372C22C2×D4×D5311yescharacteristic
373C22C2×D4×D5312yes—
374C22C2×D4×D5312yes—
375C22#1313326211yescharacteristic
376C22#1313326311yescharacteristic
377C4—24no—
378C4—41no—
379C4—41no—
380C4—1064no—
381C4—2122no—
382C2—42no—
383C2—44no—
384C2—1068no—
385C2—2124no—
386C2#120×D5312yes—
387C2#1681118111yescharacteristic
388C2#1681125511yescharacteristic
389C2#1681125712yes—
390C2#1681125811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2403
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×#13133187C2semidirect
3. C2×#13133185C2semidirect
4. #16810532C2semidirect
5. #16810531C2semidirect
6. #16810501C2semidirect
7. #16810499C2semidirect
8. #16810478C2semidirect
9. C53×#392C2semidirect
11. C24×D53C22semidirect
13. C22×D212C22semidirect
15. C22×C22⋊D53C22semidirect
16. C22×C22⋊D53C22semidirect
20. C22×C4×D53C22semidirect
23. C2×C22⋊Dic53C22semidirect
24. C2×C22⋊Dic53C22semidirect
25. C2×#11719921C22semidirect
26. C2×#11719921C22semidirect
27. C2×#11719921C22semidirect
28. C2×#11719921C22semidirect
29. #13133124C22semidirect
30. #13133124C22semidirect
31. C22×D4×C53C22semidirect
32. C22×D4×C53C22semidirect
33. C2×C22⋊C4×C53C22semidirect
34. C2×C22⋊C4×C53C22semidirect
35. #95×C53C22semidirect
49. C23×D53D4semidirect
68. C2×C22⋊D53D4semidirect
76. C2×D212D4semidirect
99. C23×C53C2×D4semidirect
126. C22×D53C2×D4semidirect
135. C2×C4×C53C2×D4semidirect
144. C2×Dic53C2×D4semidirect
180. C2×D53#120semidirect
201. #392D53semidirect
205. C53#2403semidirect
216. C22×D4C2×D53semidirect
217. C22×D4C2×D53semidirect
227. C2×C22⋊C4C2×D53semidirect
228. C2×C22⋊C4C2×D53semidirect
231. #95C2×D53semidirect
321. C23D4×D53semidirect
343. C2×C4D4×D53semidirect