Finite Group Databasebeta

← #20280426

Subgroups — #20280426

Order 6784.

1134 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 153 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×#13133199C211yesmaximal, characteristic, complemented
3C2×#13133185C212yesmaximal, complemented
4#16810541C211yesmaximal, characteristic, complemented
5#16810539C212yesmaximal, complemented
6#16810537C211yesmaximal, characteristic, complemented
7#16810532C211yesmaximal, characteristic, complemented
8#16810531C212yesmaximal, complemented
9#16810485C211yesmaximal, characteristic, complemented
10#16810478C212yesmaximal, complemented
11C53×#392C211yesmaximal, characteristic, complemented
12C24×D53C2212yes—
13C22×D212C2211yescharacteristic
14C22×C22⋊D53C2212yes—
15C22×C22⋊D53C2211yescharacteristic
16C22×C22⋊D53C2212yescomplemented
17C2×D4×D53—28no—
18C2×D4×D53—24no—
19C22×C4×D53C2211yescharacteristic
20#13133199—24no—
21#13133185—28no—
22C2×C4⋊Dic53C2211yescharacteristic, complemented
23C2×#11719919C2211yescharacteristic, complemented
24C2×C22⋊Dic53C2211yescharacteristic, complemented
25C2×C22⋊Dic53C2212yescomplemented
26C2×C22⋊Dic53C2212yescomplemented
27C2×#11719921C2212yes—
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2212yescomplemented
30C2×#11719921C2212yescomplemented
31C2×#11719921C2212yescomplemented
32#13133124C2212yescomplemented
33#13133124C2211yescharacteristic, complemented
34C22×D4×C53C2211yescharacteristic
35C22×D4×C53C2212yescomplemented
36C2×C22⋊C4×C53C2211yescharacteristic, complemented
37C2×C22⋊C4×C53C2212yescomplemented
38#95×C53C2211yescharacteristic, complemented
39C24×C53C2311yescharacteristic
40C24×C53C2312yes—
41C23×D53—24no—
42C23×D53—24no—
43C23×D53—42no—
44C23×D53—42no—
45C23×D53—42no—
46C23×D53—42no—
47C23×D53D418yescomplemented
48C23×D53C2312yes—
49C23×D53C2311yescharacteristic
50C2×D4×C53—24no—
51C2×D4×C53—24no—
52C2×D4×C53—22no—
53C2×D4×C53—42no—
54C2×D4×C53—42no—
55C22×C4×C53C2312yes—
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C2×C22⋊D53—24no—
59C2×C22⋊D53—24no—
60C2×C22⋊D53—24no—
61C2×C22⋊D53—24no—
62C2×C22⋊D53—22no—
63C2×C22⋊D53—24no—
64C2×C22⋊D53—42no—
65C2×C22⋊D53—42no—
66C22×Dic53C2312yes—
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70D4×D53—232no—
71C2×D212—24no—
72C2×D212—22no—
73C2×C4×D53—24no—
74C2×C4×D53D414yescomplemented
75C22⋊C4×C53—22no—
76C22⋊C4×C53—42no—
77C22⋊Dic53—24no—
78C22⋊Dic53—22no—
79C22⋊Dic53—42no—
80#11719921—24no—
81#11719921—22no—
82#11719921—24no—
83#11719921—42no—
84#11719921—42no—
85C4⋊Dic53—22no—
86#11719919—22no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—22no—
97C23×C53—21no—
98C23×C53C2×D414yes—
99C23×C53C2411yescharacteristic
100C22×D53—22no—
101C22×D53—21no—
102C22×D53—21no—
103C22×D53—22no—
104C22×D53—22no—
105C22×D53—22no—
106C22×D53—22no—
107C22×D53—22no—
108C22×D53—22no—
109C22×D53—216no—
110C22×D53—48no—
111C22×D53—48no—
112C22×D53—48no—
113C22×D53—48no—
114C22×D53—48no—
115C22×D53—48no—
116C22×D53C2×D414yes—
117C22×D53C2×D414yes—
118C22×D53C2×D412yes—
119D4×C53—28no—
120D4×C53—48no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—21no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53C2×D412yes—
128C22⋊D53—216no—
129C22⋊D53—48no—
130C22⋊D53—44no—
131C2×Dic53—22no—
132C2×Dic53—21no—
133C2×Dic53—22no—
134C2×Dic53—22no—
135C2×Dic53—21no—
136C2×Dic53—22no—
137C2×Dic53—22no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53C2×D412yes—
141D212—28no—
142C4×D53—28no—
143C22×C53—24no—
144C22×C53—24no—
145C22×C53—24no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—41no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53C22×D412yes—
156C22×C53C22×D411yescharacteristic
157C22×C53H(5,𝔽2)12yes—
158C22×C53H(5,𝔽2)11yescharacteristic
159C22×C53H(5,𝔽2)11yescharacteristic
160C2×D53—24no—
161C2×D53—24no—
162C2×D53—22no—
163C2×D53—22no—
164C2×D53—24no—
165C2×D53—24no—
166C2×D53—24no—
167C2×D53—432no—
168C2×D53—42no—
169C2×D53—42no—
170C2×D53—42no—
171C2×D53—42no—
172C2×D53#12014yes—
173C4×C53—22no—
174C4×C53—42no—
175C4×C53—41no—
176Dic53—22no—
177Dic53—41no—
178Dic53—42no—
179Dic53—41no—
180#2613—531nomaximal, Sylow
181C2×C53—44no—
182C2×C53—81no—
183C2×C53C2×#12011yescharacteristic
184C2×C53#53412yes—
185C2×C53#53412yes—
186C2×C53#53511yescharacteristic
187C2×C53#53511yescharacteristic
188D53—24no—
189D53—44no—
190C23×D4—531no—
191C2×#123—531no—
192C2×#120—532no—
193#394—531no—
194#394—532no—
195#394—531no—
196#392—532no—
197#392—531no—
198#392D5311yescharacteristic, complemented
199#381—531no—
200#379—532no—
201C53#261311yescharacteristic, complemented, Sylow
202C25—532no—
203C22×D4—531no—
204C22×D4—532no—
205C22×D4—532no—
206C22×D4—531no—
207C22×D4—1064no—
208C22×D4—1068no—
209C22×D4C2×D5311yescharacteristic
210C22×D4C2×D5312yescomplemented
211C23×C4—531no—
212#123—1064no—
213#120—1068no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—531no—
216C2×C22⋊C4—532no—
217C2×C22⋊C4—531no—
218C2×C22⋊C4—532no—
219C2×C22⋊C4—532no—
220C2×C22⋊C4—531no—
221C2×C22⋊C4—532no—
222C2×C22⋊C4—532no—
223C2×C22⋊C4—532no—
224C2×C22⋊C4C2×D5311yescharacteristic, complemented
225C2×C22⋊C4C2×D5312yescomplemented
226#95—532no—
227#95—531no—
228#95C2×D5311yescharacteristic, complemented
229C24—531no—
230C24—532no—
231C24—538no—
232C24—1064no—
233C24—1064no—
234C24—2122no—
235C24—2122no—
236C24—2122no—
237C24—2122no—
238C24C22×D5311yescharacteristic
239C24C22×D5312yes—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—1064no—
246C2×D4—1064no—
247C2×D4—10632no—
248C2×D4—1064no—
249C2×D4—1064no—
250C2×D4—1064no—
251C2×D4—1064no—
252C2×D4—1062no—
253C2×D4—1062no—
254C2×D4—2122no—
255C2×D4—2122no—
256C22×C4—531no—
257C22×C4—534no—
258C22×C4—532no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—1064no—
262C22×C4C22×D5312yes—
263C22×C4C22×D5311yescharacteristic
264C22×C4C22×D5311yescharacteristic
265C4⋊C4—1062no—
266C4⋊C4—1062no—
267C22⋊C4—22no—
268C22⋊C4—42no—
269C22⋊C4—1064no—
270C22⋊C4—1064no—
271C22⋊C4—1062no—
272C22⋊C4—1062no—
273C22⋊C4—1064no—
274C22⋊C4—2122no—
275C22⋊C4—2122no—
276C22⋊C4—2122no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—21no—
284C23—22no—
285C23—21no—
286C23—22no—
287C23—21no—
288C23—532no—
289C23—534no—
290C23—534no—
291C23—10616no—
292C23—1062no—
293C23—1061no—
294C23—1061no—
295C23—1062no—
296C23—1062no—
297C23—1062no—
298C23—1062no—
299C23—1062no—
300C23—1062no—
301C23—2128no—
302C23—2128no—
303C23—2128no—
304C23—2128no—
305C23—2128no—
306C23—2128no—
307C23D4×D5314yes—
308C23C23×D5311yescharacteristic
309D4—28no—
310D4—48no—
311D4—1068no—
312D4—10616no—
313D4—2124no—
314D4—2128no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—532no—
322C2×C4—1068no—
323C2×C4—1062no—
324C2×C4—1061no—
325C2×C4—1062no—
326C2×C4—1062no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1062no—
330C2×C4—1061no—
331C2×C4—1062no—
332C2×C4D4×D5312yes—
333C22—24no—
334C22—24no—
335C22—24no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—42no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—534no—
346C22—1064no—
347C22—1064no—
348C22—1062no—
349C22—1062no—
350C22—1064no—
351C22—1064no—
352C22—1064no—
353C22—21232no—
354C22—2122no—
355C22—2122no—
356C22—2122no—
357C22—2122no—
358C22C2×D4×D5312yes—
359C22C2×D4×D5311yescharacteristic
360C22#1313326212yes—
361C22#1313326211yescharacteristic
362C22#1313326311yescharacteristic
363C4—22no—
364C4—42no—
365C4—41no—
366C4—1062no—
367C4—2121no—
368C4—2122no—
369C4—2121no—
370C2—44no—
371C2—81no—
372C2—1064no—
373C2—2124no—
374C2#120×D5311yescharacteristic
375C2#1681118112yes—
376C2#1681118312yes—
377C2#1681118611yescharacteristic
378C2#1681118911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2613
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×#13133199C2semidirect
3. C2×#13133185C2semidirect
4. #16810541C2semidirect
5. #16810539C2semidirect
6. #16810537C2semidirect
7. #16810532C2semidirect
8. #16810531C2semidirect
9. #16810485C2semidirect
10. #16810478C2semidirect
11. C53×#392C2semidirect
16. C22×C22⋊D53C22semidirect
22. C2×C4⋊Dic53C22semidirect
23. C2×#11719919C22semidirect
24. C2×C22⋊Dic53C22semidirect
25. C2×C22⋊Dic53C22semidirect
26. C2×C22⋊Dic53C22semidirect
29. C2×#11719921C22semidirect
30. C2×#11719921C22semidirect
31. C2×#11719921C22semidirect
32. #13133124C22semidirect
33. #13133124C22semidirect
35. C22×D4×C53C22semidirect
36. C2×C22⋊C4×C53C22semidirect
37. C2×C22⋊C4×C53C22semidirect
38. #95×C53C22semidirect
47. C23×D53D4semidirect
74. C2×C4×D53D4semidirect
198. #392D53semidirect
201. C53#2613semidirect
210. C22×D4C2×D53semidirect
224. C2×C22⋊C4C2×D53semidirect
225. C2×C22⋊C4C2×D53semidirect
228. #95C2×D53semidirect