Finite Group Databasebeta

← #20278364

Subgroups — #20278364

Order 6784.

1088 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1088 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×C22⋊D53C211yesmaximal, characteristic
2C2×#13133203C212yesmaximal, complemented
3C2×#13133194C211yesmaximal, characteristic, complemented
4#16810539C212yesmaximal, complemented
5#16810538C212yesmaximal, complemented
6#16810523C212yesmaximal, complemented
7#16810479C211yesmaximal, characteristic, complemented
8#16810478C211yesmaximal, characteristic, complemented
9#16810478C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
11C53×#379C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C22×Dic106C2211yescharacteristic, complemented
14C22×C22⋊D53—28no—
15C22×C22⋊D53—24no—
16C22×C22⋊D53C2212yes—
17C22×C22⋊D53C2212yes—
18C23×Dic53C2211yescharacteristic
19#13133203—28no—
20#13133194—24no—
21C2×C4⋊Dic53C2211yescharacteristic, complemented
22C2×#11719919C2211yescharacteristic, complemented
23C2×#11719919C2211yescharacteristic, complemented
24C2×C22⋊Dic53C2211yescharacteristic
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2212yescomplemented
27C2×C22⋊Dic53C2212yescomplemented
28C2×#11719921C2212yescomplemented
29C2×#11719921C2212yescomplemented
30C2×#11719921C2212yescomplemented
31C2×#11719921C2212yescomplemented
32C2×C4×Dic53C2212yescomplemented
33#13133124C2212yescomplemented
34C25×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2211yescharacteristic
37C2×C22⋊C4×C53C2212yescomplemented
38C2×C22⋊C4×C53C2212yescomplemented
39C24×C53—24no—
40C24×C53—22no—
41C24×C53—22no—
42C24×C53—22no—
43C24×C53—42no—
44C24×C53C2312yes—
45C24×C53C2311yescharacteristic
46C23×D53—22no—
47C23×D53—24no—
48C23×D53—22no—
49C23×D53—22no—
50C23×D53—42no—
51C23×D53C2312yes—
52C22×C4×C53C2312yes—
53C22×C4×C53C2311yescharacteristic
54C22×C4×C53C2311yescharacteristic
55C2×C22⋊D53—232no—
56C2×C22⋊D53—24no—
57C2×C22⋊D53—24no—
58C2×C22⋊D53—24no—
59C2×C22⋊D53—24no—
60C2×C22⋊D53—24no—
61C2×C22⋊D53D418yes—
62C22×Dic53—24no—
63C22×Dic53D414yes—
64C22×Dic53C2312yes—
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C2×Dic106—22no—
70C2×Dic106—24no—
71C22⋊C4×C53—22no—
72C22⋊C4×C53—24no—
73C22⋊C4×C53—22no—
74C22⋊C4×C53—42no—
75C22⋊Dic53—22no—
76C22⋊Dic53—24no—
77C22⋊Dic53—22no—
78C22⋊Dic53—42no—
79#11719921—24no—
80#11719921—24no—
81#11719921—42no—
82#11719921—42no—
83C4⋊Dic53—22no—
84#11719919—22no—
85#11719919—22no—
86C4×Dic53—24no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—28no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53—22no—
96C23×C53—48no—
97C23×C53—48no—
98C23×C53—44no—
99C23×C53—44no—
100C23×C53C2×D414yes—
101C23×C53C2×D412yes—
102C23×C53#4112yes—
103C23×C53#4112yes—
104C23×C53C2411yescharacteristic
105C22×D53—28no—
106C22×D53—22no—
107C22×D53—22no—
108C22×D53—22no—
109C22×D53—22no—
110C22×D53—22no—
111C22×D53—22no—
112C22×D53—48no—
113C22×D53—48no—
114C22×D53—44no—
115C22×D53—44no—
116C22×D53C2×D414yes—
117C2×C4×C53—21no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—21no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C22⋊D53—216no—
125C22⋊D53—216no—
126C2×Dic53—22no—
127C2×Dic53—28no—
128C2×Dic53—21no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—21no—
132C2×Dic53—22no—
133C2×Dic53—22no—
134C2×Dic53—22no—
135C2×Dic53C2×D412yes—
136C2×Dic53C2×D414yes—
137C2×Dic53C2×D412yes—
138Dic106—44no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—24no—
142C22×C53—24no—
143C22×C53—24no—
144C22×C53—24no—
145C22×C53—24no—
146C22×C53—416no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—42no—
151C22×C53#12414yescomplemented
152C22×C53C22×D411yescharacteristic
153C22×C53C22×D412yes—
154C22×C53C2×#4111yescharacteristic
155C22×C53C2×#4111yescharacteristic
156C22×C53H(5,𝔽2)12yes—
157C2×D53—24no—
158C2×D53—24no—
159C2×D53—24no—
160C2×D53—416no—
161C2×D53—42no—
162C2×D53—42no—
163C2×D53—42no—
164C2×D53—42no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53—42no—
168Dic53—22no—
169Dic53—41no—
170Dic53—41no—
171Dic53—42no—
172Dic53#12014yes—
173#2435—531nomaximal, Sylow
174C2×C53—24no—
175C2×C53—44no—
176C2×C53C2×#12011yescharacteristic
177C2×C53C2×#12411yescharacteristic
178C2×C53#53511yescharacteristic
179C2×C53#54612yes—
180C2×C53#54612yes—
181D53—44no—
182C23×D4—531no—
183C2×#124—532no—
184C2×C22⋊Q8—531no—
185#394—532no—
186#394—532no—
187#390—532no—
188#379—531no—
189#379—531no—
190#379—531no—
191#379D5311yescharacteristic, complemented
192C4×C22⋊C4—531no—
193C53#243511yescharacteristic, complemented, Sylow
194C25—531no—
195C25C2×D5311yescharacteristic
196C22×Q8—531no—
197C22×D4—532no—
198C22×D4—532no—
199C22×D4—1064no—
200C22×D4—1068no—
201C23×C4—531no—
202#124—1068no—
203C22⋊Q8—1064no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C22⋊C4—531no—
208C2×C22⋊C4—532no—
209C2×C22⋊C4—531no—
210C2×C22⋊C4—532no—
211C2×C22⋊C4—532no—
212C2×C22⋊C4—532no—
213C2×C22⋊C4—532no—
214C2×C22⋊C4—532no—
215C2×C22⋊C4C2×D5311yescharacteristic
216C2×C22⋊C4C2×D5311yescharacteristic
217C2×C22⋊C4C2×D5312yescomplemented
218C2×C22⋊C4C2×D5312yescomplemented
219C2×C42—532no—
220#95—532no—
221C24—24no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—42no—
226C24—532no—
227C24—1064no—
228C24—1062no—
229C24—1062no—
230C24—1062no—
231C24—2122no—
232C24C22×D5312yes—
233C24C22×D5311yescharacteristic
234C2×Q8—1062no—
235C2×Q8—1064no—
236C2×D4—538no—
237C2×D4—1064no—
238C2×D4—1064no—
239C2×D4—1064no—
240C2×D4—1064no—
241C2×D4—10632no—
242C2×D4—1064no—
243C22×C4—531no—
244C22×C4—534no—
245C22×C4—531no—
246C22×C4—532no—
247C22×C4—531no—
248C22×C4—531no—
249C22×C4—1064no—
250C22×C4C22×D5312yes—
251C22×C4C22×D5311yescharacteristic
252C22×C4C22×D5311yescharacteristic
253C4⋊C4—1062no—
254C4⋊C4—1062no—
255C4⋊C4—1062no—
256C22⋊C4—22no—
257C22⋊C4—24no—
258C22⋊C4—22no—
259C22⋊C4—42no—
260C22⋊C4—1064no—
261C22⋊C4—1064no—
262C22⋊C4—1062no—
263C22⋊C4—1062no—
264C22⋊C4—1064no—
265C22⋊C4—2122no—
266C22⋊C4—2122no—
267C22⋊C4—2122no—
268C42—1064no—
269C23—22no—
270C23—22no—
271C23—28no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—48no—
279C23—48no—
280C23—44no—
281C23—44no—
282C23—534no—
283C23—1068no—
284C23—1062no—
285C23—1062no—
286C23—1062no—
287C23—1062no—
288C23—1062no—
289C23—1062no—
290C23—2128no—
291C23—2128no—
292C23—2124no—
293C23—2124no—
294C23#1171994512yes—
295C23D4×D5314yes—
296C23D4×D5312yes—
297C23#1171994712yes—
298C23C23×D5311yescharacteristic
299Q8—2124no—
300D4—10616no—
301D4—10616no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—534no—
310C2×C4—532no—
311C2×C4—532no—
312C2×C4—1068no—
313C2×C4—1062no—
314C2×C4—1062no—
315C2×C4—1062no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1062no—
319C2×C4—1062no—
320C2×C4—1062no—
321C22—22no—
322C22—22no—
323C22—24no—
324C22—24no—
325C22—24no—
326C22—24no—
327C22—24no—
328C22—416no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—1064no—
334C22—1064no—
335C22—1064no—
336C22—21216no—
337C22—2122no—
338C22—2122no—
339C22—2122no—
340C22—2122no—
341C22#1313320314yescomplemented
342C22C2×D4×D5311yescharacteristic
343C22C2×D4×D5312yes—
344C22C2×#1171994711yescharacteristic
345C22C2×#1171994511yescharacteristic
346C22#1313326212yes—
347C4—41no—
348C4—41no—
349C4—42no—
350C4—534no—
351C4—1062no—
352C4—2121no—
353C4—2121no—
354C4—2122no—
355C2—24no—
356C2—44no—
357C2—2124no—
358C2#120×D5311yescharacteristic
359C2C2×#1313320311yescharacteristic
360C2#1681118811yescharacteristic
361C2#1681126012yes—
362C2#1681126312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#379C2
hypercenter#379D53
hypocenterC53#2435
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#13133203C2semidirect
3. C2×#13133194C2semidirect
4. #16810539C2semidirect
5. #16810538C2semidirect
6. #16810523C2semidirect
7. #16810479C2semidirect
8. #16810478C2semidirect
9. #16810478C2semidirect
10. C22⋊C4×Dic53C2semidirect
11. C53×#379C2semidirect
13. C22×Dic106C22semidirect
21. C2×C4⋊Dic53C22semidirect
22. C2×#11719919C22semidirect
23. C2×#11719919C22semidirect
26. C2×C22⋊Dic53C22semidirect
27. C2×C22⋊Dic53C22semidirect
28. C2×#11719921C22semidirect
29. C2×#11719921C22semidirect
30. C2×#11719921C22semidirect
31. C2×#11719921C22semidirect
32. C2×C4×Dic53C22semidirect
33. #13133124C22semidirect
37. C2×C22⋊C4×C53C22semidirect
38. C2×C22⋊C4×C53C22semidirect
151. C22×C53#124semidirect
191. #379D53semidirect
193. C53#2435semidirect
217. C2×C22⋊C4C2×D53semidirect
218. C2×C22⋊C4C2×D53semidirect
341. C22#13133203semidirect