Finite Group Databasebeta

← #20284721

Subgroups — #20284721

Order 6784.

448 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811177C211yesmaximal, characteristic, complemented
2C4×D4×D53C211yesmaximal, characteristic
3#16810955C211yesmaximal, characteristic, complemented
4#16810952C211yesmaximal, characteristic, complemented
5#16810912C211yesmaximal, characteristic, complemented
6#16810911C211yesmaximal, characteristic, complemented
7#16810827C211yesmaximal, characteristic, complemented
8#16810826C211yesmaximal, characteristic, complemented
9#16810775C211yesmaximal, characteristic
10#16810772C211yesmaximal, characteristic
11C4×#11719913C211yesmaximal, characteristic, complemented
12C4×#11719924C211yesmaximal, characteristic, complemented
13C4×#11719923C211yesmaximal, characteristic, complemented
14#16810674C211yesmaximal, characteristic, complemented
15C4×SD16×C53C211yesmaximal, characteristic, complemented
16C2×D4×D53C2211yescharacteristic
17C22×C4×D53—21no—
18C2×#11719913C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719924C2211yescharacteristic
21#13133216C2211yescharacteristic, complemented
22#13133215C2211yescharacteristic, complemented
23#13133207—21no—
24#13133197C2211yescharacteristic
25#13133195—21no—
26#13133193C2211yescharacteristic
27C4×Dic106—21no—
28C4×Dic106C2211yescharacteristic, complemented
29C4×D212C2211yescharacteristic
30D4×Dic53C2211yescharacteristic
31#13133179C2211yescharacteristic
32C4×C22⋊D53—21no—
33#13133177—21no—
34#13133174—21no—
35C4⋊C4×D53C2211yescharacteristic
36C22⋊C4×D53—21no—
37C42×D53C2211yescharacteristic
38#13133149C2211yescharacteristic, complemented
39#13133148C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133146C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133141C2211yescharacteristic, complemented
45#13133140C2211yescharacteristic, complemented
46#13133139C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51C4×#88322C2211yescharacteristic, complemented
52C2×SD16×C53C2211yescharacteristic
53C4×Q8×C53C2211yescharacteristic, complemented
54C4×D4×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic, complemented
56#103×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic
58C4×C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53—21no—
63C2×C22⋊D53—21no—
64C22×Dic53—21no—
65D4×D53—22no—
66D4×D53—22no—
67C2×D212C2311yescharacteristic
68C2×C4×D53—21no—
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53—44no—
72C2×C4×D53D411yescharacteristic
73C2×C4×D53D411yescharacteristic
74C2×C4×D53C2311yescharacteristic
75C2×Dic106—21no—
76C2×Dic106C2311yescharacteristic
77SD16×C53—22no—
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C4⋊C4×C53—21no—
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C42×C53—21no—
85C42×C53C2311yescharacteristic
86C22⋊Dic53—21no—
87#11719924—22no—
88#11719923—22no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102C4×Dic53—21no—
103C4×Dic53D411yescharacteristic
104C4×Dic53D411yescharacteristic
105C4×Dic53C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108#11719913—22no—
109C23×C53—21no—
110C22×D53—24no—
111C22×D53—21no—
112C22×D53—42no—
113C22×D53—42no—
114C22×D53C2×D411yescharacteristic
115Q8×C53—21no—
116Q8×C53—21no—
117D4×C53—21no—
118D4×C53#4112yes—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—42no—
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2411yescharacteristic
126C22⋊D53—24no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—42no—
131C2×Dic53C2×D411yescharacteristic
132C2×Dic53C2×D411yescharacteristic
133C2×Dic53C2×D411yescharacteristic
134D212—21no—
135D212#4112yes—
136C4×D53—21no—
137C4×D53—21no—
138C4×D53—41no—
139C4×D53—42no—
140C4×D53—42no—
141C4×D53—42no—
142C4×D53SD1614yescomplemented
143Dic106—21no—
144Dic106—21no—
145Dic106—42no—
146C8×C53—21no—
147C8×C53—21no—
148#88322—21no—
149#88322—21no—
150C22×C53—22no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53C22×D411yescharacteristic
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—22no—
157C2×D53—48no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53C2×SD1612yes—
161C4×C53—21no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53C2×SD1612yes—
166C4×C53C2×#4111yescharacteristic
167C4×C53C2×#4111yescharacteristic
168Dic53—21no—
169Dic53—21no—
170Dic53—41no—
171Dic53—41no—
172Dic53—41no—
173Dic53C2×SD1612yes—
174#3101—531nomaximal, Sylow
175C2×C53—42no—
176C2×C53#52511yescharacteristic
177C2×C53C22×SD1611yescharacteristic
178C2×C53#57511yescharacteristic
179D53—22no—
180D53—42no—
181#533—531no—
182C2×C4×D4—531no—
183#483—531no—
184#483—531no—
185#476—531no—
186#476—531no—
187#461—531no—
188#461—531no—
189#450—531no—
190#450—531no—
191C4×SD16—531no—
192C4×SD16—531no—
193C4×SD16—531no—
194C4×SD16D5311yescharacteristic, complemented
195#431—531no—
196C53#310111yescharacteristic, complemented, Sylow
197C22×D4—531no—
198C23×C4—1061no—
199C2×SD16—531no—
200C2×SD16—531no—
201C2×SD16—531no—
202C2×SD16C2×D5311yescharacteristic
203C4⋊Q8—531no—
204C4⋊Q8—531no—
205#125—1061no—
206C22⋊Q8—531no—
207C22⋊Q8—531no—
208C22⋊Q8—1061no—
209C4×Q8—531no—
210C4×Q8—1061no—
211C4×Q8C2×D5311yescharacteristic, complemented
212C4×D4—531no—
213C4×D4—531no—
214C4×D4—531no—
215C4×D4—1061no—
216C4×D4—1061no—
217C4×D4C2×D5311yescharacteristic
218#117—1061no—
219C2×C4⋊C4—531no—
220C2×C22⋊C4—1061no—
221C2×C42—531no—
222#106—531no—
223#106—531no—
224#106—531no—
225#106C2×D5311yescharacteristic, complemented
226C4⋊C8—531no—
227C4⋊C8—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#103C2×D5311yescharacteristic, complemented
232#102—531no—
233#102—531no—
234#102—531no—
235#102C2×D5311yescharacteristic
236C22⋊C8—531no—
237C22⋊C8—531no—
238C4×C8—531no—
239C4×C8C2×D5311yescharacteristic, complemented
240C24—1061no—
241C2×Q8—531no—
242C2×Q8—1061no—
243C2×Q8C22×D5311yescharacteristic
244C2×D4—531no—
245C2×D4—1061no—
246C2×D4—1062no—
247C2×D4—1062no—
248C2×D4C22×D5311yescharacteristic
249C22×C4—21no—
250C22×C4—531no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—2124no—
258SD16—22no—
259SD16—1062no—
260SD16—1062no—
261SD16—1062no—
262C2×C8—531no—
263C2×C8—531no—
264C2×C8C22×D5311yescharacteristic
265C2×C8C22×D5311yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—531no—
268C4⋊C4—531no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4C22×D5311yescharacteristic
277C4⋊C4C22×D5311yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C42—21no—
285C42—531no—
286C42—531no—
287C42—531no—
288C42—1061no—
289C42C22×D5311yescharacteristic
290C23—21no—
291C23—531no—
292C23—1064no—
293C23—1061no—
294C23—2122no—
295C23—2122no—
296Q8—21no—
297Q8—21no—
298Q8—1061no—
299Q8—1061no—
300Q8—2122no—
301D4—21no—
302D4—532no—
303D4—1064no—
304D4—1061no—
305D4#1171994512yes—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—531no—
311C2×C4—534no—
312C2×C4—531no—
313C2×C4—531no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—2121no—
320C2×C4—2122no—
321C2×C4—2122no—
322C2×C4—2122no—
323C2×C4—2122no—
324C2×C4D4×D5311yescharacteristic
325C2×C4D4×D5311yescharacteristic
326C2×C4C23×D5311yescharacteristic
327C8—21no—
328C8—21no—
329C8—1061no—
330C8—1061no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—532no—
335C22—1061no—
336C22—1061no—
337C22—1062no—
338C22—2128no—
339C22—2121no—
340C22—2121no—
341C22C2×D4×D5311yescharacteristic
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—532no—
347C4—1061no—
348C4—1061no—
349C4—2121no—
350C4—2121no—
351C4—2121no—
352C4SD16×D5312yes—
353C4#41×D5311yescharacteristic
354C4C2×#1171994511yescharacteristic
355C2—42no—
356C2—1062no—
357C2—2122no—
358C2#1681113211yescharacteristic
359C2C2×SD16×D5311yescharacteristic
360C2#1681142011yescharacteristic
subgroupG/H
centerC2×C4D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC4×SD16×C53C2
socleC22×C53C22×D4
hypercenterC4×SD16D53
hypocenterC53#3101
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #16811177C2semidirect
3. #16810955C2semidirect
4. #16810952C2semidirect
5. #16810912C2semidirect
6. #16810911C2semidirect
7. #16810827C2semidirect
8. #16810826C2semidirect
11. C4×#11719913C2semidirect
12. C4×#11719924C2semidirect
13. C4×#11719923C2semidirect
14. #16810674C2semidirect
15. C4×SD16×C53C2semidirect
21. #13133216C22semidirect
22. #13133215C22semidirect
28. C4×Dic106C22semidirect
38. #13133149C22semidirect
39. #13133148C22semidirect
40. #13133148C22semidirect
41. #13133146C22semidirect
42. #13133146C22semidirect
43. #13133142C22semidirect
44. #13133141C22semidirect
45. #13133140C22semidirect
51. C4×#88322C22semidirect
53. C4×Q8×C53C22semidirect
55. #106×C53C22semidirect
56. #103×C53C22semidirect
58. C4×C8×C53C22semidirect
142. C4×D53SD16semidirect
194. C4×SD16D53semidirect
196. C53#3101semidirect
211. C4×Q8C2×D53semidirect
225. #106C2×D53semidirect
231. #103C2×D53semidirect
239. C4×C8C2×D53semidirect