Finite Group Databasebeta

← #20284510

Subgroups — #20284510

Order 6784.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811179C211yesmaximal, characteristic, complemented
2C4×D4×D53C211yesmaximal, characteristic
3#16810924C211yesmaximal, characteristic, complemented
4#16810923C211yesmaximal, characteristic, complemented
5#16810920C211yesmaximal, characteristic, complemented
6#16810912C211yesmaximal, characteristic, complemented
7#16810911C211yesmaximal, characteristic, complemented
8#16810902C211yesmaximal, characteristic, complemented
9#16810901C211yesmaximal, characteristic, complemented
10#16810664C211yesmaximal, characteristic, complemented
11#16810662C211yesmaximal, characteristic, complemented
12#16810645C211yesmaximal, characteristic, complemented
13#16810626C211yesmaximal, characteristic
14#16810626C211yesmaximal, characteristic
15C53×#478C211yesmaximal, characteristic, complemented
16C2×D4×D53C2211yescharacteristic
17C22×C4×D53—21no—
18C2×#11719912C2211yescharacteristic
19C2×#11719912C2211yescharacteristic
20#13133216C2211yescharacteristic, complemented
21#13133216C2211yescharacteristic, complemented
22#13133209C2211yescharacteristic, complemented
23#13133195—21no—
24#13133195—21no—
25#13133182—21no—
26#13133182—21no—
27C4×D212C2211yescharacteristic
28D4×Dic53C2211yescharacteristic
29#13133179C2211yescharacteristic
30C4×C22⋊D53—21no—
31#13133177—21no—
32#13133175C2211yescharacteristic
33#13133175C2211yescharacteristic
34C4⋊C4×D53C2211yescharacteristic
35C22⋊C4×D53—21no—
36C42×D53C2211yescharacteristic
37#13133151C2211yescharacteristic, complemented
38#13133150C2211yescharacteristic, complemented
39#13133150C2211yescharacteristic, complemented
40#13133149C2211yescharacteristic, complemented
41#13133148C2211yescharacteristic, complemented
42#13133148C2211yescharacteristic, complemented
43C4⋊#88322C2211yescharacteristic, complemented
44#13133146C2211yescharacteristic, complemented
45#13133146C2211yescharacteristic, complemented
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133138C2211yescharacteristic
50#13133137C2211yescharacteristic
51#13133137C2211yescharacteristic
52#125×C53C2211yescharacteristic, complemented
53C4×D4×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56C4⋊C8×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C23×D53—21no—
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C2×C22⋊D53—21no—
63C22×Dic53—21no—
64D4×D53—22no—
65D4×D53Q814yescomplemented
66C2×D212C2311yescharacteristic
67C2×C4×D53—21no—
68C2×C4×D53—21no—
69C2×C4×D53—21no—
70C2×C4×D53—44no—
71C2×C4×D53D411yescharacteristic
72C2×C4×D53D411yescharacteristic
73C2×C4×D53C2311yescharacteristic
74C2×Dic106—21no—
75C2×Dic106—21no—
76C2×C8×C53C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C4⋊C4×C53—21no—
79C4⋊C4×C53—21no—
80C4⋊C4×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C42×C53C2311yescharacteristic
85C22⋊Dic53—21no—
86#11719921—21no—
87#11719921—21no—
88#11719921—21no—
89#11719921—21no—
90C4⋊Dic53C2311yescharacteristic
91C4⋊Dic53C2311yescharacteristic
92C4⋊Dic53C2311yescharacteristic
93#11719919—21no—
94#11719919—21no—
95#11719919—21no—
96#11719919—21no—
97#11719919—21no—
98C4×Dic53—21no—
99C4×Dic53—21no—
100C4×Dic53D411yescharacteristic
101C4×Dic53D411yescharacteristic
102C4×Dic53C2311yescharacteristic
103C2×#88322C2311yescharacteristic
104C2×#88322C2311yescharacteristic
105#11719912—22no—
106#11719912—22no—
107C23×C53—21no—
108C22×D53—21no—
109C22×D53—24no—
110C22×D53—42no—
111C22×D53—42no—
112C22×D53C2×D411yescharacteristic
113D4×C53—21no—
114D4×C53C2×Q812yes—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—42no—
119C2×C4×C53C2×D411yescharacteristic
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2411yescharacteristic
122C22⋊D53—24no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—42no—
127C2×Dic53C2×D411yescharacteristic
128C2×Dic53C2×D411yescharacteristic
129C2×Dic53C2×D411yescharacteristic
130D212—21no—
131D212C2×Q812yes—
132C4×D53—22no—
133C4×D53—41no—
134C4×D53—42no—
135C4×D53—42no—
136C4×D53—42no—
137C4×D53C2×Q812yes—
138C4×D53#4112yes—
139Dic106—42no—
140Dic106—42no—
141C8×C53—21no—
142C8×C53—21no—
143#88322—21no—
144#88322—21no—
145C22×C53—22no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53C22×D411yescharacteristic
149C2×D53—21no—
150C2×D53—21no—
151C2×D53—22no—
152C2×D53—48no—
153C2×D53—41no—
154C2×D53—41no—
155C2×D53C22⋊Q812yes—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53C22×Q811yescharacteristic
162C4×C53C2×#4111yescharacteristic
163Dic53—21no—
164Dic53—41no—
165Dic53—41no—
166Dic53—41no—
167Dic53C22⋊Q812yes—
168Dic53AGL(1,ℤ/8)12yes—
169#3077—531nomaximal, Sylow
170C2×C53—42no—
171C2×C53C2×C22⋊Q811yescharacteristic
172C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
173C2×C53#57511yescharacteristic
174D53—22no—
175D53—42no—
176#533—531no—
177C2×C4×D4—531no—
178#478—531no—
179#478—531no—
180#478—531no—
181#478D5311yescharacteristic, complemented
182#476—531no—
183#476—531no—
184#474—531no—
185#474—531no—
186#428—531no—
187#428—531no—
188#423—531no—
189#418—531no—
190#418—531no—
191C53#307711yescharacteristic, complemented, Sylow
192C22×D4—531no—
193C23×C4—1061no—
194C2×M16—531no—
195C2×M16—531no—
196C4⋊Q8—531no—
197C4⋊Q8—531no—
198#125—531no—
199#125C2×D5311yescharacteristic, complemented
200C22⋊Q8—1061no—
201C22⋊Q8—1061no—
202C4×Q8—1061no—
203C4×Q8—1061no—
204C4×D4—531no—
205C4×D4—531no—
206C4×D4—531no—
207C4×D4—1061no—
208C4×D4—1061no—
209C4×D4C2×D5311yescharacteristic
210#117—531no—
211#117—531no—
212C2×C4⋊C4—531no—
213C2×C22⋊C4—1061no—
214C2×C42—531no—
215#107—531no—
216#107—531no—
217#107—531no—
218#107C2×D5311yescharacteristic, complemented
219#106—531no—
220#106—531no—
221#106—531no—
222#106C2×D5311yescharacteristic, complemented
223C4⋊C8—531no—
224C4⋊C8—531no—
225C4⋊C8—531no—
226C4⋊C8C2×D5311yescharacteristic, complemented
227#102—531no—
228#102—531no—
229#102—531no—
230#102—531no—
231#102—531no—
232#102—531no—
233#102C2×D5311yescharacteristic
234#102C2×D5311yescharacteristic
235C24—1061no—
236C2×Q8—1061no—
237C2×Q8—1061no—
238C2×D4—534no—
239C2×D4—531no—
240C2×D4—1062no—
241C2×D4—1061no—
242C2×D4C22×D5311yescharacteristic
243C22×C4—21no—
244C22×C4—531no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—1061no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4—2124no—
252M16—1062no—
253M16—1062no—
254C2×C8—531no—
255C2×C8—531no—
256C2×C8C22×D5311yescharacteristic
257C2×C8C22×D5311yescharacteristic
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—531no—
261C4⋊C4—531no—
262C4⋊C4—531no—
263C4⋊C4—1061no—
264C4⋊C4—1061no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4C22×D5311yescharacteristic
269C4⋊C4C22×D5311yescharacteristic
270C4⋊C4C22×D5311yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—1061no—
273C22⋊C4—1061no—
274C22⋊C4—1061no—
275C22⋊C4—1061no—
276C22⋊C4—1061no—
277C42—531no—
278C42—531no—
279C42—531no—
280C42—1061no—
281C42—1061no—
282C42C22×D5311yescharacteristic
283C23—21no—
284C23—531no—
285C23—1064no—
286C23—1061no—
287C23—2122no—
288C23—2122no—
289Q8—2122no—
290Q8—2122no—
291D4—21no—
292D4—532no—
293D4—1064no—
294D4—1061no—
295D4Q8×D5312yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—531no—
301C2×C4—531no—
302C2×C4—532no—
303C2×C4—532no—
304C2×C4—531no—
305C2×C4—1062no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—2121no—
310C2×C4—2122no—
311C2×C4—2122no—
312C2×C4—2122no—
313C2×C4—2122no—
314C2×C4D4×D5311yescharacteristic
315C2×C4D4×D5311yescharacteristic
316C2×C4C23×D5311yescharacteristic
317C8—21no—
318C8—21no—
319C8—1061no—
320C8—1061no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—532no—
325C22—1061no—
326C22—1061no—
327C22—1062no—
328C22—2128no—
329C22—2121no—
330C22—2121no—
331C22C2×D4×D5311yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—532no—
338C4—532no—
339C4—1061no—
340C4—2121no—
341C4—2121no—
342C4—2121no—
343C4C2×Q8×D5311yescharacteristic
344C4#41×D5311yescharacteristic
345C2—42no—
346C2—1062no—
347C2—2122no—
348C2C22⋊Q8×D5311yescharacteristic
349C2AGL(1,ℤ/8)×D5311yescharacteristic
350C2#1681142011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#478C2
socleC22×C53C22×D4
hypercenter#478D53
hypocenterC53#3077
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #16811179C2semidirect
3. #16810924C2semidirect
4. #16810923C2semidirect
5. #16810920C2semidirect
6. #16810912C2semidirect
7. #16810911C2semidirect
8. #16810902C2semidirect
9. #16810901C2semidirect
10. #16810664C2semidirect
11. #16810662C2semidirect
12. #16810645C2semidirect
15. C53×#478C2semidirect
20. #13133216C22semidirect
21. #13133216C22semidirect
22. #13133209C22semidirect
37. #13133151C22semidirect
38. #13133150C22semidirect
39. #13133150C22semidirect
40. #13133149C22semidirect
41. #13133148C22semidirect
42. #13133148C22semidirect
43. C4⋊#88322C22semidirect
44. #13133146C22semidirect
45. #13133146C22semidirect
52. #125×C53C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. C4⋊C8×C53C22semidirect
65. D4×D53Q8semidirect
181. #478D53semidirect
191. C53#3077semidirect
199. #125C2×D53semidirect
218. #107C2×D53semidirect
222. #106C2×D53semidirect
226. C4⋊C8C2×D53semidirect