Finite Group Databasebeta

← #20284654

Subgroups — #20284654

Order 6784.

462 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×D53C211yesmaximal, characteristic
2#117×D53C211yesmaximal, characteristic, complemented
3#16810948C211yesmaximal, characteristic, complemented
4#16810947C211yesmaximal, characteristic, complemented
5#16810944C211yesmaximal, characteristic, complemented
6#16810943C211yesmaximal, characteristic, complemented
7#16810940C211yesmaximal, characteristic, complemented
8#16810938C211yesmaximal, characteristic, complemented
9#16810935C211yesmaximal, characteristic, complemented
10#16810664C211yesmaximal, characteristic, complemented
11#16810662C211yesmaximal, characteristic, complemented
12#16810626C211yesmaximal, characteristic, complemented
13#16810626C211yesmaximal, characteristic, complemented
14#16810585C211yesmaximal, characteristic, complemented
15C53×#482C211yesmaximal, characteristic, complemented
16C2×D4×D53—21no—
17C2×D4×D53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19C2×#11719912C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21#13133192C2211yescharacteristic
22#13133191C2211yescharacteristic
23#13133190—21no—
24#13133189—21no—
25#13133188C2211yescharacteristic
26#13133176C2211yescharacteristic, complemented
27#13133175C2211yescharacteristic
28#13133175C2211yescharacteristic
29#13133174—21no—
30#13133173—21no—
31C4⋊C4×D53C2211yescharacteristic
32C4⋊C4×D53C2211yescharacteristic, complemented
33C4⋊C4×D53C2211yescharacteristic, complemented
34C22⋊C4×D53—21no—
35C22⋊C4×D53—21no—
36C42×D53—21no—
37#13133151C2211yescharacteristic, complemented
38#13133150C2211yescharacteristic, complemented
39#13133150C2211yescharacteristic, complemented
40#13133149C2211yescharacteristic, complemented
41#13133148C2211yescharacteristic, complemented
42#13133148C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic, complemented
46#13133138C2211yescharacteristic, complemented
47#13133137C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic, complemented
52#121×C53C2211yescharacteristic
53#117×C53C2211yescharacteristic, complemented
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C23×D53D411yescharacteristic
61C2×D4×C53—21no—
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C22×Dic53D411yescharacteristic
68D4×D53—22no—
69D4×D53—22no—
70D4×D53—44no—
71C2×D212—21no—
72C2×D212C2311yescharacteristic
73C2×C4×D53—22no—
74C2×C4×D53—22no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53D411yescharacteristic
79C2×C4×D53D411yescharacteristic
80C2×C4×D53C2311yescharacteristic
81C2×C4×D53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C22⋊C4×C53—21no—
88C22⋊C4×C53—21no—
89C42×C53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105C2×#88322C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107#11719912—22no—
108#11719912—22no—
109C23×C53—21no—
110C23×C53C2×D411yescharacteristic
111C22×D53—22no—
112C22×D53—22no—
113C22×D53—21no—
114C22×D53—41no—
115C22×D53—42no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53C2×D411yescharacteristic
119C22×D53C2×D411yescharacteristic
120D4×C53—21no—
121D4×C53—21no—
122D4×C53—42no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2411yescharacteristic
130C22⋊D53—42no—
131C22⋊D53—42no—
132C22⋊D53—42no—
133C2×Dic53—22no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139D212—21no—
140D212—21no—
141D212—42no—
142C4×D53—22no—
143C4×D53—21no—
144C4×D53—21no—
145C4×D53—22no—
146C4×D53—42no—
147C4×D53—42no—
148C4×D53—42no—
149C4×D53#4112yes—
150C4×D53#4112yes—
151C8×C53—21no—
152C8×C53—21no—
153#88322—21no—
154#88322—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53C22×D411yescharacteristic
162C2×D53—21no—
163C2×D53—21no—
164C2×D53—21no—
165C2×D53—21no—
166C2×D53—44no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—84no—
171C2×D53#12312yes—
172C2×D53AGL(1,ℤ/8)12yes—
173C4×C53—21no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53C2×#4111yescharacteristic
178C4×C53C2×#4111yescharacteristic
179Dic53—21no—
180Dic53—41no—
181Dic53—41no—
182Dic53—41no—
183Dic53#12312yes—
184#3094—531nomaximal, Sylow
185C2×C53—41no—
186C2×C53—81no—
187C2×C53C2×#12311yescharacteristic
188C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C53#57511yescharacteristic
190D53—22no—
191D53—22no—
192D53—81no—
193C2×#121—531no—
194C2×#117—531no—
195#482—531no—
196#482—531no—
197#482—531no—
198#482D5311yescharacteristic, complemented
199#481—531no—
200#481—531no—
201#480—531no—
202#480—531no—
203#428—531no—
204#428—531no—
205#418—531no—
206#418—531no—
207#407—531no—
208C53#309411yescharacteristic, complemented, Sylow
209C22×D4—531no—
210C22×D4—1061no—
211C23×C4—531no—
212C2×M16—531no—
213C2×M16—531no—
214#121—531no—
215#121—531no—
216#121—531no—
217#121—1061no—
218#121—1061no—
219#121C2×D5311yescharacteristic
220#117—531no—
221#117—531no—
222#117—531no—
223#117—1061no—
224#117—1061no—
225#117C2×D5311yescharacteristic, complemented
226C2×C4⋊C4—531no—
227C2×C4⋊C4—531no—
228C2×C4⋊C4—531no—
229C2×C22⋊C4—1061no—
230C2×C22⋊C4—1061no—
231C2×C42—1061no—
232#107—531no—
233#107—531no—
234#107—531no—
235#107C2×D5311yescharacteristic, complemented
236#106—531no—
237#106—531no—
238#106—531no—
239#106C2×D5311yescharacteristic, complemented
240#102—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102C2×D5311yescharacteristic, complemented
247#102C2×D5311yescharacteristic, complemented
248C22⋊C8—531no—
249C22⋊C8—531no—
250C22⋊C8—531no—
251C22⋊C8C2×D5311yescharacteristic, complemented
252C24—531no—
253C24—1061no—
254C2×D4—21no—
255C2×D4—531no—
256C2×D4—1061no—
257C2×D4—1062no—
258C2×D4—1062no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—2124no—
263C2×D4C22×D5311yescharacteristic
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—1062no—
270C22×C4—1062no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4C22×D5311yescharacteristic
275M16—1062no—
276M16—1062no—
277C2×C8—531no—
278C2×C8—531no—
279C2×C8C22×D5311yescharacteristic
280C2×C8C22×D5311yescharacteristic
281C4⋊C4—531no—
282C4⋊C4—531no—
283C4⋊C4—531no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4C22×D5311yescharacteristic
288C4⋊C4C22×D5311yescharacteristic
289C4⋊C4C22×D5311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C42—21no—
299C42—1061no—
300C42—1061no—
301C42—1061no—
302C23—21no—
303C23—531no—
304C23—531no—
305C23—1062no—
306C23—1062no—
307C23—1061no—
308C23—2121no—
309C23—2122no—
310C23—2122no—
311C23—2122no—
312C23D4×D5311yescharacteristic
313D4—21no—
314D4—21no—
315D4—42no—
316D4—1061no—
317D4—1061no—
318D4—2122no—
319D4—2122no—
320D4—2122no—
321D4—2122no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—531no—
328C2×C4—532no—
329C2×C4—532no—
330C2×C4—531no—
331C2×C4—1062no—
332C2×C4—1062no—
333C2×C4—1062no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4D4×D5311yescharacteristic
343C2×C4C23×D5311yescharacteristic
344C8—21no—
345C8—21no—
346C8—1061no—
347C8—1061no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—532no—
355C22—532no—
356C22—1061no—
357C22—1061no—
358C22—1061no—
359C22—1061no—
360C22—4244no—
361C22—2124no—
362C22—2121no—
363C22—2121no—
364C22—2121no—
365C22C2×D4×D5311yescharacteristic
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—532no—
371C4—1061no—
372C4—2121no—
373C4—2121no—
374C4—2121no—
375C4#41×D5311yescharacteristic
376C4#41×D5311yescharacteristic
377C2—41no—
378C2—81no—
379C2—1062no—
380C2—1062no—
381C2—4241no—
382C2#123×D5311yescharacteristic
383C2AGL(1,ℤ/8)×D5311yescharacteristic
384C2#1681142011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#482C2
socleC22×C53C22×D4
hypercenter#482D53
hypocenterC53#3094
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #117×D53C2semidirect
3. #16810948C2semidirect
4. #16810947C2semidirect
5. #16810944C2semidirect
6. #16810943C2semidirect
7. #16810940C2semidirect
8. #16810938C2semidirect
9. #16810935C2semidirect
10. #16810664C2semidirect
11. #16810662C2semidirect
12. #16810626C2semidirect
13. #16810626C2semidirect
14. #16810585C2semidirect
15. C53×#482C2semidirect
26. #13133176C22semidirect
32. C4⋊C4×D53C22semidirect
33. C4⋊C4×D53C22semidirect
37. #13133151C22semidirect
38. #13133150C22semidirect
39. #13133150C22semidirect
40. #13133149C22semidirect
41. #13133148C22semidirect
42. #13133148C22semidirect
43. #13133139C22semidirect
44. #13133139C22semidirect
45. #13133138C22semidirect
46. #13133138C22semidirect
47. #13133137C22semidirect
48. #13133137C22semidirect
49. C22⋊#88322C22semidirect
50. #13133129C22semidirect
51. #13133129C22semidirect
53. #117×C53C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. #102×C53C22semidirect
57. #102×C53C22semidirect
58. C22⋊C8×C53C22semidirect
198. #482D53semidirect
208. C53#3094semidirect
225. #117C2×D53semidirect
235. #107C2×D53semidirect
239. #106C2×D53semidirect
246. #102C2×D53semidirect
247. #102C2×D53semidirect
251. C22⋊C8C2×D53semidirect