Finite Group Databasebeta

← #20285516

Subgroups — #20285516

Order 6784.

462 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D53C211yesmaximal, characteristic, complemented
2#121×D53C211yesmaximal, characteristic
3#16810960C211yesmaximal, characteristic, complemented
4#16810957C211yesmaximal, characteristic, complemented
5#16810938C211yesmaximal, characteristic, complemented
6#16810935C211yesmaximal, characteristic, complemented
7#16810871C211yesmaximal, characteristic, complemented
8#16810869C211yesmaximal, characteristic, complemented
9#16810867C211yesmaximal, characteristic, complemented
10#16810775C211yesmaximal, characteristic, complemented
11#16810772C211yesmaximal, characteristic, complemented
12#16810763C211yesmaximal, characteristic, complemented
13#16810760C211yesmaximal, characteristic, complemented
14#16810586C211yesmaximal, characteristic, complemented
15C53×#468C211yesmaximal, characteristic, complemented
16C2×Q8×D53C2211yescharacteristic, complemented
17C2×D4×D53—21no—
18C2×D4×D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20C2×#11719913C2211yescharacteristic, complemented
21C2×#11719923C2211yescharacteristic, complemented
22C2×#11719924C2211yescharacteristic, complemented
23C2×#11719917C2211yescharacteristic, complemented
24#13133197C2211yescharacteristic
25#13133196C2211yescharacteristic, complemented
26#13133195—21no—
27#13133194—21no—
28#13133193C2211yescharacteristic
29#13133192C2211yescharacteristic
30#13133191C2211yescharacteristic
31#13133190—21no—
32#13133189—21no—
33#13133188C2211yescharacteristic
34C4⋊C4×D53—21no—
35C4⋊C4×D53C2211yescharacteristic
36C4⋊C4×D53C2211yescharacteristic, complemented
37C22⋊C4×D53—21no—
38C22⋊C4×D53—21no—
39#13133151C2211yescharacteristic, complemented
40#13133150C2211yescharacteristic, complemented
41#13133150C2211yescharacteristic, complemented
42#13133142C2211yescharacteristic, complemented
43#13133141C2211yescharacteristic, complemented
44#13133140C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic, complemented
46#13133138C2211yescharacteristic, complemented
47#13133137C2211yescharacteristic, complemented
48#13133129C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic, complemented
52C2×SD16×C53C2211yescharacteristic, complemented
53C2×M16×C53C2211yescharacteristic, complemented
54C22⋊Q8×C53C2211yescharacteristic, complemented
55#121×C53C2211yescharacteristic
56#107×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C23×D53D411yescharacteristic
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C22×Dic53D411yescharacteristic
69Q8×D53—41no—
70Q8×D53—41no—
71D4×D53—41no—
72D4×D53—41no—
73D4×D53—44no—
74C2×D212—21no—
75C2×D212C2311yescharacteristic
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C2×C4×D53D411yescharacteristic
82C2×C4×D53D411yescharacteristic
83C2×C4×D53C2311yescharacteristic
84C2×C4×D53C2311yescharacteristic
85C2×Dic106—21no—
86C2×Dic106C2311yescharacteristic
87SD16×C53—41no—
88M16×C53—41no—
89C2×C8×C53C2311yescharacteristic
90C2×C8×C53C2311yescharacteristic
91C4⋊C4×C53—21no—
92C4⋊C4×C53C2311yescharacteristic
93C4⋊C4×C53C2311yescharacteristic
94C22⋊C4×C53—21no—
95C22⋊C4×C53—21no—
96C22⋊Dic53—21no—
97C22⋊Dic53—21no—
98#11719924—41no—
99#11719923—41no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104C4⋊Dic53—21no—
105C4⋊Dic53C2311yescharacteristic
106C4⋊Dic53C2311yescharacteristic
107#11719919—21no—
108#11719919—21no—
109#11719919—21no—
110#11719919—21no—
111#11719917—41no—
112C2×#88322C2311yescharacteristic
113C2×#88322C2311yescharacteristic
114#11719913—41no—
115C23×C53—21no—
116C23×C53C2×D411yescharacteristic
117C22×D53—22no—
118C22×D53—22no—
119C22×D53—21no—
120C22×D53—41no—
121C22×D53—42no—
122C22×D53—42no—
123C22×D53—42no—
124C22×D53C2×D411yescharacteristic
125C22×D53C2×D411yescharacteristic
126Q8×C53—21no—
127Q8×C53—21no—
128D4×C53—21no—
129D4×C53—21no—
130D4×C53—42no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2411yescharacteristic
138C22⋊D53—42no—
139C22⋊D53—42no—
140C22⋊D53—42no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—41no—
145C2×Dic53C2×D411yescharacteristic
146C2×Dic53C2×D411yescharacteristic
147D212—21no—
148D212—21no—
149D212—42no—
150C4×D53—21no—
151C4×D53—21no—
152C4×D53—21no—
153C4×D53—21no—
154C4×D53—41no—
155C4×D53—41no—
156C4×D53—42no—
157C4×D53—42no—
158C4×D53—42no—
159Dic106—21no—
160Dic106—21no—
161Dic106—42no—
162C8×C53—21no—
163C8×C53—21no—
164#88322—21no—
165#88322—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53C22×D411yescharacteristic
173C2×D53—21no—
174C2×D53—21no—
175C2×D53—21no—
176C2×D53—21no—
177C2×D53—44no—
178C2×D53—41no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—84no—
182C2×D53AGL(1,ℤ/8)12yes—
183C2×D53#13712yes—
184C4×C53—21no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53H(5,𝔽2)11yescharacteristic
189C4×C53H(5,𝔽2)11yescharacteristic
190Dic53—21no—
191Dic53—21no—
192Dic53—41no—
193Dic53—41no—
194Dic53—41no—
195#3193—531nomaximal, Sylow
196C2×C53—41no—
197C2×C53—81no—
198C2×C53#53411yescharacteristic
199C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
200C2×C53C2×#13711yescharacteristic
201D53—22no—
202D53—22no—
203D53—81no—
204C2×C22⋊Q8—531no—
205C2×#121—531no—
206#484—531no—
207#484—531no—
208#480—531no—
209#480—531no—
210#468—531no—
211#468—531no—
212#468—531no—
213#468D5311yescharacteristic, complemented
214#450—531no—
215#450—531no—
216#448—531no—
217#448—531no—
218#407—531no—
219C53#319311yescharacteristic, complemented, Sylow
220C22×Q8—531no—
221C22×D4—531no—
222C22×D4—1061no—
223C23×C4—531no—
224C2×SD16—531no—
225C2×SD16—531no—
226C2×SD16—531no—
227C2×SD16C2×D5311yescharacteristic, complemented
228C2×M16—531no—
229C2×M16C2×D5311yescharacteristic, complemented
230C22⋊Q8—531no—
231C22⋊Q8—531no—
232C22⋊Q8—531no—
233C22⋊Q8—1061no—
234C22⋊Q8—1061no—
235C22⋊Q8C2×D5311yescharacteristic, complemented
236#121—531no—
237#121—531no—
238#121—531no—
239#121—1061no—
240#121—1061no—
241#121C2×D5311yescharacteristic
242C2×C4⋊C4—531no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4—1061no—
245C2×C22⋊C4—1061no—
246C2×C22⋊C4—1061no—
247#107—531no—
248#107—531no—
249#107—531no—
250#107C2×D5311yescharacteristic, complemented
251#103—531no—
252#103—531no—
253#103—531no—
254#103C2×D5311yescharacteristic, complemented
255#102—531no—
256#102—531no—
257#102—531no—
258#102C2×D5311yescharacteristic, complemented
259C22⋊C8—531no—
260C22⋊C8—531no—
261C22⋊C8—531no—
262C22⋊C8—531no—
263C24—531no—
264C24—1061no—
265C2×Q8—531no—
266C2×Q8—1061no—
267C2×Q8—2121no—
268C2×Q8—2121no—
269C2×Q8C22×D5311yescharacteristic
270C2×D4—21no—
271C2×D4—531no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—2121no—
277C2×D4—2121no—
278C2×D4—2124no—
279C2×D4C22×D5311yescharacteristic
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—531no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—2121no—
289C22×C4—2121no—
290C22×C4C22×D5311yescharacteristic
291SD16—41no—
292SD16—2121no—
293SD16—2121no—
294SD16—2121no—
295M16—41no—
296M16—2121no—
297C2×C8—531no—
298C2×C8—531no—
299C2×C8C22×D5311yescharacteristic
300C2×C8C22×D5311yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—531no—
303C4⋊C4—531no—
304C4⋊C4—1061no—
305C4⋊C4—1061no—
306C4⋊C4—1061no—
307C4⋊C4—1061no—
308C4⋊C4—1061no—
309C4⋊C4C22×D5311yescharacteristic
310C4⋊C4C22×D5311yescharacteristic
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—1061no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C22⋊C4—1061no—
318C22⋊C4—1061no—
319C23—21no—
320C23—531no—
321C23—531no—
322C23—1062no—
323C23—1062no—
324C23—1061no—
325C23—2121no—
326C23—2122no—
327C23—2122no—
328C23—2122no—
329C23D4×D5311yescharacteristic
330Q8—21no—
331Q8—21no—
332Q8—1061no—
333Q8—1061no—
334Q8—2122no—
335D4—21no—
336D4—21no—
337D4—42no—
338D4—1061no—
339D4—1061no—
340D4—2122no—
341D4—2122no—
342D4—2122no—
343D4—2122no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—531no—
350C2×C4—531no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4—2122no—
363C2×C4—2121no—
364C2×C4D4×D5311yescharacteristic
365C2×C4C23×D5311yescharacteristic
366C8—21no—
367C8—21no—
368C8—1061no—
369C8—1061no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—532no—
377C22—532no—
378C22—1061no—
379C22—1061no—
380C22—1061no—
381C22—1061no—
382C22—4244no—
383C22—2124no—
384C22—2121no—
385C22—2121no—
386C22—2121no—
387C22C2×D4×D5311yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—1061no—
393C4—1061no—
394C4—2121no—
395C4—2121no—
396C4—2121no—
397C4#1313326211yescharacteristic
398C4#1313326311yescharacteristic
399C2—41no—
400C2—81no—
401C2—1062no—
402C2—1062no—
403C2—4241no—
404C2#1681118111yescharacteristic
405C2AGL(1,ℤ/8)×D5311yescharacteristic
406C2#137×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#468C2
socleC22×C53C22×D4
hypercenter#468D53
hypocenterC53#3193
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D53C2semidirect
3. #16810960C2semidirect
4. #16810957C2semidirect
5. #16810938C2semidirect
6. #16810935C2semidirect
7. #16810871C2semidirect
8. #16810869C2semidirect
9. #16810867C2semidirect
10. #16810775C2semidirect
11. #16810772C2semidirect
12. #16810763C2semidirect
13. #16810760C2semidirect
14. #16810586C2semidirect
15. C53×#468C2semidirect
16. C2×Q8×D53C22semidirect
20. C2×#11719913C22semidirect
21. C2×#11719923C22semidirect
22. C2×#11719924C22semidirect
23. C2×#11719917C22semidirect
25. #13133196C22semidirect
36. C4⋊C4×D53C22semidirect
39. #13133151C22semidirect
40. #13133150C22semidirect
41. #13133150C22semidirect
42. #13133142C22semidirect
43. #13133141C22semidirect
44. #13133140C22semidirect
45. #13133139C22semidirect
46. #13133138C22semidirect
47. #13133137C22semidirect
50. #13133129C22semidirect
51. #13133129C22semidirect
52. C2×SD16×C53C22semidirect
53. C2×M16×C53C22semidirect
54. C22⋊Q8×C53C22semidirect
56. #107×C53C22semidirect
57. #103×C53C22semidirect
58. #102×C53C22semidirect
213. #468D53semidirect
219. C53#3193semidirect
227. C2×SD16C2×D53semidirect
229. C2×M16C2×D53semidirect
235. C22⋊Q8C2×D53semidirect
250. #107C2×D53semidirect
254. #103C2×D53semidirect
258. #102C2×D53semidirect