Finite Group Databasebeta

← #20286754

Subgroups — #20286754

Order 6784.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811341C211yesmaximal, characteristic, complemented
2#16811264C211yesmaximal, characteristic
3#16810984C211yesmaximal, characteristic, complemented
4#16810950C211yesmaximal, characteristic, complemented
5#16810912C211yesmaximal, characteristic, complemented
6#16810879C211yesmaximal, characteristic, complemented
7#16810853C211yesmaximal, characteristic, complemented
8#16810848C211yesmaximal, characteristic, complemented
9#16810788C211yesmaximal, characteristic, complemented
10#16810776C211yesmaximal, characteristic, complemented
11#16810722C211yesmaximal, characteristic, complemented
12#16810697C211yesmaximal, characteristic, complemented
13#16810627C211yesmaximal, characteristic, complemented
14#16810618C211yesmaximal, characteristic, complemented
15C53×#485C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C2×#11719925C2211yescharacteristic, complemented
19C2×#11719913C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21C2×#11719917C2211yescharacteristic
22C2×C8×D53C2211yescharacteristic, complemented
23#13133216C2211yescharacteristic, complemented
24#13133211—21no—
25#13133210—21no—
26#13133206C2211yescharacteristic
27#13133203—21no—
28#13133202—21no—
29#13133197C2211yescharacteristic, complemented
30#13133196C2211yescharacteristic
31#13133192—21no—
32#13133192C2211yescharacteristic
33#13133187—21no—
34C4×Dic106—21no—
35#13133182C2211yescharacteristic, complemented
36#13133179C2211yescharacteristic
37C4×C22⋊D53C2211yescharacteristic
38#13133175C2211yescharacteristic, complemented
39C22⋊C4×D53—21no—
40#13133148C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133141C2211yescharacteristic, complemented
43#13133140C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51#13133127C2211yescharacteristic, complemented
52C22⋊Q8×C53C2211yescharacteristic
53#117×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56#103×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53D411yescharacteristic
63#11719949—22no—
64#11719949—41no—
65C2×D212—22no—
66C2×D212—21no—
67C2×D212—21no—
68C2×D212—21no—
69C2×D212—41no—
70C2×D212C2311yescharacteristic
71C2×D212C2311yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53C2311yescharacteristic
75C2×Dic106—21no—
76C2×Dic106C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C4⋊C4×C53—21no—
80C4⋊C4×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C42×C53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53D411yescharacteristic
88#11719925—41no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921D411yescharacteristic
96C4⋊Dic53—21no—
97C4⋊Dic53C2311yescharacteristic
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919D411yescharacteristic
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53C2311yescharacteristic
105#11719917—41no—
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108#11719913—22no—
109#11719913—41no—
110C8×D53—41no—
111C23×C53C2×D411yescharacteristic
112C22×D53—21no—
113C22×D53—21no—
114C22×D53—42no—
115C22×D53—42no—
116C22×D53—81no—
117C22×D53C2×D411yescharacteristic
118Q8×C53—21no—
119Q8×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2411yescharacteristic
128C22⋊D53—41no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53C2×D411yescharacteristic
132C2×Dic53C2×D411yescharacteristic
133C2×Dic53C2×D411yescharacteristic
134D212—21no—
135D212—21no—
136D212—21no—
137D212—42no—
138D212—42no—
139D212—81no—
140D212#4112yes—
141C4×D53—21no—
142C4×D53—21no—
143C4×D53—42no—
144C4×D53—81no—
145Dic106—21no—
146Dic106—21no—
147Dic106—42no—
148C8×C53—21no—
149C8×C53—21no—
150#88322—21no—
151#88322—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53C22×D411yescharacteristic
156C2×D53—21no—
157C2×D53—22no—
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—84no—
166C4×C53—21no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53C2×#4111yescharacteristic
172C4×C53H(5,𝔽2)11yescharacteristic
173Dic53—21no—
174Dic53—21no—
175Dic53—21no—
176Dic53—41no—
177Dic53—41no—
178#3311—531nomaximal, Sylow
179C2×C53—41no—
180C2×C53#54611yescharacteristic
181C2×C53#57511yescharacteristic
182C2×C53#57711yescharacteristic
183D53—41no—
184D53—42no—
185D53—81no—
186#556—531no—
187#546—531no—
188#489—531no—
189#485D5311yescharacteristic, complemented
190#482—531no—
191#476—531no—
192#470—531no—
193#465—531no—
194#464—531no—
195#452—531no—
196#450—531no—
197#440—531no—
198#435—531no—
199#418—531no—
200#416—531no—
201C53#331111yescharacteristic, complemented, Sylow
202C2×#41—531no—
203C22×D4—531no—
204C2×Q16—531no—
205C2×SD16—531no—
206C2×SD16—531no—
207C2×M16—531no—
208C22×C8—531no—
209C4⋊Q8—531no—
210#126—1061no—
211#126—1061no—
212#124—531no—
213#124—1061no—
214#123—1061no—
215C22⋊Q8—531no—
216C22⋊Q8—531no—
217C22⋊Q8C2×D5311yescharacteristic
218#121—531no—
219#121—1061no—
220#120—1061no—
221C4×Q8—531no—
222C4×Q8—1061no—
223C4×D4—531no—
224C4×D4—531no—
225#117—531no—
226#117C2×D5311yescharacteristic
227C2×C22⋊C4—1061no—
228#107C2×D5311yescharacteristic, complemented
229#106—531no—
230#106C2×D5311yescharacteristic, complemented
231C4⋊C8—531no—
232#103—531no—
233#103—531no—
234#103C2×D5311yescharacteristic, complemented
235#103C2×D5311yescharacteristic, complemented
236#102—531no—
237#102—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241#102—531no—
242C22⋊C8—531no—
243C22⋊C8C2×D5311yescharacteristic
244#97—531no—
245C24—1061no—
246#41—1062no—
247#41—2121no—
248C2×Q8—531no—
249C2×Q8—1061no—
250C2×Q8C22×D5311yescharacteristic
251C2×D4—531no—
252C2×D4—531no—
253C2×D4—531no—
254C2×D4—1062no—
255C2×D4—1061no—
256C2×D4—1061no—
257C2×D4—1061no—
258C2×D4—2121no—
259C22×C4—531no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4C22×D5311yescharacteristic
263Q16—2121no—
264SD16—1062no—
265SD16—2121no—
266M16—2121no—
267C2×C8—531no—
268C2×C8—531no—
269C2×C8—2121no—
270C2×C8C22×D5311yescharacteristic
271C2×C8C22×D5311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4C22×D5311yescharacteristic
280C4⋊C4C22×D5311yescharacteristic
281C4⋊C4C22×D5311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—531no—
285C22⋊C4—531no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C42—21no—
294C42—531no—
295C42—1061no—
296C42—1061no—
297C23—531no—
298C23—1061no—
299C23—1061no—
300C23—4241no—
301C23—2122no—
302C23—2122no—
303C23D4×D5311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—1061no—
307Q8—1061no—
308Q8—2122no—
309D4—532no—
310D4—1061no—
311D4—1061no—
312D4—1061no—
313D4—4241no—
314D4—2122no—
315D4—2122no—
316D4—2121no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—531no—
324C2×C4—531no—
325C2×C4—531no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—4241no—
331C2×C4—2122no—
332C2×C4D4×D5311yescharacteristic
333C2×C4C23×D5311yescharacteristic
334C8—21no—
335C8—21no—
336C8—1061no—
337C8—1061no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—1061no—
342C22—1062no—
343C22—1061no—
344C22—1061no—
345C22—4244no—
346C22—2121no—
347C22—2121no—
348C22—2121no—
349C22—2121no—
350C22—2121no—
351C22C2×D4×D5311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—1061no—
358C4—1061no—
359C4—1061no—
360C4—2121no—
361C4—2121no—
362C4#41×D5311yescharacteristic
363C4#1313326311yescharacteristic
364C2—41no—
365C2—4241no—
366C2—2121no—
367C2—2122no—
368C2#1681126511yescharacteristic
369C2#1681142211yescharacteristic
370C2#1681143011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#485C2
socleC22×C53C22×D4
hypercenter#485D53
hypocenterC53#3311
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #16811341C2semidirect
3. #16810984C2semidirect
4. #16810950C2semidirect
5. #16810912C2semidirect
6. #16810879C2semidirect
7. #16810853C2semidirect
8. #16810848C2semidirect
9. #16810788C2semidirect
10. #16810776C2semidirect
11. #16810722C2semidirect
12. #16810697C2semidirect
13. #16810627C2semidirect
14. #16810618C2semidirect
15. C53×#485C2semidirect
18. C2×#11719925C22semidirect
22. C2×C8×D53C22semidirect
23. #13133216C22semidirect
29. #13133197C22semidirect
35. #13133182C22semidirect
38. #13133175C22semidirect
40. #13133148C22semidirect
41. #13133146C22semidirect
42. #13133141C22semidirect
43. #13133140C22semidirect
50. #13133129C22semidirect
51. #13133127C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. #103×C53C22semidirect
57. #103×C53C22semidirect
189. #485D53semidirect
201. C53#3311semidirect
228. #107C2×D53semidirect
230. #106C2×D53semidirect
234. #103C2×D53semidirect
235. #103C2×D53semidirect