Finite Group Databasebeta

← #20286755

Subgroups — #20286755

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#16811344C211yesmaximal, characteristic, complemented
2#16811261C211yesmaximal, characteristic
3#16810984C211yesmaximal, characteristic, complemented
4#16810963C211yesmaximal, characteristic, complemented
5#16810948C211yesmaximal, characteristic, complemented
6#16810911C211yesmaximal, characteristic, complemented
7#16810878C211yesmaximal, characteristic, complemented
8#16810854C211yesmaximal, characteristic, complemented
9#16810847C211yesmaximal, characteristic, complemented
10#16810772C211yesmaximal, characteristic, complemented
11#16810724C211yesmaximal, characteristic, complemented
12#16810691C211yesmaximal, characteristic, complemented
13#16810626C211yesmaximal, characteristic, complemented
14#16810619C211yesmaximal, characteristic, complemented
15C53×#452C211yesmaximal, characteristic, complemented
16C2×D4×D53C2211yescharacteristic
17C2×#11719924C2211yescharacteristic
18C2×#11719912C2211yescharacteristic
19C22×#88322C2211yescharacteristic, complemented
20#13133216C2211yescharacteristic, complemented
21#13133212—21no—
22#13133210—21no—
23#13133206C2211yescharacteristic
24#13133203—21no—
25#13133200—21no—
26#13133193C2211yescharacteristic
27#13133193C2211yescharacteristic
28#13133191C2211yescharacteristic
29#13133189—21no—
30#13133185—21no—
31Q8×Dic53C2211yescharacteristic, complemented
32#13133182—21no—
33D4×Dic53C2211yescharacteristic
34C4×C22⋊D53C2211yescharacteristic
35#13133176C2211yescharacteristic, complemented
36#13133175C2211yescharacteristic
37C2×#11719921—21no—
38#13133151C2211yescharacteristic, complemented
39#13133149C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133141C2211yescharacteristic, complemented
43#13133141C2211yescharacteristic, complemented
44#13133141C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133127C2211yescharacteristic, complemented
52C2×#41×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic, complemented
56#103×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60#41×C53—22no—
61#41×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×Q8×C53C2311yescharacteristic
64C2×D4×C53—21no—
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53—21no—
67C22×C4×C53C2311yescharacteristic
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53D411yescharacteristic
71C22×Dic53—21no—
72D4×D53—22no—
73D4×D53—41no—
74C2×D212C2311yescharacteristic
75C2×C4×D53C2311yescharacteristic
76C2×Dic106—21no—
77Q16×C53—41no—
78SD16×C53—22no—
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊Dic53—21no—
85C22⋊Dic53—21no—
86C22⋊Dic53D411yescharacteristic
87#11719924—41no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921D411yescharacteristic
95C4⋊Dic53—21no—
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919D411yescharacteristic
103C4×Dic53—21no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53C2311yescharacteristic
107C2×#88322—41no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719912—41no—
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114C22×D53—42no—
115C22×D53—42no—
116C22×D53—81no—
117C22×D53C2×D411yescharacteristic
118Q8×C53—21no—
119Q8×C53—21no—
120Q8×C53—21no—
121Q8×C53—21no—
122D4×C53—21no—
123D4×C53—42no—
124D4×C53#4112yes—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—42no—
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2411yescharacteristic
133C22⋊D53—41no—
134C22⋊D53—42no—
135C22⋊D53—81no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—81no—
140C2×Dic53C2×D411yescharacteristic
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143D212—21no—
144D212—21no—
145C4×D53—21no—
146C4×D53—21no—
147Dic106—42no—
148C8×C53—21no—
149C8×C53—21no—
150#88322—21no—
151#88322—21no—
152C22×C53—22no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53C22×D411yescharacteristic
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—21no—
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×C53C2×#4111yescharacteristic
171C4×C53H(5,𝔽2)11yescharacteristic
172Dic53—21no—
173Dic53—21no—
174Dic53—21no—
175Dic53—41no—
176Dic53—41no—
177Dic53—41no—
178#3311—531nomaximal, Sylow
179C2×C53—42no—
180C2×C53—41no—
181C2×C53#54611yescharacteristic
182C2×C53#57511yescharacteristic
183C2×C53#57711yescharacteristic
184D53—41no—
185D53—81no—
186#556—531no—
187#546—531no—
188#489—531no—
189#485—531no—
190#482—531no—
191#476—531no—
192#470—531no—
193#465—531no—
194#464—531no—
195#452D5311yescharacteristic, complemented
196#450—531no—
197#440—531no—
198#435—531no—
199#418—531no—
200#416—531no—
201C53#331111yescharacteristic, complemented, Sylow
202C2×#41C2×D5311yescharacteristic
203C22×D4—531no—
204C2×Q16C2×D5311yescharacteristic, complemented
205C2×SD16—531no—
206C2×SD16C2×D5311yescharacteristic
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, complemented
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#117—531no—
227C2×C22⋊C4—1061no—
228#107—531no—
229#106—531no—
230#106—531no—
231C4⋊C8—531no—
232#103—531no—
233#103—531no—
234#103—531no—
235#103C2×D5311yescharacteristic, complemented
236#102—531no—
237#102—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241#102C2×D5311yescharacteristic
242C22⋊C8—531no—
243C22⋊C8C2×D5311yescharacteristic, complemented
244#97—531no—
245C24—1061no—
246#41—22no—
247#41—41no—
248C2×Q8—1061no—
249C2×Q8C22×D5311yescharacteristic
250C2×Q8C22×D5311yescharacteristic
251C2×D4—21no—
252C2×D4—531no—
253C2×D4—531no—
254C2×D4—1061no—
255C2×D4—1062no—
256C2×D4—1061no—
257C2×D4—2121no—
258C2×D4C22×D5311yescharacteristic
259C22×C4—21no—
260C22×C4—531no—
261C22×C4—1061no—
262C22×C4C22×D5311yescharacteristic
263Q16—41no—
264SD16—22no—
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—531no—
276C4⋊C4—531no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4C22×D5311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—531no—
284C22⋊C4—531no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C42—531no—
294C42—1061no—
295C42—1061no—
296C42—1061no—
297C23—21no—
298C23—531no—
299C23—1061no—
300C23—4241no—
301C23—2122no—
302C23—2122no—
303C23D4×D5311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308Q8—2122no—
309D4—21no—
310D4—42no—
311D4—1061no—
312D4—1061no—
313D4—4241no—
314D4—2121no—
315D4—2122no—
316D4#1171994712yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—531no—
324C2×C4—531no—
325C2×C4—531no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—4241no—
332C2×C4D4×D5311yescharacteristic
333C2×C4C23×D5311yescharacteristic
334C8—21no—
335C8—21no—
336C8—1061no—
337C8—1061no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—1061no—
345C22—1061no—
346C22—1061no—
347C22—4244no—
348C22—2121no—
349C22—2121no—
350C22—2121no—
351C22C2×D4×D5311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—1061no—
357C4—1061no—
358C4—1061no—
359C4—2121no—
360C4—2121no—
361C4—2121no—
362C4C2×#1171994711yescharacteristic
363C4#1313326211yescharacteristic
364C2—41no—
365C2—42no—
366C2—4241no—
367C2—2121no—
368C2#1681126011yescharacteristic
369C2#1681142011yescharacteristic
370C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#452C2
socleC22×C53C22×D4
hypercenter#452D53
hypocenterC53#3311
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #16811344C2semidirect
3. #16810984C2semidirect
4. #16810963C2semidirect
5. #16810948C2semidirect
6. #16810911C2semidirect
7. #16810878C2semidirect
8. #16810854C2semidirect
9. #16810847C2semidirect
10. #16810772C2semidirect
11. #16810724C2semidirect
12. #16810691C2semidirect
13. #16810626C2semidirect
14. #16810619C2semidirect
15. C53×#452C2semidirect
19. C22×#88322C22semidirect
20. #13133216C22semidirect
31. Q8×Dic53C22semidirect
35. #13133176C22semidirect
38. #13133151C22semidirect
39. #13133149C22semidirect
40. #13133148C22semidirect
41. #13133146C22semidirect
42. #13133141C22semidirect
43. #13133141C22semidirect
44. #13133141C22semidirect
51. #13133127C22semidirect
53. C2×Q16×C53C22semidirect
55. C22⋊Q8×C53C22semidirect
56. #103×C53C22semidirect
58. C22⋊C8×C53C22semidirect
195. #452D53semidirect
201. C53#3311semidirect
204. C2×Q16C2×D53semidirect
217. C22⋊Q8C2×D53semidirect
235. #103C2×D53semidirect
243. C22⋊C8C2×D53semidirect