Finite Group Databasebeta

← #20286757

Subgroups — #20286757

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#16811270C211yesmaximal, characteristic
3#16810981C211yesmaximal, characteristic, complemented
4#16810967C211yesmaximal, characteristic, complemented
5#16810944C211yesmaximal, characteristic, complemented
6#16810912C211yesmaximal, characteristic, complemented
7#16810880C211yesmaximal, characteristic, complemented
8#16810843C211yesmaximal, characteristic, complemented
9#16810783C211yesmaximal, characteristic, complemented
10#16810772C211yesmaximal, characteristic, complemented
11#16810726C211yesmaximal, characteristic, complemented
12#16810694C211yesmaximal, characteristic, complemented
13#16810626C211yesmaximal, characteristic, complemented
14#16810620C211yesmaximal, characteristic, complemented
15C53×#465C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18C2×#11719925C2211yescharacteristic, complemented
19C2×#11719924C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21#13133216C2211yescharacteristic, complemented
22#13133212—21no—
23#13133210—21no—
24#13133204C2211yescharacteristic
25#13133203—21no—
26#13133199—21no—
27#13133197C2211yescharacteristic
28#13133193C2211yescharacteristic
29#13133188—21no—
30#13133185—21no—
31Q8×Dic53C2211yescharacteristic, complemented
32#13133182—21no—
33#13133179C2211yescharacteristic
34C4×C22⋊D53C2211yescharacteristic
35#13133176C2211yescharacteristic, complemented
36#13133175C2211yescharacteristic
37C22⋊C4×D53—21no—
38#13133150C2211yescharacteristic, complemented
39#13133148C2211yescharacteristic, complemented
40#13133146C2211yescharacteristic, complemented
41#13133142C2211yescharacteristic, complemented
42#13133142C2211yescharacteristic, complemented
43#13133140C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic
51#13133127C2211yescharacteristic, complemented
52C2×SD16×C53C2211yescharacteristic
53C22×C8×C53C2211yescharacteristic, complemented
54C22⋊Q8×C53C2211yescharacteristic, complemented
55#121×C53C2211yescharacteristic
56#106×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53—21no—
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53D411yescharacteristic
67D4×D53—22no—
68D4×D53—41no—
69#11719945—22no—
70#11719945—41no—
71C2×D212C2311yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53C2311yescharacteristic
75C2×Dic106—21no—
76C2×Dic106C2311yescharacteristic
77SD16×C53—41no—
78C2×C8×C53—41no—
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C22⋊C4×C53—21no—
85C22⋊C4×C53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53D411yescharacteristic
90#11719925—41no—
91#11719924—22no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921D411yescharacteristic
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101#11719919—21no—
102#11719919—21no—
103#11719919D411yescharacteristic
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107C4×Dic53C2311yescharacteristic
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—
120D4×C53—21no—
121D4×C53—21no—
122D4×C53—81no—
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—41no—
131C22⋊D53—42no—
132C22⋊D53—42no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139D212—21no—
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×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53C22×D411yescharacteristic
159C2×D53—21no—
160C2×D53—22no—
161C2×D53—21no—
162C2×D53—21no—
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—41no—
180C2×C53—81no—
181C2×C53#54611yescharacteristic
182C2×C53#57511yescharacteristic
183C2×C53#57711yescharacteristic
184D53—41no—
185D53—42no—
186#556—531no—
187#546—531no—
188#489—531no—
189#485—531no—
190#482—531no—
191#476—531no—
192#470—531no—
193#465D5311yescharacteristic, complemented
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×SD16C2×D5311yescharacteristic
207C2×M16—531no—
208C22×C8C2×D5311yescharacteristic, complemented
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—1061no—
219#121C2×D5311yescharacteristic
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#106C2×D5311yescharacteristic, complemented
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⋊C8—531no—
244#97—531no—
245C24—1061no—
246#41—1062no—
247#41—2121no—
248C2×Q8—531no—
249C2×Q8—1061no—
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—531no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4C22×D5311yescharacteristic
263Q16—2121no—
264SD16—41no—
265SD16—1062no—
266M16—2121no—
267C2×C8—41no—
268C2×C8—531no—
269C2×C8—531no—
270C2×C8C22×D5311yescharacteristic
271C2×C8C22×D5311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
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—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—1061no—
307Q8—1061no—
308Q8—2122no—
309D4—21no—
310D4—21no—
311D4—81no—
312D4—532no—
313D4—1061no—
314D4—2121no—
315D4—2122no—
316D4—2122no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—531no—
323C2×C4—531no—
324C2×C4—531no—
325C2×C4—1061no—
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—41no—
342C22—41no—
343C22—41no—
344C22—1061no—
345C22—1061no—
346C22—1061no—
347C22—1062no—
348C22—4244no—
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—
362C4#41×D5311yescharacteristic
363C4#1313326211yescharacteristic
364C2—41no—
365C2—81no—
366C2—2121no—
367C2—2122no—
368C2#1681126211yescharacteristic
369C2#1681142011yescharacteristic
370C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#465C2
socleC22×C53C22×D4
hypercenter#465D53
hypocenterC53#3311
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #16811344C2semidirect
3. #16810981C2semidirect
4. #16810967C2semidirect
5. #16810944C2semidirect
6. #16810912C2semidirect
7. #16810880C2semidirect
8. #16810843C2semidirect
9. #16810783C2semidirect
10. #16810772C2semidirect
11. #16810726C2semidirect
12. #16810694C2semidirect
13. #16810626C2semidirect
14. #16810620C2semidirect
15. C53×#465C2semidirect
18. C2×#11719925C22semidirect
21. #13133216C22semidirect
31. Q8×Dic53C22semidirect
35. #13133176C22semidirect
38. #13133150C22semidirect
39. #13133148C22semidirect
40. #13133146C22semidirect
41. #13133142C22semidirect
42. #13133142C22semidirect
43. #13133140C22semidirect
49. C22⋊#88322C22semidirect
51. #13133127C22semidirect
53. C22×C8×C53C22semidirect
54. C22⋊Q8×C53C22semidirect
56. #106×C53C22semidirect
57. #103×C53C22semidirect
193. #465D53semidirect
201. C53#3311semidirect
208. C22×C8C2×D53semidirect
217. C22⋊Q8C2×D53semidirect
230. #106C2×D53semidirect
235. #103C2×D53semidirect