Finite Group Databasebeta

← #21434965

Subgroups — #21434965

Order 7552.

410 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16837975C211yesmaximal, characteristic, complemented
2#16837969C211yesmaximal, characteristic
3#16837744C211yesmaximal, characteristic, complemented
4#16837740C211yesmaximal, characteristic, complemented
5#16837725C211yesmaximal, characteristic, complemented
6#16837724C211yesmaximal, characteristic, complemented
7#16837647C211yesmaximal, characteristic, complemented
8#16837645C211yesmaximal, characteristic, complemented
9#16837630C211yesmaximal, characteristic, complemented
10#16837628C211yesmaximal, characteristic, complemented
11#16837553C211yesmaximal, characteristic, complemented
12#16837539C211yesmaximal, characteristic, complemented
13#16837539C211yesmaximal, characteristic, complemented
14#16837367C211yesmaximal, characteristic, complemented
15C59×#450C211yesmaximal, characteristic, complemented
16C2×#11745685C2211yescharacteristic
17C2×Q8×D59C2211yescharacteristic, complemented
18C2×#11745670C2211yescharacteristic
19C2×#11745670C2211yescharacteristic
20C2×#11745659C2211yescharacteristic, complemented
21C2×C8×D59C2211yescharacteristic, complemented
22#14273726—21no—
23#14273723—21no—
24#14273714—21no—
25#14273713—21no—
26#14273712—21no—
27#14273709—21no—
28#14273707C2211yescharacteristic, complemented
29#14273706C2211yescharacteristic
30#14273703C2211yescharacteristic, complemented
31#14273698C2211yescharacteristic
32#14273698C2211yescharacteristic
33#14273698C2211yescharacteristic
34#14273698C2211yescharacteristic
35#14273696—21no—
36#14273696—21no—
37#14273686C2211yescharacteristic
38#14273685C2211yescharacteristic, complemented
39#14273661C2211yescharacteristic, complemented
40#14273660C2211yescharacteristic, complemented
41#14273659C2211yescharacteristic, complemented
42#14273658C2211yescharacteristic, complemented
43#14273652C2211yescharacteristic, complemented
44#14273652C2211yescharacteristic, complemented
45#14273651C2211yescharacteristic, complemented
46#14273651C2211yescharacteristic, complemented
47#14273648C2211yescharacteristic
48#14273648C2211yescharacteristic
49C22⋊(C59⋊C8)C2211yescharacteristic
50(C2×D59)⋊C8C2211yescharacteristic, complemented
51(C2×D59)⋊C8C2211yescharacteristic, complemented
52C22×D4×C59C2211yescharacteristic
53C2×SD16×C59C2211yescharacteristic
54C2×SD16×C59C2211yescharacteristic
55C22⋊Q8×C59C2211yescharacteristic
56#102×C59C2211yescharacteristic
57#102×C59C2211yescharacteristic
58C22⋊C8×C59C2211yescharacteristic
59C24×C59—21no—
60C2×Q8×C59C2311yescharacteristic
61C2×D4×C59—21no—
62C2×D4×C59—41no—
63C2×D4×C59—41no—
64C2×D4×C59C2311yescharacteristic
65C2×D4×C59C2311yescharacteristic
66C22×C4×C59C2311yescharacteristic
67C2×C22⋊D59—21no—
68C2×C22⋊D59—21no—
69C2×C22⋊D59—21no—
70C2×C22⋊D59—21no—
71C2×C22⋊D59D411yescharacteristic
72C2×C22⋊D59D411yescharacteristic
73Q8×D59—41no—
74Q8×D59—41no—
75#11745685—41no—
76#11745685—41no—
77C2×D236D411yescharacteristic
78C2×C4×D59—21no—
79C2×C4×D59C2311yescharacteristic
80C2×C4×D59C2311yescharacteristic
81C2×Dic118—21no—
82C2×Dic118D411yescharacteristic
83C2×Dic118C2311yescharacteristic
84SD16×C59—41no—
85SD16×C59—41no—
86C2×C8×C59C2311yescharacteristic
87C2×C8×C59C2311yescharacteristic
88C4⋊C4×C59—21no—
89C4⋊C4×C59C2311yescharacteristic
90C22⋊C4×C59—21no—
91C22⋊Dic59—21no—
92C22⋊Dic59—21no—
93C22⋊Dic59—21no—
94C22⋊Dic59—21no—
95C22⋊Dic59—21no—
96C22⋊Dic59—21no—
97#11745670—41no—
98#11745670—41no—
99(C2×D59)⋊C4—21no—
100(C2×D59)⋊C4—21no—
101(C2×D59)⋊C4—21no—
102C4⋊Dic59—21no—
103C4⋊Dic59C2311yescharacteristic
104C4⋊Dic59C2311yescharacteristic
105C4⋊Dic59C2311yescharacteristic
106Dic59⋊C4—21no—
107Dic59⋊C4—21no—
108C4×Dic59—21no—
109C4×Dic59—21no—
110C2×C59⋊C8C2311yescharacteristic
111C2×C59⋊C8C2311yescharacteristic
112#11745659—41no—
113C8×D59—41no—
114C23×C59—21no—
115C23×C59—21no—
116C23×C59—42no—
117C23×C59—81no—
118C23×C59—81no—
119C23×C59C2×D411yescharacteristic
120C22×D59C2×D411yescharacteristic
121C22×D59C2×D411yescharacteristic
122Q8×C59—21no—
123Q8×C59—21no—
124D4×C59—21no—
125D4×C59—21no—
126D4×C59—21no—
127D4×C59—21no—
128D4×C59—42no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59C2×D411yescharacteristic
134C2×C4×C59C2411yescharacteristic
135C22⋊D59—41no—
136C22⋊D59—41no—
137C22⋊D59—81no—
138C22⋊D59—81no—
139C22⋊D59—81no—
140C22⋊D59—81no—
141C2×Dic59—21no—
142C2×Dic59—21no—
143C2×Dic59—21no—
144C2×Dic59—21no—
145C2×Dic59C2×D411yescharacteristic
146C2×Dic59C2×D411yescharacteristic
147D236—41no—
148C4×D59—21no—
149C4×D59—21no—
150C4×D59—21no—
151C4×D59—21no—
152C4×D59—42no—
153Dic118—21no—
154Dic118—21no—
155Dic118—41no—
156Dic118—42no—
157C8×C59—21no—
158C8×C59—21no—
159C59⋊C8—21no—
160C59⋊C8—21no—
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—41no—
168C22×C59—41no—
169C22×C59—41no—
170C22×C59—84no—
171C22×C59C22×D411yescharacteristic
172C2×D59—21no—
173C2×D59—21no—
174C2×D59—21no—
175C2×D59—21no—
176C2×D59—21no—
177C2×D59—21no—
178C4×C59—21no—
179C4×C59—41no—
180C4×C59—41no—
181C4×C59H(5,𝔽2)11yescharacteristic
182C4×C59H(5,𝔽2)11yescharacteristic
183Dic59—21no—
184Dic59—21no—
185Dic59—41no—
186Dic59—41no—
187Dic59—41no—
188Dic59—41no—
189#3200—591nomaximal, Sylow
190C2×C59—41no—
191C2×C59—81no—
192C2×C59—81no—
193C2×C59#53411yescharacteristic
194C2×C59#57711yescharacteristic
195C2×C59#57711yescharacteristic
196D59—41no—
197D59—41no—
198#536—591no—
199#535—591no—
200#485—591no—
201#485—591no—
202#482—591no—
203#482—591no—
204#468—591no—
205#468—591no—
206#465—591no—
207#465—591no—
208#450—591no—
209#450D5911yescharacteristic, complemented
210#448—591no—
211#448—591no—
212#408—591no—
213C59#320011yescharacteristic, complemented, Sylow
214C2×#41—591no—
215C22×Q8—591no—
216C22×D4C2×D5911yescharacteristic
217C2×SD16—591no—
218C2×SD16—591no—
219C2×SD16C2×D5911yescharacteristic
220C2×SD16C2×D5911yescharacteristic
221C2×M16—591no—
222C22×C8—591no—
223C4⋊Q8—1181no—
224#127—1181no—
225#124—1181no—
226#124—1181no—
227#123—1181no—
228#123—1181no—
229C22⋊Q8—591no—
230C22⋊Q8—591no—
231C22⋊Q8—591no—
232C22⋊Q8C2×D5911yescharacteristic
233#121—591no—
234#121—591no—
235#121—591no—
236#121—591no—
237#120—1181no—
238#120—1181no—
239#117—591no—
240#117—591no—
241#107—591no—
242#107—591no—
243#106—591no—
244#106—591no—
245#103—591no—
246#103—591no—
247#103—591no—
248#103—591no—
249#102—591no—
250#102—591no—
251#102C2×D5911yescharacteristic
252#102C2×D5911yescharacteristic
253C22⋊C8—591no—
254C22⋊C8—591no—
255C22⋊C8—591no—
256C22⋊C8C2×D5911yescharacteristic
257C24—21no—
258#41—2361no—
259#41—2361no—
260C2×Q8—591no—
261C2×Q8—591no—
262C2×Q8—1181no—
263C2×Q8—2361no—
264C2×Q8—2361no—
265C2×Q8C22×D5911yescharacteristic
266C2×D4—21no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—591no—
270C2×D4—591no—
271C2×D4—591no—
272C2×D4—1181no—
273C2×D4—1181no—
274C2×D4—1181no—
275C2×D4—1181no—
276C2×D4C22×D5911yescharacteristic
277C2×D4C22×D5911yescharacteristic
278C22×C4—591no—
279C22×C4—591no—
280C22×C4—1181no—
281C22×C4C22×D5911yescharacteristic
282SD16—41no—
283SD16—41no—
284SD16—2361no—
285SD16—2361no—
286M16—2361no—
287C2×C8—591no—
288C2×C8—591no—
289C2×C8—2361no—
290C2×C8C22×D5911yescharacteristic
291C2×C8C22×D5911yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—591no—
294C4⋊C4—591no—
295C4⋊C4—591no—
296C4⋊C4—1181no—
297C4⋊C4—1181no—
298C4⋊C4—1181no—
299C4⋊C4C22×D5911yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—1181no—
302C22⋊C4—1181no—
303C22⋊C4—1181no—
304C22⋊C4—1181no—
305C22⋊C4—1181no—
306C22⋊C4—1181no—
307C22⋊C4—1181no—
308C22⋊C4—1181no—
309C22⋊C4—1181no—
310C42—1181no—
311C42—1181no—
312C23—21no—
313C23—21no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—591no—
318C23—591no—
319C23D4×D5911yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—1181no—
323Q8—1181no—
324Q8—2361no—
325Q8—2362no—
326D4—21no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—42no—
331D4—4721no—
332D4—4721no—
333D4—4721no—
334D4—4721no—
335D4—2361no—
336D4—2361no—
337D4—2361no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—591no—
343C2×C4—591no—
344C2×C4—1181no—
345C2×C4—1181no—
346C2×C4—1181no—
347C2×C4—1181no—
348C2×C4—1181no—
349C2×C4—1181no—
350C2×C4—1181no—
351C2×C4—1181no—
352C2×C4—2362no—
353C2×C4D4×D5911yescharacteristic
354C2×C4C23×D5911yescharacteristic
355C8—21no—
356C8—21no—
357C8—1181no—
358C8—1181no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—1181no—
370C22—1181no—
371C22—1181no—
372C22—1181no—
373C22—1181no—
374C22—1181no—
375C22C2×D4×D5911yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—1181no—
380C4—1181no—
381C4—2361no—
382C4—2361no—
383C4—2361no—
384C4—2361no—
385C4#1427377211yescharacteristic
386C4#1427377211yescharacteristic
387C2—41no—
388C2—81no—
389C2—81no—
390C2—2361no—
391C2—2361no—
392C2#1683796211yescharacteristic
393C2#1683820811yescharacteristic
394C2#1683821211yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#450C2
socleC22×C59C22×D4
hypercenter#450D59
hypocenterC59#3200
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #16837975C2semidirect
3. #16837744C2semidirect
4. #16837740C2semidirect
5. #16837725C2semidirect
6. #16837724C2semidirect
7. #16837647C2semidirect
8. #16837645C2semidirect
9. #16837630C2semidirect
10. #16837628C2semidirect
11. #16837553C2semidirect
12. #16837539C2semidirect
13. #16837539C2semidirect
14. #16837367C2semidirect
15. C59×#450C2semidirect
17. C2×Q8×D59C22semidirect
20. C2×#11745659C22semidirect
21. C2×C8×D59C22semidirect
28. #14273707C22semidirect
30. #14273703C22semidirect
38. #14273685C22semidirect
39. #14273661C22semidirect
40. #14273660C22semidirect
41. #14273659C22semidirect
42. #14273658C22semidirect
43. #14273652C22semidirect
44. #14273652C22semidirect
45. #14273651C22semidirect
46. #14273651C22semidirect
50. (C2×D59)⋊C8C22semidirect
51. (C2×D59)⋊C8C22semidirect
209. #450D59semidirect
213. C59#3200semidirect