Finite Group Databasebeta

← #21434281

Subgroups — #21434281

Order 7552.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16838025C211yesmaximal, characteristic, complemented
2#16837877C211yesmaximal, characteristic, complemented
3#16837740C211yesmaximal, characteristic, complemented
4#16837727C211yesmaximal, characteristic, complemented
5#16837690C211yesmaximal, characteristic, complemented
6#16837680C211yesmaximal, characteristic, complemented
7#16837619C211yesmaximal, characteristic, complemented
8#16837619C211yesmaximal, characteristic, complemented
9#16837536C211yesmaximal, characteristic
10#16837511C211yesmaximal, characteristic, complemented
11#16837503C211yesmaximal, characteristic, complemented
12SD16×Dic59C211yesmaximal, characteristic, complemented
13#16837482C211yesmaximal, characteristic, complemented
14#16837460C211yesmaximal, characteristic, complemented
15C59×#450C211yesmaximal, characteristic, complemented
16C2×#11745669C2211yescharacteristic
17C2×#11745669C2211yescharacteristic
18#14273726C2211yescharacteristic, complemented
19#14273722—21no—
20#14273716C2211yescharacteristic, complemented
21#14273713—21no—
22#14273709—21no—
23#14273707—21no—
24#14273701C2211yescharacteristic
25Q8×Dic59C2211yescharacteristic, complemented
26D4×Dic59—21no—
27D4×Dic59—21no—
28D4×Dic59C2211yescharacteristic
29D4×Dic59C2211yescharacteristic
30D4×Dic59C2211yescharacteristic
31D4×Dic59C2211yescharacteristic
32#14273689C2211yescharacteristic, complemented
33#14273686C2211yescharacteristic
34#14273686C2211yescharacteristic
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37#14273661C2211yescharacteristic, complemented
38#14273660C2211yescharacteristic, complemented
39#14273659C2211yescharacteristic, complemented
40#14273658C2211yescharacteristic, complemented
41Dic59⋊C8C2211yescharacteristic, complemented
42Dic59⋊C8C2211yescharacteristic, complemented
43#14273651C2211yescharacteristic, complemented
44#14273651C2211yescharacteristic, complemented
45#14273649C2211yescharacteristic, complemented
46#14273649C2211yescharacteristic, complemented
47#14273648C2211yescharacteristic
48#14273648C2211yescharacteristic
49C22⋊(C59⋊C8)C2211yescharacteristic
50#14273637C2211yescharacteristic, complemented
51C8×Dic59C2211yescharacteristic, 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—22no—
62C2×D4×C59—22no—
63C2×D4×C59—21no—
64C2×D4×C59C2311yescharacteristic
65C2×D4×C59C2311yescharacteristic
66C22×C4×C59C2311yescharacteristic
67C2×C22⋊D59—21no—
68C22×Dic59—21no—
69C22×Dic59—21no—
70C22×Dic59—21no—
71C22×Dic59—21no—
72C2×D236C2311yescharacteristic
73C2×C4×D59—21no—
74C2×Dic118—21no—
75SD16×C59—22no—
76SD16×C59—22no—
77C2×C8×C59C2311yescharacteristic
78C2×C8×C59C2311yescharacteristic
79C4⋊C4×C59—21no—
80C4⋊C4×C59C2311yescharacteristic
81C22⋊C4×C59—21no—
82C22⋊Dic59—21no—
83C22⋊Dic59—21no—
84C22⋊Dic59—21no—
85C22⋊Dic59—21no—
86C22⋊Dic59—21no—
87C22⋊Dic59D411yescharacteristic
88C22⋊Dic59D411yescharacteristic
89#11745669—22no—
90#11745669—22no—
91(C2×D59)⋊C4—21no—
92(C2×D59)⋊C4—21no—
93(C2×D59)⋊C4—21no—
94(C2×D59)⋊C4—21no—
95C4⋊Dic59—21no—
96C4⋊Dic59D411yescharacteristic
97C4⋊Dic59D411yescharacteristic
98C4⋊Dic59C2311yescharacteristic
99C4⋊Dic59C2311yescharacteristic
100C4⋊Dic59C2311yescharacteristic
101Dic59⋊C4—21no—
102C4×Dic59—21no—
103C4×Dic59—21no—
104C4×Dic59C2311yescharacteristic
105C4×Dic59C2311yescharacteristic
106C2×C59⋊C8C2311yescharacteristic
107C2×C59⋊C8C2311yescharacteristic
108C23×C59—24no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59—81no—
112C23×C59—81no—
113C23×C59C2×D411yescharacteristic
114C22×D59—21no—
115Q8×C59—21no—
116Q8×C59—21no—
117D4×C59—24no—
118D4×C59—21no—
119D4×C59—21no—
120D4×C59#4112yes—
121D4×C59#4112yes—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59C2×D411yescharacteristic
127C2×C4×C59C2411yescharacteristic
128C22⋊D59—81no—
129C2×Dic59—21no—
130C2×Dic59—21no—
131C2×Dic59—21no—
132C2×Dic59—81no—
133C2×Dic59—81no—
134C2×Dic59—81no—
135C2×Dic59—81no—
136C2×Dic59C2×D411yescharacteristic
137C2×Dic59C2×D411yescharacteristic
138C2×Dic59C2×D411yescharacteristic
139C2×Dic59C2×D411yescharacteristic
140D236—21no—
141D236—21no—
142C4×D59—81no—
143Dic118—81no—
144C8×C59—21no—
145C8×C59—21no—
146C59⋊C8—21no—
147C59⋊C8—21no—
148C22×C59—21no—
149C22×C59—22no—
150C22×C59—22no—
151C22×C59—21no—
152C22×C59—21no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—84no—
158C22×C59C22×D411yescharacteristic
159C2×D59—41no—
160C2×D59—41no—
161C2×D59—41no—
162C4×C59—21no—
163C4×C59—41no—
164C4×C59—41no—
165C4×C59C2×#4111yescharacteristic
166C4×C59C2×#4111yescharacteristic
167Dic59—21no—
168Dic59—21no—
169Dic59—21no—
170Dic59—21no—
171Dic59—41no—
172Dic59—41no—
173Dic59—41no—
174#3118—591nomaximal, Sylow
175C2×C59—41no—
176C2×C59—42no—
177C2×C59—42no—
178C2×C59#52511yescharacteristic
179C2×C59#57611yescharacteristic
180C2×C59#57711yescharacteristic
181D59—81no—
182#542—591no—
183#518—591no—
184#485—591no—
185#482—591no—
186#476—591no—
187#474—591no—
188#463—591no—
189#463—591no—
190#450D5911yescharacteristic, complemented
191#447—591no—
192#442—591no—
193#440—591no—
194C4×SD16—591no—
195C4×D8—591no—
196#433—591no—
197C59#311811yescharacteristic, complemented, Sylow
198C22×D4C2×D5911yescharacteristic
199C2×SD16C2×D5911yescharacteristic
200C2×SD16C2×D5911yescharacteristic
201C2×D8—591no—
202C2×D8—591no—
203C4⋊Q8—591no—
204#126—1181no—
205#124—591no—
206#124—1181no—
207#123—1181no—
208C22⋊Q8—1181no—
209C22⋊Q8C2×D5911yescharacteristic
210#121—591no—
211C4×Q8—591no—
212C4×D4—591no—
213C4×D4—591no—
214C4×D4—591no—
215C4×D4—591no—
216C4×D4—591no—
217C4×D4—1181no—
218C4×D4—1181no—
219#117—591no—
220#117—591no—
221C2×C22⋊C4—1181no—
222C2×C22⋊C4—1181no—
223#107—591no—
224#107—591no—
225#106—591no—
226#106—591no—
227C4⋊C8—591no—
228C4⋊C8—591no—
229#103—591no—
230#103—591no—
231#102—591no—
232#102—591no—
233#102—591no—
234#102—591no—
235#102C2×D5911yescharacteristic
236#102C2×D5911yescharacteristic
237C22⋊C8—591no—
238C22⋊C8C2×D5911yescharacteristic
239#97—591no—
240C4×C8—591no—
241C24—21no—
242C2×Q8—1181no—
243C2×Q8C22×D5911yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—591no—
248C2×D4—1181no—
249C2×D4C22×D5911yescharacteristic
250C2×D4C22×D5911yescharacteristic
251C22×C4—1181no—
252C22×C4—1181no—
253C22×C4—1181no—
254C22×C4—1181no—
255C22×C4—1181no—
256C22×C4C22×D5911yescharacteristic
257SD16—22no—
258SD16—22no—
259D8—1182no—
260D8—1182no—
261C2×C8—591no—
262C2×C8—591no—
263C2×C8C22×D5911yescharacteristic
264C2×C8C22×D5911yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—591no—
267C4⋊C4—591no—
268C4⋊C4—591no—
269C4⋊C4—591no—
270C4⋊C4—591no—
271C4⋊C4—1181no—
272C4⋊C4—1181no—
273C4⋊C4C22×D5911yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—591no—
276C22⋊C4—591no—
277C22⋊C4—1181no—
278C22⋊C4—1181no—
279C22⋊C4—1181no—
280C22⋊C4—1181no—
281C22⋊C4—1181no—
282C22⋊C4—1181no—
283C22⋊C4—1181no—
284C22⋊C4—1181no—
285C22⋊C4—1181no—
286C42—591no—
287C42—591no—
288C42—1181no—
289C42—1181no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—1181no—
296C23D4×D5911yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—4721no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—1181no—
304D4—1181no—
305D4—4721no—
306D4#1174568712yes—
307D4#1174568712yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—591no—
313C2×C4—591no—
314C2×C4—591no—
315C2×C4—591no—
316C2×C4—1181no—
317C2×C4—1181no—
318C2×C4—1181no—
319C2×C4—4721no—
320C2×C4—4721no—
321C2×C4—4721no—
322C2×C4—4721no—
323C2×C4—4721no—
324C2×C4D4×D5911yescharacteristic
325C2×C4C23×D5911yescharacteristic
326C8—21no—
327C8—21no—
328C8—1181no—
329C8—1181no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—2361no—
341C22—2361no—
342C22—2361no—
343C22C2×D4×D5911yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—1181no—
348C4—1181no—
349C4—1181no—
350C4—1181no—
351C4—2361no—
352C4—2361no—
353C4—2361no—
354C4C2×#1174568711yescharacteristic
355C4C2×#1174568711yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—4721no—
360C2#1683790811yescharacteristic
361C2#1683820611yescharacteristic
362C2#1683820811yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#450C2
socleC22×C59C22×D4
hypercenter#450D59
hypocenterC59#3118
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. #16838025C2semidirect
2. #16837877C2semidirect
3. #16837740C2semidirect
4. #16837727C2semidirect
5. #16837690C2semidirect
6. #16837680C2semidirect
7. #16837619C2semidirect
8. #16837619C2semidirect
10. #16837511C2semidirect
11. #16837503C2semidirect
12. SD16×Dic59C2semidirect
13. #16837482C2semidirect
14. #16837460C2semidirect
15. C59×#450C2semidirect
18. #14273726C22semidirect
20. #14273716C22semidirect
25. Q8×Dic59C22semidirect
32. #14273689C22semidirect
37. #14273661C22semidirect
38. #14273660C22semidirect
39. #14273659C22semidirect
40. #14273658C22semidirect
41. Dic59⋊C8C22semidirect
42. Dic59⋊C8C22semidirect
43. #14273651C22semidirect
44. #14273651C22semidirect
45. #14273649C22semidirect
46. #14273649C22semidirect
50. #14273637C22semidirect
51. C8×Dic59C22semidirect
190. #450D59semidirect
197. C59#3118semidirect