Finite Group Databasebeta

← #509×D59 #21437216

Subgroups — #509×D59 #21437216

Order 7552.

400 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#135×D59C211yesmaximal, characteristic, complemented
2C2×D8×D59C211yesmaximal, characteristic, complemented
3#16837840C211yesmaximal, characteristic, complemented
4#16837839C212yesmaximal, complemented
5#16837838C211yesmaximal, characteristic, complemented
6#16837837C211yesmaximal, characteristic, complemented
7#16837836C212yesmaximal, complemented
8SD32×D59C212yesmaximal, complemented
9D16×D59C212yesmaximal, complemented
10M32×D59C211yesmaximal, characteristic, complemented
11C59×#509C211yesmaximal, characteristic, complemented
12#41×D59—21no—
13C2×D4×D59—21no—
14#14273750C2211yescharacteristic, complemented
15#14273749C2211yescharacteristic, complemented
16#14273748—21no—
17#14273747—21no—
18#14273746C2211yescharacteristic, complemented
19Q16×D59C2211yescharacteristic, complemented
20SD16×D59—21no—
21C2×D472C2211yescharacteristic, complemented
22C2×#11745669—21no—
23D8×D59—21no—
24D8×D59—21no—
25D8×D59C2212yes—
26D8×D59C2212yescomplemented
27D8×D59C2211yescharacteristic, complemented
28C2×C8×D59C2211yescharacteristic
29#14273672C2212yescomplemented
30#14273671C2212yescomplemented
31#14273670C2212yescomplemented
32D944C2212yescomplemented
33#14273668C2212yescomplemented
34#14273668C2212yescomplemented
35#14273667C2211yescharacteristic, complemented
36#14273666C2212yescomplemented
37C16×D59C2212yescomplemented
38#135×C59C2211yescharacteristic, complemented
39C2×D8×C59C2211yescharacteristic, complemented
40SD32×C59C2212yescomplemented
41D16×C59C2212yescomplemented
42M32×C59C2211yescharacteristic, complemented
43C23×D59—41no—
44#41×C59—21no—
45C2×D4×C59—21no—
46C2×C22⋊D59—41no—
47#11745689—21no—
48Q8×D59—21no—
49#11745687—21no—
50#11745687—41no—
51D4×D59—22no—
52D4×D59—22no—
53D4×D59—21no—
54D4×D59—41no—
55D4×D59—41no—
56D4×D59—41no—
57#11745685—21no—
58#11745685—41no—
59C2×D236—21no—
60C2×C4×D59—41no—
61C2×C4×D59D411yescharacteristic
62Q16×C59C2311yescharacteristic, complemented
63SD16×C59—21no—
64D8×C59—21no—
65D8×C59C2312yes—
66D8×C59C2311yescharacteristic, complemented
67C2×C8×C59C2311yescharacteristic
68#11745672—21no—
69#11745671—21no—
70#11745670—21no—
71#11745669—22no—
72#11745669—22no—
73#11745669—21no—
74C2×C59⋊C8D411yescharacteristic
75Dic236C2311yescharacteristic, complemented
76D472—21no—
77D472C2312yes—
78D472C2311yescharacteristic, complemented
79#11745660—21no—
80C8×D59D411yescharacteristic
81C8×D59D411yescharacteristic
82C8×D59C2311yescharacteristic
83C8×D59C2311yescharacteristic
84C16×C59C2312yescomplemented
85C59⋊C16C2312yescomplemented
86C23×C59—41no—
87C22×D59—41no—
88C22×D59—41no—
89C22×D59—42no—
90C22×D59—42no—
91C22×D59—84no—
92C22×D59D811yescharacteristic
93Q8×C59—21no—
94D4×C59—22no—
95D4×C59—21no—
96D4×C59—41no—
97D4×C59—41no—
98C2×C4×C59—41no—
99C2×C4×C59C2×D411yescharacteristic
100C22⋊D59—41no—
101C22⋊D59—41no—
102C22⋊D59—41no—
103C22⋊D59—42no—
104C22⋊D59—42no—
105C2×Dic59—41no—
106C2×Dic59D811yescharacteristic
107D236—21no—
108D236—22no—
109D236—41no—
110D236—41no—
111C4×D59—41no—
112C4×D59—41no—
113C4×D59—41no—
114C4×D59D811yescharacteristic
115C4×D59D811yescharacteristic
116C4×D59C2×D411yescharacteristic
117C4×D59C2×D411yescharacteristic
118Dic118—21no—
119Dic118—41no—
120C8×C59C2×D411yescharacteristic
121C8×C59C2411yescharacteristic
122C59⋊C8C2×D411yescharacteristic
123C59⋊C8C2×D411yescharacteristic
124C22×C59—41no—
125C22×C59—42no—
126C22×C59—82no—
127C22×C59C2×D811yescharacteristic
128C2×D59—22no—
129C2×D59—41no—
130C2×D59—42no—
131C2×D59—84no—
132C2×D59—82no—
133C2×D59—82no—
134C2×D59—84no—
135C2×D59C2×D811yescharacteristic
136C2×D59C2×D811yescharacteristic
137C4×C59—41no—
138C4×C59C2×D811yescharacteristic
139C4×C59C22×D411yescharacteristic
140Dic59—41no—
141Dic59C2×D811yescharacteristic
142Dic59C2×D811yescharacteristic
143C2×#509—591nomaximal, Sylow
144C2×C59—21no—
145C2×C59—82no—
146C2×C59—81no—
147C2×C59C22×D811yescharacteristic
148D59—21no—
149D59—81no—
150D59—82no—
151D59#50912yesdirect product factor
152C2×#135—591no—
153C22×D8—591no—
154#509—591no—
155#509—591no—
156#509—592no—
157#509—592no—
158#509—591no—
159#509D5911yescharacteristic, direct product factor
160C2×SD32—592no—
161C2×D16—592no—
162C2×M32—591no—
163C59C2×#50911yescharacteristic, complemented, Sylow
164C2×#41—1181no—
165C22×D4—1181no—
166#135—591no—
167#135—591no—
168#135—591no—
169#135—1181no—
170#135—1181no—
171#135C2×D5911yescharacteristic, complemented
172C2×Q16—591no—
173C2×SD16—1181no—
174C2×D8—592no—
175C2×D8—592no—
176C2×D8—591no—
177C2×D8—591no—
178C2×D8—1181no—
179C2×D8—1181no—
180C2×D8—1181no—
181C2×D8C2×D5911yescharacteristic, complemented
182C22×C8—591no—
183SD32—592no—
184SD32—592no—
185SD32—592no—
186SD32C2×D5912yescomplemented
187D16—592no—
188D16—592no—
189D16—592no—
190D16C2×D5912yescomplemented
191M32—591no—
192M32—592no—
193M32C2×D5911yescharacteristic, complemented
194C2×C16—592no—
195C24—2361no—
196#41—21no—
197#41—1181no—
198#41—1181no—
199#41—1181no—
200#41—2361no—
201#41—2361no—
202C2×Q8—1181no—
203C2×D4—21no—
204C2×D4—1182no—
205C2×D4—1182no—
206C2×D4—1181no—
207C2×D4—1181no—
208C2×D4—2361no—
209C2×D4—2361no—
210C2×D4—2361no—
211C2×D4—2361no—
212C22×C4—591no—
213C22×C4—2361no—
214Q16—591no—
215Q16—1181no—
216Q16C22×D5911yescharacteristic, complemented
217SD16—21no—
218SD16—1181no—
219SD16—1181no—
220SD16—1181no—
221D8—21no—
222D8—591no—
223D8—592no—
224D8—1182no—
225D8—1182no—
226D8—1181no—
227D8—1181no—
228D8C22×D5912yes—
229D8C22×D5911yescharacteristic, complemented
230C2×C8—591no—
231C2×C8—591no—
232C2×C8—591no—
233C2×C8—591no—
234C2×C8—591no—
235C2×C8C22×D5911yescharacteristic
236C16—592no—
237C16C22×D5912yescomplemented
238C23—41no—
239C23—591no—
240C23—4724no—
241C23—2361no—
242C23—2362no—
243C23—2362no—
244C23—2361no—
245Q8—21no—
246Q8—1181no—
247Q8—2361no—
248D4—22no—
249D4—21no—
250D4—41no—
251D4—41no—
252D4—1182no—
253D4—1181no—
254D4—2361no—
255D4—2362no—
256D4—2362no—
257D4—2361no—
258D4—2361no—
259D4—2361no—
260D4—2361no—
261C2×C4—41no—
262C2×C4—591no—
263C2×C4—591no—
264C2×C4—591no—
265C2×C4—591no—
266C2×C4—591no—
267C2×C4—2361no—
268C2×C4—2361no—
269C2×C4—2361no—
270C2×C4—2361no—
271C2×C4D4×D5911yescharacteristic
272C8—591no—
273C8—591no—
274C8D4×D5911yescharacteristic
275C8C23×D5911yescharacteristic
276C22—41no—
277C22—42no—
278C22—82no—
279C22—591no—
280C22—591no—
281C22—1182no—
282C22—4722no—
283C22—4724no—
284C22—4722no—
285C22—4724no—
286C22—2361no—
287C22—2362no—
288C22D8×D5911yescharacteristic
289C4—41no—
290C4—591no—
291C4—591no—
292C4—2361no—
293C4D8×D5911yescharacteristic
294C4C2×D4×D5911yescharacteristic
295C2—21no—
296C2—82no—
297C2—81no—
298C2—592no—
299C2—1181no—
300C2—4722no—
301C2—4721no—
302C2C2×D8×D5911yescharacteristic
subgroupG/H
centerC2C2×D8×D59
commutatorC8×C59C24
FrattiniC8C23×D59
FittingC59×#509C2
socleC2×C59C22×D8
hypercenter#509D59
hypocenterC59C2×#509
LCSG ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #135×D59C2semidirect
2. C2×D8×D59C2semidirect
3. #16837840C2semidirect
4. #16837839C2semidirect
5. #16837838C2semidirect
6. #16837837C2semidirect
7. #16837836C2semidirect
8. SD32×D59C2semidirect
9. D16×D59C2semidirect
10. M32×D59C2semidirect
11. C59×#509C2semidirect
14. #14273750C22semidirect
15. #14273749C22semidirect
18. #14273746C22semidirect
19. Q16×D59C22semidirect
21. C2×D472C22semidirect
26. D8×D59C22semidirect
27. D8×D59C22semidirect
29. #14273672C22semidirect
30. #14273671C22semidirect
31. #14273670C22semidirect
32. D944C22semidirect
33. #14273668C22semidirect
34. #14273668C22semidirect
35. #14273667C22semidirect
36. #14273666C22semidirect
37. C16×D59C22semidirect
38. #135×C59C22semidirect
39. C2×D8×C59C22semidirect
40. SD32×C59C22semidirect
41. D16×C59C22semidirect
42. M32×C59C22semidirect
62. Q16×C59C23semidirect
66. D8×C59C23semidirect
75. Dic236C23semidirect
78. D472C23semidirect
84. C16×C59C23semidirect
85. C59⋊C16C23semidirect
151. D59#509direct (N × H)
159. #509D59direct (N × H)
163. C59C2×#509semidirect
171. #135C2×D59semidirect
181. C2×D8C2×D59semidirect
186. SD32C2×D59semidirect
190. D16C2×D59semidirect
193. M32C2×D59semidirect
216. Q16C22×D59semidirect
229. D8C22×D59semidirect
237. C16C22×D59semidirect