Finite Group Databasebeta

← #21435890

Subgroups — #21435890

Order 7552.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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#16838214C212yesmaximal, complemented
2#16838209C212yesmaximal, complemented
3#16837775C211yesmaximal, characteristic, complemented
4#16837666C212yesmaximal, complemented
5#16837665C212yesmaximal, complemented
6#16837586C212yesmaximal, complemented
7#16837571C212yesmaximal, complemented
8#16837520C211yesmaximal, characteristic, complemented
9C59×#443C211yesmaximal, characteristic, complemented
10#14273774C2212yes—
11#14273773C2212yescomplemented
12#14273760—22no—
13#14273760C2212yescomplemented
14#14273759—22no—
15#14273757C2212yescomplemented
16#14273755—22no—
17#14273755C2212yes—
18#14273752C2212yes—
19#14273751—22no—
20#14273749—22no—
21#14273749C2211yescharacteristic
22#14273748—22no—
23C2×#11745660—22no—
24C2×#11745660C2212yescomplemented
25C2×#11745660C2212yescomplemented
26SD16×D59—22no—
27#14273733C2212yescomplemented
28#14273727C2211yescharacteristic, complemented
29#14273724C2211yescharacteristic
30#14273663C2211yescharacteristic, complemented
31#14273655C2212yescomplemented
32#14273645C2212yescomplemented
33#14273643C2212yescomplemented
34#135×C59C2211yescharacteristic
35#131×C59C2212yescomplemented
36#108×C59C2211yescharacteristic, complemented
37#104×C59C2212yescomplemented
38C4×C8×C59C2211yescharacteristic, complemented
39#41×C59C2312yes—
40#11745689—22no—
41Q8×D59—22no—
42#11745687—22no—
43#11745687—42no—
44D4×D59—22no—
45D4×D59—42no—
46#11745685—22no—
47#11745685—22no—
48#11745685C2312yes—
49C2×D236—22no—
50C2×D236—41no—
51C2×D236C2312yes—
52C2×Dic118—22no—
53C2×Dic118C2312yes—
54Q16×C59D411yescharacteristic, complemented
55SD16×C59D412yescomplemented
56D8×C59D411yescharacteristic, complemented
57M16×C59—22no—
58M16×C59C2312yes—
59C2×C8×C59—22no—
60C2×C8×C59C2311yescharacteristic
61C2×C8×C59C2311yescharacteristic
62C42×C59C2311yescharacteristic
63#11745672—22no—
64#11745671—22no—
65#11745670—22no—
66#11745669—22no—
67C4⋊Dic59—41no—
68#11745664C2312yes—
69Dic236—22no—
70Dic236D411yescharacteristic, complemented
71D472—22no—
72D472D411yescharacteristic, complemented
73#11745660—22no—
74#11745660—22no—
75#11745660—22no—
76#11745660—22no—
77#11745660—22no—
78#11745660—22no—
79#11745660D412yescomplemented
80#11745659—22no—
81C8×D59—22no—
82C22×D59—22no—
83C22×D59—42no—
84Q8×C59C2×D412yes—
85D4×C59—22no—
86D4×C59C2×D412yes—
87C2×C4×C59—21no—
88C2×C4×C59—22no—
89C2×C4×C59C2411yescharacteristic
90C22⋊D59—22no—
91C22⋊D59—42no—
92C22⋊D59—42no—
93C2×Dic59—22no—
94C2×Dic59—42no—
95D236—22no—
96D236—22no—
97D236—42no—
98D236—82no—
99D236C2×D412yes—
100C4×D59—22no—
101C4×D59—42no—
102C4×D59—42no—
103Dic118—22no—
104Dic118—22no—
105Dic118—42no—
106Dic118C2×D412yes—
107C8×C59—22no—
108C8×C59C2×D411yescharacteristic
109C8×C59C2×D411yescharacteristic
110C8×C59C2×D411yescharacteristic
111C8×C59C2×D411yescharacteristic
112C59⋊C8—22no—
113C22×C59—22no—
114C22×C59H(5,𝔽2)11yescharacteristic
115C2×D59—22no—
116C2×D59—42no—
117C2×D59—82no—
118C2×D59—162no—
119C4×C59—22no—
120C4×C59—22no—
121C4×C59C22×D411yescharacteristic
122C4×C59C22×D411yescharacteristic
123Dic59—22no—
124Dic59—42no—
125#3291—591nomaximal, Sylow
126C2×C59—21no—
127C2×C59—42no—
128C2×C59D4211yescharacteristic
129D59—42no—
130D59—82no—
131#577—592no—
132#577—592no—
133#492—591no—
134#471—592no—
135#471—592no—
136#456—592no—
137#453—592no—
138#443—591no—
139#443D5911yescharacteristic, complemented
140C59#329111yescharacteristic, complemented, Sylow
1412−1+4—592no—
142H(5,𝔽2)—592no—
143#137—592no—
144#137—592no—
145#137—1182no—
146#137—1182no—
147AGL(1,ℤ/8)—592no—
148AGL(1,ℤ/8)—592no—
149AGL(1,ℤ/8)—1182no—
150AGL(1,ℤ/8)—1182no—
151#135—591no—
152#135—1182no—
153#135—1182no—
154#135C2×D5911yescharacteristic
155C2×SD16—592no—
156C2×SD16—592no—
157C2×SD16—1182no—
158C2×SD16—1182no—
159#131—592no—
160#131C2×D5912yescomplemented
161C4⋊Q8—591no—
162#127—591no—
163#108—591no—
164#108C2×D5911yescharacteristic, complemented
165#104—592no—
166#104C2×D5912yescomplemented
167#101—592no—
168#100—592no—
169C4×C8C2×D5911yescharacteristic, complemented
170#41—592no—
171#41—1182no—
172#41—1182no—
173#41—1182no—
174#41—1182no—
175#41—2362no—
176#41C22×D5912yes—
177C2×Q8—592no—
178C2×Q8—1182no—
179C2×Q8—1182no—
180C2×D4—592no—
181C2×D4—1182no—
182C2×D4—1182no—
183C2×D4—2361no—
184C2×D4—2362no—
185Q16—591no—
186Q16—1182no—
187Q16—1182no—
188Q16D23611yescharacteristic, complemented
189SD16—592no—
190SD16—1182no—
191SD16—1182no—
192SD16—1182no—
193SD16—1182no—
194SD16—1182no—
195SD16—1182no—
196SD16—1182no—
197SD16—1182no—
198SD16D23612yescomplemented
199D8—591no—
200D8—1182no—
201D8—1182no—
202D8D23611yescharacteristic, complemented
203M16—22no—
204M16—592no—
205M16—1182no—
206M16C22×D5912yes—
207C2×C8—22no—
208C2×C8—1182no—
209C2×C8C22×D5911yescharacteristic
210C2×C8C22×D5911yescharacteristic
211C4⋊C4—2361no—
212C42C22×D5911yescharacteristic
213C23—1182no—
214C23—2362no—
215Q8—592no—
216Q8—1182no—
217Q8—1182no—
218Q8—2362no—
219Q8C2×D23612yes—
220D4—22no—
221D4—592no—
222D4—1182no—
223D4—1182no—
224D4—1182no—
225D4—4722no—
226D4—2362no—
227D4—2362no—
228D4—2362no—
229D4C2×D23612yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—1182no—
233C2×C4—1182no—
234C2×C4—2362no—
235C2×C4—2362no—
236C2×C4—2362no—
237C2×C4C23×D5911yescharacteristic
238C8—22no—
239C8—1182no—
240C8C2×D23611yescharacteristic
241C8C2×D23611yescharacteristic
242C8D4×D5911yescharacteristic
243C8D4×D5911yescharacteristic
244C22—22no—
245C22—1182no—
246C22—9442no—
247C22—4722no—
248C22—2362no—
249C22#1427377311yescharacteristic
250C4—22no—
251C4—22no—
252C4—1182no—
253C4—2362no—
254C4C2×D4×D5911yescharacteristic
255C4C22×D23611yescharacteristic
256C2—21no—
257C2—42no—
258C2—4722no—
259C2—2362no—
260C2D4×D23611yescharacteristic
subgroupG/H
centerC2D4×D236
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#443C2
socleC2×C59D42
hypercenter#443D59
hypocenterC59#3291
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #16838214C2semidirect
2. #16838209C2semidirect
3. #16837775C2semidirect
4. #16837666C2semidirect
5. #16837665C2semidirect
6. #16837586C2semidirect
7. #16837571C2semidirect
8. #16837520C2semidirect
9. C59×#443C2semidirect
11. #14273773C22semidirect
13. #14273760C22semidirect
15. #14273757C22semidirect
24. C2×#11745660C22semidirect
25. C2×#11745660C22semidirect
27. #14273733C22semidirect
28. #14273727C22semidirect
30. #14273663C22semidirect
31. #14273655C22semidirect
32. #14273645C22semidirect
33. #14273643C22semidirect
35. #131×C59C22semidirect
36. #108×C59C22semidirect
37. #104×C59C22semidirect
38. C4×C8×C59C22semidirect
54. Q16×C59D4semidirect
55. SD16×C59D4semidirect
56. D8×C59D4semidirect
70. Dic236D4semidirect
72. D472D4semidirect
79. #11745660D4semidirect
139. #443D59semidirect
140. C59#3291semidirect
160. #131C2×D59semidirect
164. #108C2×D59semidirect
166. #104C2×D59semidirect
169. C4×C8C2×D59semidirect
188. Q16D236semidirect
198. SD16D236semidirect
202. D8D236semidirect