Finite Group Databasebeta

← #21435896

Subgroups — #21435896

Order 7552.

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

normal subgroups only

This group has 476 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#16838207C212yesmaximal, complemented
2#16838204C212yesmaximal, complemented
3#16837779C211yesmaximal, characteristic, complemented
4#16837673C212yesmaximal, complemented
5#16837672C212yesmaximal, complemented
6#16837571C214yesmaximal, complemented
7#16837520C211yesmaximal, characteristic, complemented
8C59×#443C211yesmaximal, characteristic, complemented
9#14273773C2214yescomplemented
10#14273755—24no—
11#14273755C2214yescomplemented
12#14273754—24no—
13#14273752C2214yescomplemented
14#14273750—22no—
15#14273749—22no—
16#14273749C2211yescharacteristic
17C2×D472—22no—
18C2×D472C2212yescomplemented
19C2×D472C2212yescomplemented
20D8×D59—22no—
21#14273733C2212yescomplemented
22#14273724C2212yescomplemented
23#14273663C2211yescharacteristic, complemented
24#14273655C2212yescomplemented
25#14273643C2214yescomplemented
26#135×C59C2211yescharacteristic
27#131×C59C2212yescomplemented
28#108×C59C2211yescharacteristic, complemented
29#104×C59C2212yescomplemented
30C4×C8×C59C2211yescharacteristic, complemented
31#41×C59C2312yes—
32#11745689—24no—
33D4×D59—24no—
34D4×D59—44no—
35#11745685—24no—
36#11745685C2312yes—
37C2×D236—24no—
38C2×D236—42no—
39C2×D236C2314yes—
40Q16×C59D411yescharacteristic, complemented
41SD16×C59D412yescomplemented
42D8×C59D411yescharacteristic, complemented
43M16×C59—22no—
44M16×C59C2312yes—
45C2×C8×C59—22no—
46C2×C8×C59C2311yescharacteristic
47C2×C8×C59C2311yescharacteristic
48C42×C59C2311yescharacteristic
49#11745671—24no—
50#11745669—24no—
51#11745664C2312yes—
52Dic236D411yescharacteristic, complemented
53D472—24no—
54D472—22no—
55D472—22no—
56D472—22no—
57D472—22no—
58D472D411yescharacteristic, complemented
59#11745660—24no—
60#11745660D412yescomplemented
61#11745659—22no—
62C8×D59—22no—
63C22×D59—24no—
64C22×D59—44no—
65Q8×C59C2×D412yes—
66D4×C59—22no—
67D4×C59C2×D412yes—
68C2×C4×C59—21no—
69C2×C4×C59—22no—
70C2×C4×C59C2411yescharacteristic
71C22⋊D59—22no—
72C22⋊D59—44no—
73D236—24no—
74D236—24no—
75D236—44no—
76D236—84no—
77D236C2×D412yes—
78C4×D59—22no—
79C4×D59—44no—
80Dic118C2×D412yes—
81C8×C59—22no—
82C8×C59C2×D411yescharacteristic
83C8×C59C2×D411yescharacteristic
84C8×C59C2×D411yescharacteristic
85C8×C59C2×D411yescharacteristic
86C59⋊C8—22no—
87C22×C59—22no—
88C22×C59H(5,𝔽2)11yescharacteristic
89C2×D59—22no—
90C2×D59—44no—
91C2×D59—84no—
92C2×D59—164no—
93C4×C59—22no—
94C4×C59—22no—
95C4×C59C22×D411yescharacteristic
96C4×C59C22×D411yescharacteristic
97Dic59—22no—
98#3292—591nomaximal, Sylow
99C2×C59—21no—
100C2×C59—42no—
101C2×C59D4211yescharacteristic
102D59—42no—
103D59—84no—
104#576—592no—
105#576—592no—
106#493—591no—
107#472—592no—
108#472—592no—
109#453—594no—
110#443—591no—
111#443D5911yescharacteristic, complemented
112C59#329211yescharacteristic, complemented, Sylow
113H(5,𝔽2)—594no—
114AGL(1,ℤ/8)—594no—
115AGL(1,ℤ/8)—594no—
116AGL(1,ℤ/8)—1184no—
117AGL(1,ℤ/8)—1184no—
118#135—591no—
119#135—1182no—
120#135—1182no—
121#135C2×D5911yescharacteristic
122C2×D8—592no—
123C2×D8—592no—
124C2×D8—1182no—
125C2×D8—1182no—
126#131—592no—
127#131C2×D5912yescomplemented
128#127—592no—
129#108—591no—
130#108C2×D5911yescharacteristic, complemented
131#104—592no—
132#104C2×D5912yescomplemented
133#100—594no—
134C4×C8C2×D5911yescharacteristic, complemented
135#41—592no—
136#41—1184no—
137#41—1184no—
138#41C22×D5912yes—
139C2×D4—594no—
140C2×D4—1184no—
141C2×D4—1184no—
142C2×D4—2362no—
143C2×D4—2364no—
144Q16—591no—
145Q16D23611yescharacteristic, complemented
146SD16—592no—
147SD16—1184no—
148SD16—1184no—
149SD16D23612yescomplemented
150D8—591no—
151D8—1184no—
152D8—1184no—
153D8—1182no—
154D8—1182no—
155D8—1182no—
156D8—1182no—
157D8D23611yescharacteristic, complemented
158M16—22no—
159M16—592no—
160M16—1182no—
161M16C22×D5912yes—
162C2×C8—22no—
163C2×C8—1182no—
164C2×C8C22×D5911yescharacteristic
165C2×C8C22×D5911yescharacteristic
166C42C22×D5911yescharacteristic
167C23—1184no—
168C23—2364no—
169Q8—592no—
170Q8C2×D23612yes—
171D4—22no—
172D4—592no—
173D4—1182no—
174D4—1184no—
175D4—1184no—
176D4—4724no—
177D4—2364no—
178D4—2364no—
179D4C2×D23612yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—1182no—
183C2×C4—2364no—
184C2×C4C23×D5911yescharacteristic
185C8—22no—
186C8—1182no—
187C8C2×D23611yescharacteristic
188C8C2×D23611yescharacteristic
189C8D4×D5911yescharacteristic
190C8D4×D5911yescharacteristic
191C22—22no—
192C22—1182no—
193C22—9444no—
194C22—4724no—
195C22—2364no—
196C22#1427377311yescharacteristic
197C4—22no—
198C4—22no—
199C4—1182no—
200C4C2×D4×D5911yescharacteristic
201C4C22×D23611yescharacteristic
202C2—21no—
203C2—42no—
204C2—4724no—
205C2—2362no—
206C2D4×D23611yescharacteristic
subgroupG/H
centerC2D4×D236
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#443C2
socleC2×C59D42
hypercenter#443D59
hypocenterC59#3292
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. #16838207C2semidirect
2. #16838204C2semidirect
3. #16837779C2semidirect
4. #16837673C2semidirect
5. #16837672C2semidirect
6. #16837571C2semidirect
7. #16837520C2semidirect
8. C59×#443C2semidirect
9. #14273773C22semidirect
11. #14273755C22semidirect
13. #14273752C22semidirect
18. C2×D472C22semidirect
19. C2×D472C22semidirect
21. #14273733C22semidirect
22. #14273724C22semidirect
23. #14273663C22semidirect
24. #14273655C22semidirect
25. #14273643C22semidirect
27. #131×C59C22semidirect
28. #108×C59C22semidirect
29. #104×C59C22semidirect
30. C4×C8×C59C22semidirect
40. Q16×C59D4semidirect
41. SD16×C59D4semidirect
42. D8×C59D4semidirect
52. Dic236D4semidirect
58. D472D4semidirect
60. #11745660D4semidirect
111. #443D59semidirect
112. C59#3292semidirect
127. #131C2×D59semidirect
130. #108C2×D59semidirect
132. #104C2×D59semidirect
134. C4×C8C2×D59semidirect
145. Q16D236semidirect
149. SD16D236semidirect
157. D8D236semidirect