Finite Group Databasebeta

← #45656045

Subgroups — #45656045

Order 32576.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 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#30654262C211yesmaximal, characteristic, complemented
2#30654261C211yesmaximal, characteristic, complemented
3#30654247C212yesmaximal, complemented
4#30654245C211yesmaximal, characteristic, complemented
5#30654240C211yesmaximal, characteristic, complemented
6#30654239C212yesmaximal, complemented
7#30654236C212yesmaximal, complemented
8C2×#22550850C211yesmaximal, characteristic, complemented
9SD16×D509C212yesmaximal, complemented
10#30654221C211yesmaximal, characteristic, complemented
11#135×C509C211yesmaximal, characteristic, complemented
12#22550862C2212yes—
13#22550861—21no—
14#22550861—21no—
15#22550861C2211yescharacteristic
16#22550860—22no—
17#22550860C2212yescomplemented
18C2×Dic1018—21no—
19C2×Dic1018C2211yescharacteristic, complemented
20Q8×D509C2212yescomplemented
21C2×D2036—21no—
22C2×D2036C2211yescharacteristic
23D4×D509—22no—
24D4×D509C2212yes—
25#22550851C2212yescomplemented
26#22550850C2212yescomplemented
27#22550850C2212yescomplemented
28#22550849C2212yescomplemented
29#22550848C2212yescomplemented
30#22550846C2212yescomplemented
31#22550845C2211yescharacteristic, complemented
32#22550844C2212yescomplemented
33C8×D509C2212yescomplemented
34#41×C509C2211yescharacteristic
35#41×C509C2211yescharacteristic, complemented
36Q16×C509C2211yescharacteristic, complemented
37SD16×C509C2211yescharacteristic, complemented
38SD16×C509C2211yescharacteristic, complemented
39D8×C509C2211yescharacteristic, complemented
40C2×C8×C509C2211yescharacteristic, complemented
41C22×D509—21no—
42C22×D509—22no—
43Dic1018—21no—
44Dic1018—22no—
45Dic1018D411yescharacteristic
46Dic1018C2312yes—
47D2036—22no—
48D2036—21no—
49D2036D411yescharacteristic
50D2036C2312yes—
51C22⋊D509—22no—
52C22⋊D509—22no—
53C22⋊D509D412yes—
54C2×Dic509—21no—
55C2×Dic509—22no—
56C4×D509—22no—
57C4×D509—22no—
58C4×D509C2312yes—
59#16861325C2312yes—
60Q8×C509C2311yescharacteristic
61Q8×C509C2311yescharacteristic
62D4×C509—21no—
63D4×C509—21no—
64D4×C509C2311yescharacteristic
65D4×C509C2311yescharacteristic
66C2×C4×C509—21no—
67C2×C4×C509—21no—
68C2×C4×C509C2311yescharacteristic
69C8×C509C2311yescharacteristic
70C8×C509C2311yescharacteristic
71C22×C509—21no—
72C22×C509—21no—
73C22×C509C2×D411yescharacteristic
74C2×D509—22no—
75C2×D509—82no—
76C2×D509—81no—
77C2×D509C2×D412yes—
78C4×C509—21no—
79C4×C509—21no—
80C4×C509C2×D411yescharacteristic
81C4×C509C2411yescharacteristic
82Dic509—22no—
83Dic509C2×D412yes—
84C2×C509—21no—
85C2×C509—41no—
86C2×C509—41no—
87C2×C509C22×D411yescharacteristic
88D509—22no—
89D509—42no—
90C509#57711yescharacteristic, complemented, Sylow
91#577—5091nomaximal, Sylow
922−1+4—5091no—
93H(5,𝔽2)—5091no—
94#137—5091no—
95#137—5092no—
96AGL(1,ℤ/8)—5091no—
97AGL(1,ℤ/8)—5092no—
98#135—5092no—
99#135D50911yescharacteristic, complemented
100C2×SD16—5092no—
101C2×SD16—5091no—
102#131—5091no—
103#41—10181no—
104#41—10181no—
105#41—10182no—
106#41—5092no—
107#41—5092no—
108#41—5091no—
109#41C2×D50911yescharacteristic
110#41C2×D50911yescharacteristic, complemented
111C2×Q8—10181no—
112C2×Q8—5092no—
113C2×Q8—5091no—
114C2×D4—10181no—
115C2×D4—10182no—
116C2×D4—5092no—
117C2×D4—5091no—
118Q16—5092no—
119Q16C2×D50911yescharacteristic, complemented
120SD16—5092no—
121SD16—5092no—
122SD16—5092no—
123SD16—5092no—
124SD16C2×D50911yescharacteristic, complemented
125SD16C2×D50911yescharacteristic, complemented
126D8—5092no—
127D8C2×D50911yescharacteristic, complemented
128M16—5091no—
129M16—5092no—
130C2×C8—5092no—
131C2×C8C2×D50911yescharacteristic, complemented
132C23—10182no—
133C23—10181no—
134Q8—10182no—
135Q8—10181no—
136Q8—5092no—
137Q8—5091no—
138Q8C22×D50911yescharacteristic
139Q8C22×D50911yescharacteristic
140D4—21no—
141D4—21no—
142D4—10182no—
143D4—10182no—
144D4—10182no—
145D4—10181no—
146D4—5092no—
147D4—5091no—
148D4—5092no—
149D4C22×D50911yescharacteristic
150D4C22×D50911yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—10182no—
154C2×C4—10182no—
155C2×C4—10181no—
156C2×C4—10182no—
157C2×C4—5092no—
158C2×C4C22×D50911yescharacteristic
159C8—5092no—
160C8C22×D50911yescharacteristic
161C8C22×D50911yescharacteristic
162C22—21no—
163C22—21no—
164C22—40721no—
165C22—40722no—
166C22—10182no—
167C22—5092no—
168C22D4×D50911yescharacteristic
169C4—21no—
170C4—21no—
171C4—10182no—
172C4—5092no—
173C4D4×D50911yescharacteristic
174C4C23×D50911yescharacteristic
175C2—21no—
176C2—41no—
177C2—41no—
178C2—20362no—
179C2—10182no—
180C2C2×D4×D50911yescharacteristic
subgroupG/H
centerC2C2×D4×D509
commutator, cosocleC4×C509C24
FrattiniC4C23×D509
Fitting#135×C509C2
socleC2×C509C22×D4
hypercenter#135D509
hypocenterC509#577
LCSG ▷ C4×C509 ▷ C2×C509 ▷ C509
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #30654262C2semidirect
2. #30654261C2semidirect
3. #30654247C2semidirect
4. #30654245C2semidirect
5. #30654240C2semidirect
6. #30654239C2semidirect
7. #30654236C2semidirect
8. C2×#22550850C2semidirect
9. SD16×D509C2semidirect
10. #30654221C2semidirect
11. #135×C509C2semidirect
17. #22550860C22semidirect
19. C2×Dic1018C22semidirect
20. Q8×D509C22semidirect
25. #22550851C22semidirect
26. #22550850C22semidirect
27. #22550850C22semidirect
28. #22550849C22semidirect
29. #22550848C22semidirect
30. #22550846C22semidirect
31. #22550845C22semidirect
32. #22550844C22semidirect
33. C8×D509C22semidirect
35. #41×C509C22semidirect
36. Q16×C509C22semidirect
37. SD16×C509C22semidirect
38. SD16×C509C22semidirect
39. D8×C509C22semidirect
40. C2×C8×C509C22semidirect
90. C509#577semidirect
99. #135D509semidirect
110. #41C2×D509semidirect
119. Q16C2×D509semidirect
124. SD16C2×D509semidirect
125. SD16C2×D509semidirect
127. D8C2×D509semidirect
131. C2×C8C2×D509semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#306542622
#306542612
#306542472
#306542452
#306542402
#306542392
#306542362
C2×#225508502
SD16×D5092
#306542212
#135×C5092
#577509