Finite Group Databasebeta

← #45656047

Subgroups — #45656047

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#30654263C211yesmaximal, characteristic, complemented
2#30654261C211yesmaximal, characteristic, complemented
3#30654247C212yesmaximal, complemented
4#30654243C211yesmaximal, characteristic, complemented
5#30654242C212yesmaximal, complemented
6#30654238C212yesmaximal, complemented
7#30654235C211yesmaximal, characteristic, complemented
8C2×#22550848C211yesmaximal, characteristic, complemented
9SD16×D509C212yesmaximal, complemented
10#30654223C211yesmaximal, characteristic, complemented
11#137×C509C211yesmaximal, characteristic, complemented
12#22550862—21no—
13#22550862C2212yes—
14#22550862C2212yescomplemented
15#22550861—21no—
16#22550861—22no—
17#22550861C2211yescharacteristic
18#22550861C2211yescharacteristic, complemented
19Q8×D509—21no—
20Q8×D509C2212yescomplemented
21C2×D2036—21no—
22C2×D2036C2211yescharacteristic
23D4×D509—22no—
24D4×D509C2212yes—
25#22550851C2211yescharacteristic, complemented
26#22550850C2212yescomplemented
27#22550849C2211yescharacteristic, complemented
28#22550848C2212yescomplemented
29#22550848C2212yescomplemented
30#22550848C2211yescharacteristic, complemented
31D4072C2212yescomplemented
32#22550846C2211yescharacteristic, complemented
33#22550844C2212yescomplemented
34C2×#16861325C2211yescharacteristic, complemented
35C8×D509C2212yescomplemented
36#41×C509C2211yescharacteristic
37C2×Q8×C509C2211yescharacteristic, complemented
38Q16×C509C2212yescomplemented
39SD16×C509C2212yescomplemented
40M16×C509C2211yescharacteristic, complemented
41C22×D509—21no—
42C22×D509—22no—
43Dic1018—22no—
44Dic1018D411yescharacteristic
45Dic1018C2311yescharacteristic
46D2036—22no—
47D2036—22no—
48D2036—21no—
49D2036D411yescharacteristic
50D2036C2311yescharacteristic
51D2036C2312yes—
52C22⋊D509—21no—
53C22⋊D509—22no—
54C22⋊D509D412yes—
55C4×D509—22no—
56C4×D509—21no—
57C4×D509—22no—
58C4×D509—22no—
59C4×D509C2312yes—
60#16861325C2311yescharacteristic
61#16861325C2311yescharacteristic
62Q8×C509—21no—
63Q8×C509C2312yes—
64Q8×C509C2311yescharacteristic
65D4×C509—21no—
66D4×C509C2311yescharacteristic
67C2×C4×C509—21no—
68C2×C4×C509—21no—
69C2×C4×C509C2311yescharacteristic
70C8×C509C2312yes—
71C22×C509—21no—
72C22×C509C2×D411yescharacteristic
73C2×D509—21no—
74C2×D509—22no—
75C2×D509—82no—
76C2×D509—81no—
77C2×D509C2×D412yes—
78C4×C509—21no—
79C4×C509—22no—
80C4×C509C2×D411yescharacteristic
81C4×C509C2411yescharacteristic
82Dic509—21no—
83Dic509C2×D412yes—
84C2×C509—21no—
85C2×C509—41no—
86C2×C509C22×D411yescharacteristic
87D509—22no—
88D509—41no—
89D509—42no—
90C509#57711yescharacteristic, complemented, Sylow
91#577—5091nomaximal, Sylow
922−1+4—5091no—
93H(5,𝔽2)—5091no—
94#137—5092no—
95#137D50911yescharacteristic, complemented
96AGL(1,ℤ/8)—5092no—
97AGL(1,ℤ/8)—5091no—
98#135—5091no—
99#135—5092no—
100C2×SD16—5091no—
101C2×SD16—5092no—
102#131—5091no—
103#41—10181no—
104#41—10182no—
105#41—10181no—
106#41—5092no—
107#41—5092no—
108#41—5091no—
109#41—5091no—
110#41C2×D50911yescharacteristic
111C2×Q8—10181no—
112C2×Q8—5092no—
113C2×Q8C2×D50911yescharacteristic, complemented
114C2×D4—10181no—
115C2×D4—10182no—
116C2×D4—5092no—
117C2×D4—5091no—
118Q16—5091no—
119Q16C2×D50912yescomplemented
120SD16—5092no—
121SD16—5092no—
122SD16—5091no—
123SD16—5091no—
124SD16—5092no—
125SD16C2×D50912yescomplemented
126D8—5091no—
127D8—5092no—
128M16—5092no—
129M16C2×D50911yescharacteristic, complemented
130C2×C8—5091no—
131C2×C8—5092no—
132C23—10182no—
133C23—10181no—
134Q8—21no—
135Q8—10182no—
136Q8—5091no—
137Q8—5091no—
138Q8C22×D50912yes—
139Q8C22×D50911yescharacteristic
140D4—21no—
141D4—10182no—
142D4—10182no—
143D4—10181no—
144D4—10182no—
145D4—10181no—
146D4—5092no—
147D4—5091no—
148D4—5091no—
149D4—5092no—
150D4C22×D50911yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—10182no—
154C2×C4—10182no—
155C2×C4—10182no—
156C2×C4—10181no—
157C2×C4—5092no—
158C2×C4C22×D50911yescharacteristic
159C8—5091no—
160C8—5091no—
161C8C22×D50912yes—
162C22—21no—
163C22—40721no—
164C22—40722no—
165C22—10181no—
166C22—10182no—
167C22—5092no—
168C22D4×D50911yescharacteristic
169C4—21no—
170C4—22no—
171C4—10181no—
172C4—5092no—
173C4D4×D50911yescharacteristic
174C4C23×D50911yescharacteristic
175C2—21no—
176C2—41no—
177C2—20362no—
178C2—20361no—
179C2—10182no—
180C2C2×D4×D50911yescharacteristic
subgroupG/H
centerC2C2×D4×D509
commutator, cosocleC4×C509C24
FrattiniC4C23×D509
Fitting#137×C509C2
socleC2×C509C22×D4
hypercenter#137D509
hypocenterC509#577
LCSG ▷ C4×C509 ▷ C2×C509 ▷ C509
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #30654263C2semidirect
2. #30654261C2semidirect
3. #30654247C2semidirect
4. #30654243C2semidirect
5. #30654242C2semidirect
6. #30654238C2semidirect
7. #30654235C2semidirect
8. C2×#22550848C2semidirect
9. SD16×D509C2semidirect
10. #30654223C2semidirect
11. #137×C509C2semidirect
14. #22550862C22semidirect
18. #22550861C22semidirect
20. Q8×D509C22semidirect
25. #22550851C22semidirect
26. #22550850C22semidirect
27. #22550849C22semidirect
28. #22550848C22semidirect
29. #22550848C22semidirect
30. #22550848C22semidirect
31. D4072C22semidirect
32. #22550846C22semidirect
33. #22550844C22semidirect
34. C2×#16861325C22semidirect
35. C8×D509C22semidirect
37. C2×Q8×C509C22semidirect
38. Q16×C509C22semidirect
39. SD16×C509C22semidirect
40. M16×C509C22semidirect
90. C509#577semidirect
95. #137D509semidirect
113. C2×Q8C2×D509semidirect
119. Q16C2×D509semidirect
125. SD16C2×D509semidirect
129. M16C2×D509semidirect

Maximal subgroups

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

subgroupindex
#306542632
#306542612
#306542472
#306542432
#306542422
#306542382
#306542352
C2×#225508482
SD16×D5092
#306542232
#137×C5092
#577509