Finite Group Databasebeta

← #45655744

Subgroups — #45655744

Order 32576.

330 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#22550860C211yesmaximal, characteristic, complemented
2C23×Dic509C211yesmaximal, characteristic, complemented
3#30654199C211yesmaximal, characteristic, complemented
4#30654196C211yesmaximal, characteristic, complemented
5#30654192C212yesmaximal, complemented
6#30654187C212yesmaximal, complemented
7#30654184C211yesmaximal, characteristic, complemented
8D4×Dic509C212yesmaximal, complemented
9#30654175C212yesmaximal, complemented
10#30654171C211yesmaximal, characteristic, complemented
11#120×C509C211yesmaximal, characteristic, complemented
12#22550860—24no—
13C2×Dic1018C2211yescharacteristic, complemented
14C2×C22⋊D509C2211yescharacteristic
15C2×C22⋊D509C2212yes—
16C2×C22⋊D509C2211yescharacteristic, complemented
17C22×Dic509—24no—
18C22×Dic509C2212yes—
19C22×Dic509C2212yes—
20C22×Dic509C2211yescharacteristic
21C22×Dic509C2211yescharacteristic
22C22×Dic509C2211yescharacteristic, complemented
23C2×C4×D509C2211yescharacteristic, complemented
24C4⋊Dic509C2212yescomplemented
25#22550840C2212yescomplemented
26#22550840C2212yescomplemented
27C22⋊Dic509C2211yescharacteristic
28C22⋊Dic509C2212yes—
29C22⋊Dic509C2212yescomplemented
30#22550838C2212yescomplemented
31C4×Dic509C2212yescomplemented
32C24×C509C2211yescharacteristic
33C2×D4×C509C2211yescharacteristic
34C2×D4×C509C2212yes—
35C22⋊C4×C509C2211yescharacteristic
36C22⋊C4×C509C2212yes—
37C22×D509C2311yescharacteristic
38Dic1018—22no—
39C22⋊D509—22no—
40C22⋊D509—24no—
41C22⋊D509—22no—
42C2×Dic509—22no—
43C2×Dic509—24no—
44C2×Dic509—24no—
45C2×Dic509—22no—
46C2×Dic509D414yescomplemented
47C2×Dic509C2312yes—
48C2×Dic509C2312yes—
49C2×Dic509C2311yescharacteristic
50C2×Dic509C2311yescharacteristic
51C2×Dic509C2311yescharacteristic
52C4×D509—22no—
53C23×C509—22no—
54C23×C509—24no—
55C23×C509C2311yescharacteristic
56C23×C509C2311yescharacteristic
57C23×C509C2312yes—
58D4×C509—22no—
59D4×C509—24no—
60C2×C4×C509C2312yes—
61C2×C4×C509C2311yescharacteristic
62C22×C509—28no—
63C22×C509—22no—
64C22×C509—22no—
65C22×C509—22no—
66C22×C509—21no—
67C22×C509—22no—
68C22×C509C2×D412yes—
69C22×C509#4114yescomplemented
70C22×C509C2411yescharacteristic
71C2×D509—21no—
72C2×D509—22no—
73C4×C509—21no—
74C4×C509—22no—
75Dic509—22no—
76Dic509—21no—
77Dic509—22no—
78Dic509C2×D412yes—
79Dic509C2×D412yes—
80C2×C509—22no—
81C2×C509—24no—
82C2×C509—41no—
83C2×C509C22×D411yescharacteristic
84C2×C509C2×#4112yes—
85D509—41no—
86C509#52511yescharacteristic, complemented, Sylow
87#525—5091nomaximal, Sylow
88C2×#41—5091no—
89C23×C4—5091no—
90#123—5091no—
91#123—5091no—
92C22⋊Q8—5092no—
93#121—5092no—
94#120—5091no—
95#120D50911yescharacteristic, complemented
96C4×D4—5092no—
97C4×D4—5092no—
98#117—5091no—
99C24C2×D50911yescharacteristic
100#41—10184no—
101C2×Q8—5091no—
102C2×D4—5092no—
103C2×D4—5091no—
104C2×D4—5091no—
105C2×D4C2×D50912yes—
106C2×D4C2×D50911yescharacteristic
107C22×C4—10184no—
108C22×C4—5092no—
109C22×C4—5092no—
110C22×C4—5091no—
111C22×C4—5091no—
112C22×C4—5091no—
113C22×C4—5091no—
114C4⋊C4—5092no—
115C4⋊C4—5092no—
116C4⋊C4—5092no—
117C22⋊C4—5091no—
118C22⋊C4—5092no—
119C22⋊C4—5092no—
120C22⋊C4—5092no—
121C22⋊C4C2×D50911yescharacteristic
122C22⋊C4C2×D50912yes—
123C42—5092no—
124C23—22no—
125C23—24no—
126C23—5091no—
127C23C22×D50911yescharacteristic
128C23C22×D50911yescharacteristic
129C23C22×D50912yes—
130Q8—10182no—
131D4—22no—
132D4—24no—
133D4—10182no—
134D4—10184no—
135D4—10182no—
136C2×C4—10182no—
137C2×C4—10182no—
138C2×C4—10184no—
139C2×C4—10184no—
140C2×C4—10182no—
141C2×C4—5092no—
142C2×C4—5092no—
143C2×C4—5091no—
144C2×C4—5094no—
145C2×C4—5091no—
146C2×C4—5091no—
147C2×C4C22×D50912yes—
148C2×C4C22×D50911yescharacteristic
149C22—28no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—22no—
155C22—10181no—
156C22—10182no—
157C22D4×D50912yes—
158C22#2255086014yescomplemented
159C22C23×D50911yescharacteristic
160C4—21no—
161C4—22no—
162C4—10182no—
163C4—10181no—
164C4—10182no—
165C4—5092no—
166C4—5092no—
167C2—22no—
168C2—24no—
169C2—41no—
170C2—20361no—
171C2C2×D4×D50911yescharacteristic
172C2C2×#2255086012yes—
subgroupG/H
center, FrattiniC22C23×D509
commutator, socle, cosocleC22×C509C24
Fitting#120×C509C2
hypercenter#120D509
hypocenterC509#525
LCSG ▷ C22×C509 ▷ C509
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#22550860C2semidirect
2. C23×Dic509C2semidirect
3. #30654199C2semidirect
4. #30654196C2semidirect
5. #30654192C2semidirect
6. #30654187C2semidirect
7. #30654184C2semidirect
8. D4×Dic509C2semidirect
9. #30654175C2semidirect
10. #30654171C2semidirect
11. #120×C509C2semidirect
13. C2×Dic1018C22semidirect
16. C2×C22⋊D509C22semidirect
22. C22×Dic509C22semidirect
23. C2×C4×D509C22semidirect
24. C4⋊Dic509C22semidirect
25. #22550840C22semidirect
26. #22550840C22semidirect
29. C22⋊Dic509C22semidirect
30. #22550838C22semidirect
31. C4×Dic509C22semidirect
46. C2×Dic509D4semidirect
69. C22×C509#41semidirect
86. C509#525semidirect
95. #120D509semidirect
158. C22#22550860semidirect

Maximal subgroups

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

subgroupindex
C2×#225508602
C23×Dic5092
#306541992
#306541962
#306541922
#306541872
#306541842
D4×Dic5092
#306541752
#306541712
#120×C5092
#525509