Finite Group Databasebeta

← #45655429

Subgroups — #45655429

Order 32576.

160 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 39 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#30654260C211yesmaximal, characteristic
2#30654199C212yesmaximal, complemented
3#30654196C211yesmaximal, characteristic, complemented
4#30654129C212yesmaximal, complemented
5#99×C509C211yesmaximal, characteristic, complemented
6#22550861—21no—
7#22550860—22no—
8C2×C22⋊D509—21no—
9C2×C22⋊D509C2212yes—
10D4×D509—22no—
11C22×Dic509C2211yescharacteristic, complemented
12C4⋊Dic509—22no—
13#22550840—21no—
14C22⋊Dic509—21no—
15C22⋊Dic509C2211yescharacteristic, complemented
16#22550838—22no—
17C2×D4×C509C2211yescharacteristic
18C22⋊C4×C509C2212yescomplemented
19C22×D509—21no—
20C22×D509D412yes—
21Dic1018—21no—
22D2036—21no—
23C22⋊D509—22no—
24C22⋊D509—22no—
25C22⋊D509—22no—
26C2×Dic509—21no—
27C2×Dic509—22no—
28C2×Dic509—21no—
29C2×Dic509—21no—
30C2×Dic509D412yes—
31C4×D509—22no—
32C23×C509D411yescharacteristic
33C23×C509C2311yescharacteristic
34D4×C509—22no—
35C2×C4×C509—22no—
36C2×C4×C509D411yescharacteristic
37C22×C509—21no—
38C22×C509—41no—
39C22×C509—41no—
40C22×C509C2×D412yes—
41C22×C509C2×D411yescharacteristic
42C2×D509—22no—
43C2×D509—42no—
44C2×D509—81no—
45C4×C509—21no—
46C4×C509—42no—
47Dic509—22no—
48Dic509—41no—
49Dic509—41no—
50C2×C509—21no—
51C2×C509—22no—
52C2×C509—41no—
53C2×C509#12011yescharacteristic
54D509—42no—
55C509#45811yescharacteristic, complemented, Sylow
56#458—5091nomaximal, Sylow
57H(5,𝔽2)—5091no—
58#123—5091no—
59#123—5092no—
60#99—5092no—
61#99D50911yescharacteristic, complemented
62#41—10182no—
63#41—10181no—
64C2×D4—10182no—
65C2×D4—10181no—
66C2×D4—5092no—
67C2×D4C2×D50911yescharacteristic
68C22×C4—5091no—
69C4⋊C4—10182no—
70C4⋊C4—10181no—
71C22⋊C4—10182no—
72C22⋊C4—10181no—
73C22⋊C4—5091no—
74C22⋊C4C2×D50912yescomplemented
75C23—10181no—
76C23—5092no—
77C23D203611yescharacteristic
78C23C22×D50911yescharacteristic
79Q8—10181no—
80D4—22no—
81D4—10181no—
82D4—10182no—
83D4—10182no—
84D4—10182no—
85C2×C4—22no—
86C2×C4—10182no—
87C2×C4—10182no—
88C2×C4—10181no—
89C2×C4—10181no—
90C2×C4—10181no—
91C2×C4—5092no—
92C2×C4D203611yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—40721no—
97C22—20362no—
98C22—10182no—
99C22D4×D50912yes—
100C22C2×D203611yescharacteristic
101C4—21no—
102C4—42no—
103C4—20361no—
104C4—20361no—
105C4—10182no—
106C2—22no—
107C2—21no—
108C2—41no—
109C2—20362no—
110C2#3065418511yescharacteristic
subgroupG/H
centerC2#30654185
commutator, cosocleC23×C509C23
FrattiniC23C22×D509
Fitting#99×C509C2
socleC2×C509#120
prime radical#30654260C2
hypercenter#99D509
hypocenterC509#458
LCSG ▷ C23×C509 ▷ C2×C509 ▷ C509
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
2. #30654199C2semidirect
3. #30654196C2semidirect
4. #30654129C2semidirect
5. #99×C509C2semidirect
11. C22×Dic509C22semidirect
15. C22⋊Dic509C22semidirect
18. C22⋊C4×C509C22semidirect
55. C509#458semidirect
61. #99D509semidirect
74. C22⋊C4C2×D509semidirect

Maximal subgroups

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

subgroupindex
#306542602
#306541992
#306541962
#306541292
#99×C5092
#458509