Finite Group Databasebeta

← #30682129

Subgroups — #30682129

Order 16832.

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#22554515C211yesmaximal, characteristic
2#22554454C212yesmaximal, complemented
3#22554451C211yesmaximal, characteristic, complemented
4#22554384C212yesmaximal, complemented
5#99×C263C211yesmaximal, characteristic, complemented
6#16862376—21no—
7#16862375—22no—
8C2×C22⋊D263—21no—
9C2×C22⋊D263C2212yes—
10D4×D263—22no—
11C22×Dic263C2211yescharacteristic, complemented
12C4⋊Dic263—22no—
13Dic263⋊C4—21no—
14C22⋊Dic263—21no—
15C22⋊Dic263C2211yescharacteristic, complemented
16(C2×D263)⋊C4—22no—
17C2×D4×C263C2211yescharacteristic
18C22⋊C4×C263C2212yescomplemented
19C22×D263—21no—
20C22×D263D412yes—
21Dic526—21no—
22D1052—21no—
23C22⋊D263—22no—
24C22⋊D263—22no—
25C22⋊D263—22no—
26C2×Dic263—21no—
27C2×Dic263—22no—
28C2×Dic263—21no—
29C2×Dic263—21no—
30C2×Dic263D412yes—
31C4×D263—22no—
32C23×C263D411yescharacteristic
33C23×C263C2311yescharacteristic
34D4×C263—22no—
35C2×C4×C263—22no—
36C2×C4×C263D411yescharacteristic
37C22×C263—21no—
38C22×C263—41no—
39C22×C263—41no—
40C22×C263C2×D412yes—
41C22×C263C2×D411yescharacteristic
42C2×D263—22no—
43C2×D263—42no—
44C2×D263—81no—
45C4×C263—21no—
46C4×C263—42no—
47Dic263—22no—
48Dic263—41no—
49Dic263—41no—
50C2×C263—21no—
51C2×C263—22no—
52C2×C263—41no—
53C2×C263#12011yescharacteristic
54D263—42no—
55C263#45811yescharacteristic, complemented, Sylow
56#458—2631nomaximal, Sylow
57H(5,𝔽2)—2631no—
58#123—2631no—
59#123—2632no—
60#99—2632no—
61#99D26311yescharacteristic, complemented
62#41—5262no—
63#41—5261no—
64C2×D4—2632no—
65C2×D4—5262no—
66C2×D4—5261no—
67C2×D4C2×D26311yescharacteristic
68C22×C4—2631no—
69C4⋊C4—5262no—
70C4⋊C4—5261no—
71C22⋊C4—2631no—
72C22⋊C4—5262no—
73C22⋊C4—5261no—
74C22⋊C4C2×D26312yescomplemented
75C23—2632no—
76C23—5261no—
77C23D105211yescharacteristic
78C23C22×D26311yescharacteristic
79Q8—5261no—
80D4—22no—
81D4—5261no—
82D4—5262no—
83D4—5262no—
84D4—5262no—
85C2×C4—22no—
86C2×C4—2632no—
87C2×C4—5262no—
88C2×C4—5261no—
89C2×C4—5262no—
90C2×C4—5261no—
91C2×C4—5261no—
92C2×C4D105211yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—5262no—
97C22—10522no—
98C22—21041no—
99C22D4×D26312yes—
100C22C2×D105211yescharacteristic
101C4—21no—
102C4—42no—
103C4—5262no—
104C4—10521no—
105C4—10521no—
106C2—22no—
107C2—21no—
108C2—41no—
109C2—10522no—
110C2#2255444011yescharacteristic
subgroupG/H
centerC2#22554440
commutator, cosocleC23×C263C23
FrattiniC23C22×D263
Fitting#99×C263C2
socleC2×C263#120
prime radical#22554515C2
hypercenter#99D263
hypocenterC263#458
LCSG ▷ C23×C263 ▷ C2×C263 ▷ C263
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
2. #22554454C2semidirect
3. #22554451C2semidirect
4. #22554384C2semidirect
5. #99×C263C2semidirect
11. C22×Dic263C22semidirect
15. C22⋊Dic263C22semidirect
18. C22⋊C4×C263C22semidirect
55. C263#458semidirect
61. #99D263semidirect
74. C22⋊C4C2×D263semidirect

Maximal subgroups

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

subgroupindex
#225545152
#225544542
#225544512
#225543842
#99×C2632
#458263