Finite Group Databasebeta

← #28411670

Subgroups — #28411670

Order 14656.

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#20301033C211yesmaximal, characteristic
2#20300971C212yesmaximal, complemented
3#20300903C211yesmaximal, characteristic, complemented
4#20300902C212yesmaximal, complemented
5#123×C229C211yesmaximal, characteristic, complemented
6#16814666—21no—
7#16814665—22no—
8C2×C22⋊D229—21no—
9C2×C22⋊D229C2212yes—
10D4×D229—22no—
11#16814645—22no—
12C22⋊Dic229C2212yescomplemented
13#16814643—22no—
14C2×D4×C229C2211yescharacteristic
15C22×C4×C229C2211yescharacteristic, complemented
16C4⋊C4×C229—21no—
17C22⋊C4×C229—21no—
18C22⋊C4×C229C2211yescharacteristic, complemented
19C22×D229—21no—
20C22×D229D412yes—
21Dic458—21no—
22D916—21no—
23C22⋊D229—22no—
24C22⋊D229—22no—
25C22⋊D229—22no—
26C2×Dic229—21no—
27C2×Dic229—22no—
28C2×Dic229D412yes—
29C4×D229—22no—
30C23×C229D411yescharacteristic
31C23×C229C2311yescharacteristic
32D4×C229—22no—
33C2×C4×C229—21no—
34C2×C4×C229—21no—
35C2×C4×C229—22no—
36C2×C4×C229D411yescharacteristic
37C22×C229—21no—
38C22×C229—41no—
39C22×C229—41no—
40C22×C229C2×D412yes—
41C22×C229C2×D411yescharacteristic
42C2×D229—22no—
43C2×D229—42no—
44C2×D229—81no—
45C4×C229—21no—
46C4×C229—41no—
47C4×C229—41no—
48Dic229—22no—
49Dic229—42no—
50C2×C229—22no—
51C2×C229—21no—
52C2×C229—41no—
53C2×C229#12011yescharacteristic
54D229—42no—
55C229#45811yescharacteristic, complemented, Sylow
56#458—2291nomaximal, Sylow
57H(5,𝔽2)—2291no—
58#123—2292no—
59#123D22911yescharacteristic, complemented
60#99—2291no—
61#99—2292no—
62#41—4582no—
63#41—4581no—
64C2×D4—4582no—
65C2×D4—4581no—
66C2×D4—2292no—
67C2×D4C2×D22911yescharacteristic
68C22×C4C2×D22911yescharacteristic, complemented
69C4⋊C4—21no—
70C4⋊C4—4582no—
71C22⋊C4—21no—
72C22⋊C4—4582no—
73C22⋊C4—2292no—
74C22⋊C4C2×D22911yescharacteristic, complemented
75C23—4581no—
76C23—2292no—
77C23C22⋊D22911yescharacteristic
78C23C22×D22911yescharacteristic
79Q8—4581no—
80D4—22no—
81D4—4581no—
82D4—4582no—
83D4—4582no—
84D4—4582no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—4582no—
89C2×C4—4582no—
90C2×C4—4581no—
91C2×C4—2292no—
92C2×C4C22⋊D22911yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—18321no—
97C22—9162no—
98C22—4582no—
99C22D4×D22912yes—
100C22C2×C22⋊D22911yescharacteristic
101C4—21no—
102C4—41no—
103C4—41no—
104C4—9162no—
105C4—4582no—
106C2—21no—
107C2—22no—
108C2—41no—
109C2—9162no—
110C2#2030095611yescharacteristic
subgroupG/H
centerC2#20300956
commutator, cosocleC23×C229C23
FrattiniC23C22×D229
Fitting#123×C229C2
socleC2×C229#120
prime radical#20301033C2
hypercenter#123D229
hypocenterC229#458
LCSG ▷ C23×C229 ▷ C2×C229 ▷ C229
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #20300971C2semidirect
3. #20300903C2semidirect
4. #20300902C2semidirect
5. #123×C229C2semidirect
12. C22⋊Dic229C22semidirect
15. C22×C4×C229C22semidirect
18. C22⋊C4×C229C22semidirect
55. C229#458semidirect
59. #123D229semidirect
68. C22×C4C2×D229semidirect
74. C22⋊C4C2×D229semidirect

Maximal subgroups

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

subgroupindex
#203010332
#203009712
#203009032
#203009022
#123×C2292
#458229