Finite Group Databasebeta

← #42122000

Subgroups — #42122000

Order 27712.

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#28357265C211yesmaximal, characteristic
2#28357203C212yesmaximal, complemented
3#28357135C211yesmaximal, characteristic, complemented
4#28357134C212yesmaximal, complemented
5#123×C433C211yesmaximal, characteristic, complemented
6#20294097—21no—
7#20294096—22no—
8C2×C22⋊D433—21no—
9C2×C22⋊D433C2212yes—
10D4×D433—22no—
11#20294076—22no—
12C22⋊Dic433C2212yescomplemented
13#20294074—22no—
14C2×D4×C433C2211yescharacteristic
15C22×C4×C433C2211yescharacteristic, complemented
16C4⋊C4×C433—21no—
17C22⋊C4×C433—21no—
18C22⋊C4×C433C2211yescharacteristic, complemented
19C22×D433—21no—
20C22×D433D412yes—
21Dic866—21no—
22D1732—21no—
23C22⋊D433—22no—
24C22⋊D433—22no—
25C22⋊D433—22no—
26C2×Dic433—21no—
27C2×Dic433—22no—
28C2×Dic433D412yes—
29C4×D433—22no—
30C23×C433D411yescharacteristic
31C23×C433C2311yescharacteristic
32D4×C433—22no—
33C2×C4×C433—21no—
34C2×C4×C433—21no—
35C2×C4×C433—22no—
36C2×C4×C433D411yescharacteristic
37C22×C433—21no—
38C22×C433—41no—
39C22×C433—41no—
40C22×C433C2×D412yes—
41C22×C433C2×D411yescharacteristic
42C2×D433—22no—
43C2×D433—42no—
44C2×D433—81no—
45C4×C433—21no—
46C4×C433—41no—
47C4×C433—41no—
48Dic433—22no—
49Dic433—42no—
50C2×C433—22no—
51C2×C433—21no—
52C2×C433—41no—
53C2×C433#12011yescharacteristic
54D433—42no—
55C433#45811yescharacteristic, complemented, Sylow
56#458—4331nomaximal, Sylow
57H(5,𝔽2)—4331no—
58#123—4332no—
59#123D43311yescharacteristic, complemented
60#99—4331no—
61#99—4332no—
62#41—8662no—
63#41—8661no—
64C2×D4—8662no—
65C2×D4—8661no—
66C2×D4—4332no—
67C2×D4C2×D43311yescharacteristic
68C22×C4C2×D43311yescharacteristic, complemented
69C4⋊C4—21no—
70C4⋊C4—8662no—
71C22⋊C4—21no—
72C22⋊C4—8662no—
73C22⋊C4—4332no—
74C22⋊C4C2×D43311yescharacteristic, complemented
75C23—8661no—
76C23—4332no—
77C23C22⋊D43311yescharacteristic
78C23C22×D43311yescharacteristic
79Q8—8661no—
80D4—22no—
81D4—8661no—
82D4—8662no—
83D4—8662no—
84D4—8662no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—8662no—
89C2×C4—8661no—
90C2×C4—8662no—
91C2×C4—4332no—
92C2×C4C22⋊D43311yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—8662no—
97C22—34641no—
98C22—17322no—
99C22D4×D43312yes—
100C22C2×C22⋊D43311yescharacteristic
101C4—21no—
102C4—41no—
103C4—41no—
104C4—8662no—
105C4—17322no—
106C2—21no—
107C2—22no—
108C2—41no—
109C2—17322no—
110C2#2835718811yescharacteristic
subgroupG/H
centerC2#28357188
commutator, cosocleC23×C433C23
FrattiniC23C22×D433
Fitting#123×C433C2
socleC2×C433#120
prime radical#28357265C2
hypercenter#123D433
hypocenterC433#458
LCSG ▷ C23×C433 ▷ C2×C433 ▷ C433
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #28357203C2semidirect
3. #28357135C2semidirect
4. #28357134C2semidirect
5. #123×C433C2semidirect
12. C22⋊Dic433C22semidirect
15. C22×C4×C433C22semidirect
18. C22⋊C4×C433C22semidirect
55. C433#458semidirect
59. #123D433semidirect
68. C22×C4C2×D433semidirect
74. C22⋊C4C2×D433semidirect

Maximal subgroups

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

subgroupindex
#283572652
#283572032
#283571352
#283571342
#123×C4332
#458433