Finite Group Databasebeta

← #42121997

Subgroups — #42121997

Order 27712.

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

normal subgroups only

This group has 198 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, complemented
2#28357189C212yesmaximal, complemented
3#28357135C211yesmaximal, characteristic, complemented
4#28357134C212yesmaximal, complemented
5#120×C433C211yesmaximal, characteristic, complemented
6#20294097—21no—
7#20294096—22no—
8C2×C22⋊D433—22no—
9C2×C22⋊D433—21no—
10C2×C22⋊D433C2212yes—
11D4×D433—22no—
12C22⋊Dic433—22no—
13C22⋊Dic433C2212yescomplemented
14C24×C433C2211yescharacteristic, complemented
15C2×D4×C433—21no—
16C2×D4×C433C2211yescharacteristic
17C22⋊C4×C433—21no—
18C22⋊C4×C433C2211yescharacteristic, complemented
19C22×D433—21no—
20C22×D433D412yescomplemented
21Dic866—21no—
22D1732—21no—
23C22⋊D433—22no—
24C22⋊D433—22no—
25C22⋊D433—22no—
26C22⋊D433—44no—
27C2×Dic433—21no—
28C2×Dic433—22no—
29C2×Dic433D412yescomplemented
30C4×D433—22no—
31C23×C433—21no—
32C23×C433—22no—
33C23×C433—42no—
34C23×C433D411yescharacteristic, complemented
35C23×C433C2311yescharacteristic
36D4×C433—22no—
37D4×C433—42no—
38C2×C4×C433—21no—
39C2×C4×C433D411yescharacteristic, complemented
40C22×C433—22no—
41C22×C433—24no—
42C22×C433—21no—
43C22×C433—41no—
44C22×C433—41no—
45C22×C433—42no—
46C22×C433—41no—
47C22×C433—41no—
48C22×C433C2×D412yes—
49C22×C433C2×D411yescharacteristic
50C2×D433—22no—
51C2×D433—42no—
52C2×D433—81no—
53C4×C433—21no—
54C4×C433—41no—
55Dic433—22no—
56Dic433—42no—
57C2×C433—22no—
58C2×C433—21no—
59C2×C433—41no—
60C2×C433—42no—
61C2×C433#12011yescharacteristic
62D433—42no—
63C433B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—4331nomaximal, Sylow
65H(5,𝔽2)—4331no—
66#120—4332no—
67#120D43311yescharacteristic, complemented
68#99—4331no—
69#99—4332no—
70C24C2×D43311yescharacteristic, complemented
71#41—8662no—
72#41—8661no—
73C2×D4—21no—
74C2×D4—8662no—
75C2×D4—8661no—
76C2×D4—8662no—
77C2×D4—4332no—
78C2×D4C2×D43311yescharacteristic
79C22⋊C4—21no—
80C22⋊C4—8662no—
81C22⋊C4—4332no—
82C22⋊C4C2×D43311yescharacteristic, complemented
83C23—22no—
84C23—21no—
85C23—42no—
86C23—8661no—
87C23—4332no—
88C23C22⋊D43311yescharacteristic, complemented
89C23C22×D43311yescharacteristic
90Q8—8661no—
91D4—22no—
92D4—42no—
93D4—8661no—
94D4—8662no—
95D4—8662no—
96D4—8662no—
97D4—17324no—
98C2×C4—21no—
99C2×C4—8662no—
100C2×C4—8661no—
101C2×C4—8662no—
102C2×C4—4332no—
103C2×C4C22⋊D43311yescharacteristic, complemented
104C22—22no—
105C22—24no—
106C22—21no—
107C22—41no—
108C22—41no—
109C22—41no—
110C22—42no—
111C22—41no—
112C22—8662no—
113C22—34641no—
114C22—17322no—
115C22D4×D43312yes—
116C22C2×C22⋊D43311yescharacteristic
117C4—21no—
118C4—41no—
119C4—8662no—
120C4—17322no—
121C2—21no—
122C2—22no—
123C2—42no—
124C2—41no—
125C2—17322no—
126C2#2835718811yescharacteristic
subgroupG/H
centerC2#28357188
commutator, cosocleC23×C433C23
FrattiniC23C22×D433
Fitting#120×C433C2
socleC2×C433#120
hypercenter#120D433
hypocenterC433B(4,𝔽2)
LCSG ▷ C23×C433 ▷ C2×C433 ▷ C433
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #28357265C2semidirect
2. #28357189C2semidirect
3. #28357135C2semidirect
4. #28357134C2semidirect
5. #120×C433C2semidirect
13. C22⋊Dic433C22semidirect
14. C24×C433C22semidirect
18. C22⋊C4×C433C22semidirect
20. C22×D433D4semidirect
29. C2×Dic433D4semidirect
34. C23×C433D4semidirect
39. C2×C4×C433D4semidirect
63. C433B(4,𝔽2)semidirect
67. #120D433semidirect
70. C24C2×D433semidirect
82. C22⋊C4C2×D433semidirect
88. C23C22⋊D433semidirect
103. C2×C4C22⋊D433semidirect

Maximal subgroups

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

subgroupindex
#283572652
#283571892
#283571352
#283571342
#120×C4332
B(4,𝔽2)433