Finite Group Databasebeta

← #42121990

Subgroups — #42121990

Order 27712.

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

normal subgroups only

This group has 152 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#28357266C211yesmaximal, characteristic
2#28357253C211yesmaximal, characteristic, complemented
3#28357250C211yesmaximal, characteristic, complemented
4#28357208C211yesmaximal, characteristic, complemented
5#28357148C211yesmaximal, characteristic, complemented
6#28357136C211yesmaximal, characteristic, complemented
7#104×C433C211yesmaximal, characteristic, complemented
8#20294098—21no—
9#20294097—21no—
10#20294097C2211yescharacteristic
11C2×Dic866C2211yescharacteristic, complemented
12C2×D1732—21no—
13C2×D1732C2211yescharacteristic
14D4×D433—22no—
15D4×D433—21no—
16Dic1732—21no—
17#20294087—21no—
18#20294086—21no—
19#20294085—21no—
20#20294081C2211yescharacteristic, complemented
21#20294074—22no—
22#41×C433C2211yescharacteristic
23M16×C433C2211yescharacteristic, complemented
24C42×C433C2211yescharacteristic, complemented
25C22×D433—22no—
26C22×D433D412yes—
27Dic866—21no—
28Dic866—21no—
29Dic866D411yescharacteristic
30D1732—22no—
31D1732—21no—
32D1732—21no—
33D1732D411yescharacteristic
34C22⋊D433—22no—
35C22⋊D433—21no—
36C2×Dic433—21no—
37C4×D433—22no—
38C4×D433—21no—
39#16812358—21no—
40Q8×C433D411yescharacteristic
41D4×C433—21no—
42D4×C433D411yescharacteristic
43C2×C4×C433—21no—
44C2×C4×C433—21no—
45C2×C4×C433C2311yescharacteristic
46C8×C433—21no—
47C22×C433—21no—
48C22×C433C2×D411yescharacteristic
49C2×D433—22no—
50C2×D433—21no—
51C2×D433—42no—
52C2×D433—82no—
53C4×C433—21no—
54C4×C433—41no—
55C4×C433C2×D411yescharacteristic
56C4×C433C2×D411yescharacteristic
57Dic433—21no—
58Dic433—41no—
59C2×C433—21no—
60C2×C433—41no—
61C2×C433#12011yescharacteristic
62D433—42no—
63D433—41no—
64C433#45511yescharacteristic, complemented, Sylow
65#455—4331nomaximal, Sylow
66H(5,𝔽2)—4331no—
67#137—4331no—
68#137—4331no—
69#124—4331no—
70#104—4331no—
71#104D43311yescharacteristic, complemented
72#100—4331no—
73#41—8661no—
74#41—8661no—
75#41—4331no—
76#41C2×D43311yescharacteristic
77C2×Q8—4331no—
78C2×D4—8661no—
79C2×D4—8662no—
80C2×D4—8661no—
81C2×D4—4331no—
82Q16—8661no—
83Q16—8661no—
84SD16—8661no—
85SD16—8661no—
86M16—4331no—
87M16C2×D43311yescharacteristic, complemented
88C22⋊C4—8662no—
89C42C2×D43311yescharacteristic, complemented
90C23—8662no—
91C23—4332no—
92Q8—8661no—
93Q8—8661no—
94Q8—4331no—
95Q8D173211yescharacteristic
96D4—21no—
97D4—8662no—
98D4—8662no—
99D4—8661no—
100D4—8661no—
101D4—8661no—
102D4—4331no—
103D4D173211yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—8662no—
107C2×C4—8661no—
108C2×C4—8661no—
109C2×C4C22×D43311yescharacteristic
110C8—21no—
111C8—8661no—
112C22—21no—
113C22—8662no—
114C22—8661no—
115C22—34642no—
116C22—17322no—
117C22D4×D43311yescharacteristic
118C4—21no—
119C4—41no—
120C4—8661no—
121C4—17321no—
122C4D4×D43311yescharacteristic
123C4C2×D173211yescharacteristic
124C2—21no—
125C2—41no—
126C2—17321no—
127C2—17322no—
128C2#2835719011yescharacteristic
subgroupG/H
centerC2#28357190
commutator, cosocleC2×C4×C433C23
FrattiniC2×C4C22×D433
Fitting#104×C433C2
socleC2×C433#120
prime radical#28357266C2
hypercenter#104D433
hypocenterC433#455
LCSG ▷ C2×C4×C433 ▷ C2×C433 ▷ C433
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. #28357253C2semidirect
3. #28357250C2semidirect
4. #28357208C2semidirect
5. #28357148C2semidirect
6. #28357136C2semidirect
7. #104×C433C2semidirect
11. C2×Dic866C22semidirect
20. #20294081C22semidirect
23. M16×C433C22semidirect
24. C42×C433C22semidirect
64. C433#455semidirect
71. #104D433semidirect
87. M16C2×D433semidirect
89. C42C2×D433semidirect

Maximal subgroups

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

subgroupindex
#283572662
#283572532
#283572502
#283572082
#283571482
#283571362
#104×C4332
#455433