Finite Group Databasebeta

← #42122614

Subgroups — #42122614

Order 27712.

268 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 107 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, complemented
2#28357246C213yesmaximal, complemented
3#28357245C213yesmaximal, complemented
4#28357240C213yesmaximal, complemented
5C2×#20294082C213yesmaximal, complemented
6#28357227C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C433C211yesmaximal, characteristic, complemented
8#20294098—21no—
9#20294097C2213yescomplemented
10C2×D1732C2213yescomplemented
11D4×D433—23no—
12#20294087C2211yescharacteristic, complemented
13#20294085C2213yescomplemented
14#20294084C2213yescomplemented
15#20294082C2213yescomplemented
16#20294082C2216yescomplemented
17#20294081C2213yescomplemented
18C2×#16812358C2213yescomplemented
19#41×C433—21no—
20#41×C433C2211yescharacteristic
21#41×C433C2213yescomplemented
22C2×D4×C433—23no—
23C2×D4×C433C2213yescomplemented
24C22×D433—23no—
25Dic866C2311yescharacteristic
26D1732—23no—
27D1732C2313yes—
28C22⋊D433—23no—
29C4×D433—23no—
30#16812358C2311yescharacteristic
31#16812358C2313yes—
32C23×C433—23no—
33Q8×C433D411yescharacteristic
34Q8×C433C2311yescharacteristic
35D4×C433—23no—
36D4×C433—23no—
37D4×C433D413yes—
38D4×C433C2313yes—
39C2×C4×C433—23no—
40C2×C4×C433C2313yes—
41C22×C433—23no—
42C22×C433—83no—
43C22×C433C2×D413yes—
44C2×D433—23no—
45C2×D433—83no—
46C4×C433—21no—
47C4×C433C2×D413yes—
48C4×C433C2411yescharacteristic
49Dic433—21no—
50C2×C433—23no—
51C2×C433—43no—
52C2×C433C22×D411yescharacteristic
53D433—43no—
54C433#57611yescharacteristic, complemented, Sylow
55#576—4331nomaximal, Sylow
56H(5,𝔽2)—4331no—
57H(5,𝔽2)D43311yescharacteristic, complemented
58AGL(1,ℤ/8)—4333no—
59AGL(1,ℤ/8)—4333no—
60#135—4333no—
61C2×D8—4333no—
62#131—4331no—
63#41—21no—
64#41—8661no—
65#41—4333no—
66#41C2×D43311yescharacteristic
67#41C2×D43313yescomplemented
68C2×D4—23no—
69C2×D4—8663no—
70C2×D4—4333no—
71C2×D4C2×D43313yescomplemented
72Q16—4331no—
73SD16—4333no—
74SD16—4333no—
75D8—4333no—
76D8—4336no—
77M16—4333no—
78C2×C8—4333no—
79C23—23no—
80C23—8663no—
81Q8—4331no—
82Q8C22⋊D43311yescharacteristic
83Q8C22×D43311yescharacteristic
84D4—23no—
85D4—23no—
86D4—8663no—
87D4—8663no—
88D4—4333no—
89D4C22⋊D43313yes—
90D4C22×D43313yes—
91C2×C4—23no—
92C2×C4—8663no—
93C2×C4C22×D43313yes—
94C8—4331no—
95C8—4333no—
96C22—23no—
97C22—83no—
98C22—8663no—
99C22—34643no—
100C22C2×C22⋊D43313yes—
101C4—21no—
102C4—8661no—
103C4C2×C22⋊D43313yes—
104C4C23×D43311yescharacteristic
105C2—23no—
106C2—43no—
107C2—17323no—
108C2C22×C22⋊D43311yescharacteristic
subgroupG/H
centerC2C22×C22⋊D433
commutator, cosocleC4×C433C24
FrattiniC4C23×D433
FittingH(5,𝔽2)×C433C2
socleC2×C433C22×D4
hypercenterH(5,𝔽2)D433
hypocenterC433#576
LCSG ▷ C4×C433 ▷ C2×C433 ▷ C433
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #28357266C2semidirect
2. #28357246C2semidirect
3. #28357245C2semidirect
4. #28357240C2semidirect
5. C2×#20294082C2semidirect
6. #28357227C2semidirect
7. H(5,𝔽2)×C433C2semidirect
9. #20294097C22semidirect
10. C2×D1732C22semidirect
12. #20294087C22semidirect
13. #20294085C22semidirect
14. #20294084C22semidirect
15. #20294082C22semidirect
16. #20294082C22semidirect
17. #20294081C22semidirect
18. C2×#16812358C22semidirect
21. #41×C433C22semidirect
23. C2×D4×C433C22semidirect
54. C433#576semidirect
57. H(5,𝔽2)D433semidirect
67. #41C2×D433semidirect
71. C2×D4C2×D433semidirect

Maximal subgroups

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

subgroupindex
#283572662
#283572462
#283572452
#283572402
C2×#202940822
#283572272
H(5,𝔽2)×C4332
#576433