Finite Group Databasebeta

← #27213949

Subgroups — #27213949

Order 13504.

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#20270023C211yesmaximal, characteristic, complemented
2#20270003C213yesmaximal, complemented
3#20270002C213yesmaximal, complemented
4#20269997C213yesmaximal, complemented
5C2×#16809947C213yesmaximal, complemented
6#20269984C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C211C211yesmaximal, characteristic, complemented
8#16809963—21no—
9#16809962C2213yescomplemented
10C2×D844C2213yescomplemented
11D4×D211—23no—
12#16809952C2211yescharacteristic, complemented
13#16809950C2213yescomplemented
14#16809949C2213yescomplemented
15#16809947C2213yescomplemented
16#16809947C2216yescomplemented
17#16809946C2213yescomplemented
18C2×C211⋊C8C2213yescomplemented
19#41×C211—21no—
20#41×C211C2211yescharacteristic
21#41×C211C2213yescomplemented
22C2×D4×C211—23no—
23C2×D4×C211C2213yescomplemented
24C22×D211—23no—
25Dic422C2311yescharacteristic
26D844—23no—
27D844C2313yes—
28C22⋊D211—23no—
29C4×D211—23no—
30C211⋊C8C2311yescharacteristic
31C211⋊C8C2313yes—
32C23×C211—23no—
33Q8×C211D411yescharacteristic
34Q8×C211C2311yescharacteristic
35D4×C211—23no—
36D4×C211—23no—
37D4×C211D413yes—
38D4×C211C2313yes—
39C2×C4×C211—23no—
40C2×C4×C211C2313yes—
41C22×C211—23no—
42C22×C211—83no—
43C22×C211C2×D413yes—
44C2×D211—23no—
45C2×D211—83no—
46C4×C211—21no—
47C4×C211C2×D413yes—
48C4×C211C2411yescharacteristic
49Dic211—21no—
50C2×C211—23no—
51C2×C211—43no—
52C2×C211C22×D411yescharacteristic
53D211—43no—
54C211#57611yescharacteristic, complemented, Sylow
55#576—2111nomaximal, Sylow
56H(5,𝔽2)—2111no—
57H(5,𝔽2)D21111yescharacteristic, complemented
58AGL(1,ℤ/8)—2113no—
59AGL(1,ℤ/8)—2113no—
60#135—2113no—
61C2×D8—2113no—
62#131—2111no—
63#41—21no—
64#41—4221no—
65#41—2113no—
66#41C2×D21111yescharacteristic
67#41C2×D21113yescomplemented
68C2×D4—23no—
69C2×D4—4223no—
70C2×D4—2113no—
71C2×D4C2×D21113yescomplemented
72Q16—2111no—
73SD16—2113no—
74SD16—2113no—
75D8—2113no—
76D8—2116no—
77M16—2113no—
78C2×C8—2113no—
79C23—23no—
80C23—4223no—
81Q8—2111no—
82Q8C22⋊D21111yescharacteristic
83Q8C22×D21111yescharacteristic
84D4—23no—
85D4—23no—
86D4—4223no—
87D4—4223no—
88D4—2113no—
89D4C22⋊D21113yes—
90D4C22×D21113yes—
91C2×C4—23no—
92C2×C4—4223no—
93C2×C4C22×D21113yes—
94C8—2111no—
95C8—2113no—
96C22—23no—
97C22—83no—
98C22—16883no—
99C22—4223no—
100C22C2×C22⋊D21113yes—
101C4—21no—
102C4—4221no—
103C4C2×C22⋊D21113yes—
104C4C23×D21111yescharacteristic
105C2—23no—
106C2—43no—
107C2—8443no—
108C2C22×C22⋊D21111yescharacteristic
subgroupG/H
centerC2C22×C22⋊D211
commutator, cosocleC4×C211C24
FrattiniC4C23×D211
FittingH(5,𝔽2)×C211C2
socleC2×C211C22×D4
hypercenterH(5,𝔽2)D211
hypocenterC211#576
LCSG ▷ C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20270023C2semidirect
2. #20270003C2semidirect
3. #20270002C2semidirect
4. #20269997C2semidirect
5. C2×#16809947C2semidirect
6. #20269984C2semidirect
7. H(5,𝔽2)×C211C2semidirect
9. #16809962C22semidirect
10. C2×D844C22semidirect
12. #16809952C22semidirect
13. #16809950C22semidirect
14. #16809949C22semidirect
15. #16809947C22semidirect
16. #16809947C22semidirect
17. #16809946C22semidirect
18. C2×C211⋊C8C22semidirect
21. #41×C211C22semidirect
23. C2×D4×C211C22semidirect
54. C211#576semidirect
57. H(5,𝔽2)D211semidirect
67. #41C2×D211semidirect
71. C2×D4C2×D211semidirect

Maximal subgroups

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

subgroupindex
#202700232
#202700032
#202700022
#202699972
C2×#168099472
#202699842
H(5,𝔽2)×C2112
#576211