Finite Group Databasebeta

← #29505575

Subgroups — #29505575

Order 15296.

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#21440006C211yesmaximal, characteristic, complemented
2#21439986C213yesmaximal, complemented
3#21439985C213yesmaximal, complemented
4#21439980C213yesmaximal, complemented
5C2×#16838533C213yesmaximal, complemented
6#21439967C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C239C211yesmaximal, characteristic, complemented
8#16838549—21no—
9#16838548C2213yescomplemented
10C2×D956C2213yescomplemented
11D4×D239—23no—
12#16838538C2211yescharacteristic, complemented
13#16838536C2213yescomplemented
14#16838535C2213yescomplemented
15#16838533C2213yescomplemented
16#16838533C2216yescomplemented
17#16838532C2213yescomplemented
18C2×C239⋊C8C2213yescomplemented
19#41×C239—21no—
20#41×C239C2211yescharacteristic
21#41×C239C2213yescomplemented
22C2×D4×C239—23no—
23C2×D4×C239C2213yescomplemented
24C22×D239—23no—
25Dic478C2311yescharacteristic
26D956—23no—
27D956C2313yes—
28C22⋊D239—23no—
29C4×D239—23no—
30C239⋊C8C2311yescharacteristic
31C239⋊C8C2313yes—
32C23×C239—23no—
33Q8×C239D411yescharacteristic
34Q8×C239C2311yescharacteristic
35D4×C239—23no—
36D4×C239—23no—
37D4×C239D413yes—
38D4×C239C2313yes—
39C2×C4×C239—23no—
40C2×C4×C239C2313yes—
41C22×C239—23no—
42C22×C239—83no—
43C22×C239C2×D413yes—
44C2×D239—23no—
45C2×D239—83no—
46C4×C239—21no—
47C4×C239C2×D413yes—
48C4×C239C2411yescharacteristic
49Dic239—21no—
50C2×C239—23no—
51C2×C239—43no—
52C2×C239C22×D411yescharacteristic
53D239—43no—
54C239#57611yescharacteristic, complemented, Sylow
55#576—2391nomaximal, Sylow
56H(5,𝔽2)—2391no—
57H(5,𝔽2)D23911yescharacteristic, complemented
58AGL(1,ℤ/8)—2393no—
59AGL(1,ℤ/8)—2393no—
60#135—2393no—
61C2×D8—2393no—
62#131—2391no—
63#41—21no—
64#41—4781no—
65#41—2393no—
66#41C2×D23911yescharacteristic
67#41C2×D23913yescomplemented
68C2×D4—23no—
69C2×D4—4783no—
70C2×D4—2393no—
71C2×D4C2×D23913yescomplemented
72Q16—2391no—
73SD16—2393no—
74SD16—2393no—
75D8—2393no—
76D8—2396no—
77M16—2393no—
78C2×C8—2393no—
79C23—23no—
80C23—4783no—
81Q8—2391no—
82Q8C22⋊D23911yescharacteristic
83Q8C22×D23911yescharacteristic
84D4—23no—
85D4—23no—
86D4—4783no—
87D4—4783no—
88D4—2393no—
89D4C22⋊D23913yes—
90D4C22×D23913yes—
91C2×C4—23no—
92C2×C4—4783no—
93C2×C4C22×D23913yes—
94C8—2391no—
95C8—2393no—
96C22—23no—
97C22—83no—
98C22—19123no—
99C22—4783no—
100C22C2×C22⋊D23913yes—
101C4—21no—
102C4—4781no—
103C4C2×C22⋊D23913yes—
104C4C23×D23911yescharacteristic
105C2—23no—
106C2—43no—
107C2—9563no—
108C2C22×C22⋊D23911yescharacteristic
subgroupG/H
centerC2C22×C22⋊D239
commutator, cosocleC4×C239C24
FrattiniC4C23×D239
FittingH(5,𝔽2)×C239C2
socleC2×C239C22×D4
hypercenterH(5,𝔽2)D239
hypocenterC239#576
LCSG ▷ C4×C239 ▷ C2×C239 ▷ C239
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #21440006C2semidirect
2. #21439986C2semidirect
3. #21439985C2semidirect
4. #21439980C2semidirect
5. C2×#16838533C2semidirect
6. #21439967C2semidirect
7. H(5,𝔽2)×C239C2semidirect
9. #16838548C22semidirect
10. C2×D956C22semidirect
12. #16838538C22semidirect
13. #16838536C22semidirect
14. #16838535C22semidirect
15. #16838533C22semidirect
16. #16838533C22semidirect
17. #16838532C22semidirect
18. C2×C239⋊C8C22semidirect
21. #41×C239C22semidirect
23. C2×D4×C239C22semidirect
54. C239#576semidirect
57. H(5,𝔽2)D239semidirect
67. #41C2×D239semidirect
71. C2×D4C2×D239semidirect

Maximal subgroups

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

subgroupindex
#214400062
#214399862
#214399852
#214399802
C2×#168385332
#214399672
H(5,𝔽2)×C2392
#576239