Finite Group Databasebeta

← #29505583

Subgroups — #29505583

Order 15296.

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

normal subgroups only

This group has 258 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#21440007C211yesmaximal, characteristic, complemented
2#21439990C213yesmaximal, complemented
3#21439986C213yesmaximal, complemented
4#21439980C213yesmaximal, complemented
5C2×#16838536C213yesmaximal, complemented
6#21439967C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C239C211yesmaximal, characteristic, complemented
8#16838548C2213yescomplemented
9#16838547—23no—
10C2×Dic478C2213yescomplemented
11Q8×D239—21no—
12#16838538C2213yescomplemented
13#16838536C2213yescomplemented
14#16838536C2216yescomplemented
15#16838535C2211yescharacteristic, complemented
16#16838533C2213yescomplemented
17#16838532C2213yescomplemented
18C2×C239⋊C8C2213yescomplemented
19#41×C239—21no—
20#41×C239C2213yes—
21#41×C239C2211yescharacteristic
22C2×D4×C239—23no—
23C2×D4×C239C2213yes—
24Dic478—23no—
25Dic478C2313yes—
26D956C2311yescharacteristic
27C22⋊D239—23no—
28C2×Dic239—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—21no—
45C4×C239—21no—
46C4×C239C2×D413yes—
47C4×C239C2411yescharacteristic
48Dic239—23no—
49C2×C239—23no—
50C2×C239—43no—
51C2×C239C22×D411yescharacteristic
52D239—41no—
53C239#57711yescharacteristic, complemented, Sylow
54#577—2391nomaximal, Sylow
552−1+4—2391no—
56H(5,𝔽2)D23911yescharacteristic, complemented
57#137—2393no—
58AGL(1,ℤ/8)—2393no—
59#135—2393no—
60C2×SD16—2393no—
61#131—2391no—
62#41—21no—
63#41—4783no—
64#41—2393no—
65#41C2×D23913yes—
66#41C2×D23911yescharacteristic
67C2×Q8—4781no—
68C2×Q8—2393no—
69C2×D4—23no—
70C2×D4C2×D23913yes—
71Q16—2393no—
72SD16—2391no—
73SD16—2393no—
74SD16—2396no—
75D8—2393no—
76M16—2393no—
77C2×C8—2393no—
78C23—23no—
79Q8—4783no—
80Q8—2393no—
81Q8C22⋊D23911yescharacteristic
82Q8C22×D23911yescharacteristic
83D4—23no—
84D4—23no—
85D4—4783no—
86D4—2391no—
87D4C22⋊D23913yes—
88D4C22×D23913yes—
89C2×C4—23no—
90C2×C4—4783no—
91C2×C4—4783no—
92C2×C4C22×D23913yes—
93C8—2391no—
94C8—2393no—
95C22—23no—
96C22—83no—
97C22—4781no—
98C22C2×C22⋊D23913yes—
99C4—21no—
100C4—4783no—
101C4C2×C22⋊D23913yes—
102C4C23×D23911yescharacteristic
103C2—23no—
104C2—43no—
105C2—9561no—
106C2C22×C22⋊D23911yescharacteristic
subgroupG/H
centerC2C22×C22⋊D239
commutator, cosocleC4×C239C24
FrattiniC4C23×D239
FittingH(5,𝔽2)×C239C2
socleC2×C239C22×D4
hypercenterH(5,𝔽2)D239
hypocenterC239#577
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. #21440007C2semidirect
2. #21439990C2semidirect
3. #21439986C2semidirect
4. #21439980C2semidirect
5. C2×#16838536C2semidirect
6. #21439967C2semidirect
7. H(5,𝔽2)×C239C2semidirect
8. #16838548C22semidirect
10. C2×Dic478C22semidirect
12. #16838538C22semidirect
13. #16838536C22semidirect
14. #16838536C22semidirect
15. #16838535C22semidirect
16. #16838533C22semidirect
17. #16838532C22semidirect
18. C2×C239⋊C8C22semidirect
53. C239#577semidirect
56. H(5,𝔽2)D239semidirect

Maximal subgroups

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

subgroupindex
#214400072
#214399902
#214399862
#214399802
C2×#168385362
#214399672
H(5,𝔽2)×C2392
#577239