Finite Group Databasebeta

← #29508520

Subgroups — #29508520

Order 15424.

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#21440688C211yesmaximal, characteristic, complemented
2#21440668C213yesmaximal, complemented
3#21440667C213yesmaximal, complemented
4#21440662C213yesmaximal, complemented
5C2×#16838755C213yesmaximal, complemented
6#21440649C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C241C211yesmaximal, characteristic, complemented
8#16838771—21no—
9#16838770C2213yescomplemented
10C2×D964C2213yescomplemented
11D4×D241—23no—
12#16838760C2211yescharacteristic, complemented
13#16838758C2213yescomplemented
14#16838757C2213yescomplemented
15#16838755C2213yescomplemented
16#16838755C2216yescomplemented
17#16838754C2213yescomplemented
18C2×#14515349C2213yescomplemented
19#41×C241—21no—
20#41×C241C2211yescharacteristic
21#41×C241C2213yescomplemented
22C2×D4×C241—23no—
23C2×D4×C241C2213yescomplemented
24C22×D241—23no—
25Dic482C2311yescharacteristic
26D964—23no—
27D964C2313yes—
28C22⋊D241—23no—
29C4×D241—23no—
30#14515349C2311yescharacteristic
31#14515349C2313yes—
32C23×C241—23no—
33Q8×C241D411yescharacteristic
34Q8×C241C2311yescharacteristic
35D4×C241—23no—
36D4×C241—23no—
37D4×C241D413yes—
38D4×C241C2313yes—
39C2×C4×C241—23no—
40C2×C4×C241C2313yes—
41C22×C241—23no—
42C22×C241—83no—
43C22×C241C2×D413yes—
44C2×D241—23no—
45C2×D241—83no—
46C4×C241—21no—
47C4×C241C2×D413yes—
48C4×C241C2411yescharacteristic
49Dic241—21no—
50C2×C241—23no—
51C2×C241—43no—
52C2×C241C22×D411yescharacteristic
53D241—43no—
54C241#57611yescharacteristic, complemented, Sylow
55#576—2411nomaximal, Sylow
56H(5,𝔽2)—2411no—
57H(5,𝔽2)D24111yescharacteristic, complemented
58AGL(1,ℤ/8)—2413no—
59AGL(1,ℤ/8)—2413no—
60#135—2413no—
61C2×D8—2413no—
62#131—2411no—
63#41—21no—
64#41—4821no—
65#41—2413no—
66#41C2×D24111yescharacteristic
67#41C2×D24113yescomplemented
68C2×D4—23no—
69C2×D4—4823no—
70C2×D4—2413no—
71C2×D4C2×D24113yescomplemented
72Q16—2411no—
73SD16—2413no—
74SD16—2413no—
75D8—2413no—
76D8—2416no—
77M16—2413no—
78C2×C8—2413no—
79C23—23no—
80C23—4823no—
81Q8—2411no—
82Q8C22⋊D24111yescharacteristic
83Q8C22×D24111yescharacteristic
84D4—23no—
85D4—23no—
86D4—4823no—
87D4—4823no—
88D4—2413no—
89D4C22⋊D24113yes—
90D4C22×D24113yes—
91C2×C4—23no—
92C2×C4—4823no—
93C2×C4C22×D24113yes—
94C8—2411no—
95C8—2413no—
96C22—23no—
97C22—83no—
98C22—19283no—
99C22—4823no—
100C22C2×C22⋊D24113yes—
101C4—21no—
102C4—4821no—
103C4C2×C22⋊D24113yes—
104C4C23×D24111yescharacteristic
105C2—23no—
106C2—43no—
107C2—9643no—
108C2C22×C22⋊D24111yescharacteristic
subgroupG/H
centerC2C22×C22⋊D241
commutator, cosocleC4×C241C24
FrattiniC4C23×D241
FittingH(5,𝔽2)×C241C2
socleC2×C241C22×D4
hypercenterH(5,𝔽2)D241
hypocenterC241#576
LCSG ▷ C4×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #21440688C2semidirect
2. #21440668C2semidirect
3. #21440667C2semidirect
4. #21440662C2semidirect
5. C2×#16838755C2semidirect
6. #21440649C2semidirect
7. H(5,𝔽2)×C241C2semidirect
9. #16838770C22semidirect
10. C2×D964C22semidirect
12. #16838760C22semidirect
13. #16838758C22semidirect
14. #16838757C22semidirect
15. #16838755C22semidirect
16. #16838755C22semidirect
17. #16838754C22semidirect
18. C2×#14515349C22semidirect
21. #41×C241C22semidirect
23. C2×D4×C241C22semidirect
54. C241#576semidirect
57. H(5,𝔽2)D241semidirect
67. #41C2×D241semidirect
71. C2×D4C2×D241semidirect

Maximal subgroups

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

subgroupindex
#214406882
#214406682
#214406672
#214406622
C2×#168387552
#214406492
H(5,𝔽2)×C2412
#576241