Finite Group Databasebeta

← #3415×C251 #44548002

Subgroups — #3415×C251 #44548002

Order 32128.

370 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 180 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#576×C251C212yesmaximal, complemented
2#509×C251C216yesmaximal, complemented
3#508×C251C213yesmaximal, complemented
4C2×D16×C251C213yesmaximal, complemented
5#504×C251C211yesmaximal, characteristic, complemented
6H(5,𝔽2)×C251—22no—
7AGL(1,ℤ/8)×C251—26no—
8#135×C251C2216yescomplemented
9C2×D8×C251C2216yescomplemented
10#131×C251C2211yescharacteristic
11Q32×C251C2211yescharacteristic, complemented
12SD32×C251C2216yescomplemented
13D16×C251C2213yescomplemented
14D16×C251C2216yescomplemented
15M32×C251C2213yescomplemented
16C2×C16×C251C2213yescomplemented
17#41×C251—26no—
18#41×C251—42no—
19#41×C251D411yescharacteristic
20C2×D4×C251—26no—
21C2×D4×C251—46no—
22Q16×C251C2312yes—
23SD16×C251—26no—
24D8×C251—26no—
25D8×C251C2316yes—
26M16×C251D413yes—
27C2×C8×C251C2313yes—
28C16×C251C2311yescharacteristic
29C16×C251C2313yes—
30C23×C251—46no—
31Q8×C251—22no—
32Q8×C251D811yescharacteristic
33D4×C251—26no—
34D4×C251—46no—
35D4×C251—46no—
36D4×C251D813yes—
37C2×C4×C251—46no—
38C2×C4×C251C2×D413yes—
39C8×C251C2×D413yes—
40C8×C251C2411yescharacteristic
41C22×C251—46no—
42C22×C251—166no—
43C22×C251C2×D813yes—
44C4×C251—42no—
45C4×C251C2×D813yes—
46C4×C251C22×D411yescharacteristic
47C2×C251—23no—
48C2×C251—86no—
49C2×C251C22×D811yescharacteristic
50C251#341511yescharacteristic, direct product factor, Sylow
51#3415C25111yesmaximal, characteristic, direct product factor, Sylow
52#576C2×C25112yescomplemented
53#509C2×C25116yescomplemented
54#508C2×C25113yescomplemented
55C2×D16C2×C25113yescomplemented
56#504C2×C25111yescharacteristic, complemented
57H(5,𝔽2)—22no—
58AGL(1,ℤ/8)—26no—
59#135C22×C25116yescomplemented
60C2×D8C22×C25116yescomplemented
61#131C22×C25111yescharacteristic
62Q32C22×C25111yescharacteristic, complemented
63SD32C22×C25116yescomplemented
64D16C22×C25113yescomplemented
65D16C22×C25116yescomplemented
66M32C22×C25113yescomplemented
67C2×C16C22×C25113yescomplemented
68#41—26no—
69#41—42no—
70#41D4×C25111yescharacteristic
71C2×D4—26no—
72C2×D4—46no—
73Q16C23×C25112yes—
74SD16—26no—
75D8—26no—
76D8C23×C25116yes—
77M16D4×C25113yes—
78C2×C8C23×C25113yes—
79C16C23×C25111yescharacteristic
80C16C23×C25113yes—
81C23—46no—
82Q8—22no—
83Q8D8×C25111yescharacteristic
84D4—26no—
85D4—46no—
86D4—46no—
87D4D8×C25113yes—
88C2×C4—46no—
89C2×C4C2×D4×C25113yes—
90C8C2×D4×C25113yes—
91C8C24×C25111yescharacteristic
92C22—46no—
93C22—166no—
94C22C2×D8×C25113yes—
95C4—42no—
96C4C2×D8×C25113yes—
97C4C22×D4×C25111yescharacteristic
98C2—23no—
99C2—86no—
100C2C22×D8×C25111yescharacteristic
subgroupG/H
center, socleC2×C251C22×D8
commutator, Frattini, cosocleC8C24×C251
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C251 ◁ #41×C251 ◁ #131×C251 ◁ G

Complements

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

NH ≅ G/Ntype
1. #576×C251C2semidirect
2. #509×C251C2semidirect
3. #508×C251C2semidirect
4. C2×D16×C251C2semidirect
5. #504×C251C2semidirect
8. #135×C251C22semidirect
9. C2×D8×C251C22semidirect
11. Q32×C251C22semidirect
12. SD32×C251C22semidirect
13. D16×C251C22semidirect
14. D16×C251C22semidirect
15. M32×C251C22semidirect
16. C2×C16×C251C22semidirect
50. C251#3415direct (N × H)
51. #3415C251direct (N × H)
52. #576C2×C251semidirect
53. #509C2×C251semidirect
54. #508C2×C251semidirect
55. C2×D16C2×C251semidirect
56. #504C2×C251semidirect
59. #135C22×C251semidirect
60. C2×D8C22×C251semidirect
62. Q32C22×C251semidirect
63. SD32C22×C251semidirect
64. D16C22×C251semidirect
65. D16C22×C251semidirect
66. M32C22×C251semidirect
67. C2×C16C22×C251semidirect

Maximal subgroups

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

subgroupindex
#576×C2512
#509×C2512
#508×C2512
C2×D16×C2512
#504×C2512
#3415251