Finite Group Databasebeta

← C22×H(5,𝔽2)×C199 #38766663

Subgroups — C22×H(5,𝔽2)×C199 #38766663

Order 25472.

6560 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×H(5,𝔽2)×C199C2148yesmaximal, direct product factor
2C22×#41×C199C216yesmaximal, complemented
3C23×D4×C199C219yesmaximal, complemented
4C25×C199C2216yescomplemented
5H(5,𝔽2)×C199C221256yesdirect product factor
6C2×#41×C199C221144yescomplemented
7C22×Q8×C199C2212yescomplemented
8C22×D4×C199C22118yescomplemented
9C22×D4×C199C221216yescomplemented
10C23×C4×C199C2219yescomplemented
11C24×C199—424no—
12C24×C199C2319yes—
13C24×C199C23172yescomplemented
14#41×C199C231384yescomplemented
15C2×Q8×C199C23124yescomplemented
16C2×D4×C199C231216yescomplemented
17C2×D4×C199C231576yescomplemented
18C22×C4×C199C2316yes—
19C22×C4×C199C231108yescomplemented
20C23×C199—236no—
21C23×C199—4144no—
22C23×C199C24154yes—
23C23×C199C2411yescharacteristic
24C23×C199C24196yescomplemented
25Q8×C199C24132yescomplemented
26D4×C199C241288yescomplemented
27C2×C4×C199C24136yes—
28C2×C4×C199C241144yescomplemented
29C22×C199—2108no—
30C22×C199—496no—
31C22×C199H(5,𝔽2)14yesdirect product factor
32C22×C199C25136yes—
33C22×C199C2513yes—
34C4×C199C25124yes—
35C2×C199—236no—
36C2×C199C2×H(5,𝔽2)16yesdirect product factor
37C2×C199C2611yescharacteristic
38C199C22×H(5,𝔽2)11yescharacteristic, direct product factor, Sylow
39C22×H(5,𝔽2)C19911yesmaximal, characteristic, direct product factor, Sylow
40C2×H(5,𝔽2)C2×C199148yesdirect product factor
41C22×#41C2×C19916yescomplemented
42C23×D4C2×C19919yescomplemented
43C25C22×C19916yescomplemented
44H(5,𝔽2)C22×C1991256yesdirect product factor
45C2×#41C22×C1991144yescomplemented
46C22×Q8C22×C19912yescomplemented
47C22×D4C22×C199118yescomplemented
48C22×D4C22×C1991216yescomplemented
49C23×C4C22×C19919yescomplemented
50C24—424no—
51C24C23×C19919yes—
52C24C23×C199172yescomplemented
53#41C23×C1991384yescomplemented
54C2×Q8C23×C199124yescomplemented
55C2×D4C23×C1991216yescomplemented
56C2×D4C23×C1991576yescomplemented
57C22×C4C23×C19916yes—
58C22×C4C23×C1991108yescomplemented
59C23—236no—
60C23—4144no—
61C23C24×C199154yes—
62C23C24×C19911yescharacteristic
63C23C24×C199196yescomplemented
64Q8C24×C199132yescomplemented
65D4C24×C1991288yescomplemented
66C2×C4C24×C199136yes—
67C2×C4C24×C1991144yescomplemented
68C22—2108no—
69C22—496no—
70C22H(5,𝔽2)×C19914yesdirect product factor
71C22C25×C199136yes—
72C22C25×C19913yes—
73C4C25×C199124yes—
74C2—236no—
75C2C2×H(5,𝔽2)×C19916yesdirect product factor
76C2C26×C19911yescharacteristic
subgroupG/H
center, socleC23×C199C24
commutator, Frattini, cosocleC2C26×C199
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C23×C199 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×H(5,𝔽2)×C199C2direct (N × H)
2. C22×#41×C199C2semidirect
3. C23×D4×C199C2semidirect
4. C25×C199C22semidirect
5. H(5,𝔽2)×C199C22direct (N × H)
6. C2×#41×C199C22semidirect
7. C22×Q8×C199C22semidirect
8. C22×D4×C199C22semidirect
9. C22×D4×C199C22semidirect
10. C23×C4×C199C22semidirect
13. C24×C199C23semidirect
14. #41×C199C23semidirect
15. C2×Q8×C199C23semidirect
16. C2×D4×C199C23semidirect
17. C2×D4×C199C23semidirect
19. C22×C4×C199C23semidirect
24. C23×C199C24semidirect
25. Q8×C199C24semidirect
26. D4×C199C24semidirect
28. C2×C4×C199C24semidirect
31. C22×C199H(5,𝔽2)direct (N × H)
36. C2×C199C2×H(5,𝔽2)direct (N × H)
38. C199C22×H(5,𝔽2)direct (N × H)
39. C22×H(5,𝔽2)C199direct (N × H)
40. C2×H(5,𝔽2)C2×C199direct (N × H)
41. C22×#41C2×C199semidirect
42. C23×D4C2×C199semidirect
43. C25C22×C199semidirect
44. H(5,𝔽2)C22×C199direct (N × H)
45. C2×#41C22×C199semidirect
46. C22×Q8C22×C199semidirect
47. C22×D4C22×C199semidirect
48. C22×D4C22×C199semidirect
49. C23×C4C22×C199semidirect
52. C24C23×C199semidirect
53. #41C23×C199semidirect
54. C2×Q8C23×C199semidirect
55. C2×D4C23×C199semidirect
56. C2×D4C23×C199semidirect
58. C22×C4C23×C199semidirect
63. C23C24×C199semidirect
64. Q8C24×C199semidirect
65. D4C24×C199semidirect
67. C2×C4C24×C199semidirect
70. C22H(5,𝔽2)×C199direct (N × H)
75. C2C2×H(5,𝔽2)×C199direct (N × H)

Maximal subgroups

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

subgroupindex
C2×H(5,𝔽2)×C1992
C22×#41×C1992
C23×D4×C1992
C22×H(5,𝔽2)199