Subgroups — C₂⁴×D₁₀₀₄ #44565158

Order 32128.

10552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×D₂₅₁	C₂	1	2	yes	maximal, complemented
2	C₂³×D₁₀₀₄	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×C₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₂₅₁	—	2	32	no	—
5	C₂⁴×D₂₅₁	C₂²	1	60	yes	complemented
6	C₂²×D₁₀₀₄	C₂²	1	560	yes	direct product factor
7	C₂⁵×C₂₅₁	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₂₅₁	C₂²	1	30	yes	complemented
9	C₂³×D₂₅₁	—	2	480	no	—
10	C₂³×D₂₅₁	C₂³	1	280	yes	complemented
11	C₂×D₁₀₀₄	C₂³	1	960	yes	direct product factor
12	C₂⁴×C₂₅₁	D₄	1	16	yes	complemented
13	C₂⁴×C₂₅₁	C₂³	1	15	yes	—
14	C₂²×C₄×C₂₅₁	C₂³	1	140	yes	complemented
15	C₂²×D₂₅₁	—	2	1120	no	—
16	C₂²×D₂₅₁	C₂⁴	1	240	yes	complemented
17	D₁₀₀₄	C₂⁴	1	256	yes	direct product factor
18	C₂³×C₂₅₁	C₂×D₄	1	120	yes	complemented
19	C₂³×C₂₅₁	C₂⁴	1	35	yes	—
20	C₂×C₄×C₂₅₁	C₂⁴	1	120	yes	complemented
21	C₂²×C₂₅₁	C₂²×D₄	1	140	yes	complemented
22	C₂²×C₂₅₁	C₂⁵	1	15	yes	—
23	C₂×D₂₅₁	—	2	480	no	—
24	C₂×D₂₅₁	C₂⁵	1	32	yes	complemented
25	C₄×C₂₅₁	C₂⁵	1	16	yes	complemented
26	C₂×C₂₅₁	C₂³×D₄	1	30	yes	complemented
27	C₂×C₂₅₁	C₂⁶	1	1	yes	characteristic
28	D₂₅₁	—	2	32	no	—
29	C₂₅₁	C₂⁴×D₄	1	1	yes	characteristic, complemented, Sylow
30	C₂⁴×D₄	—	251	1	no	maximal, Sylow
31	C₂⁶	—	251	2	no	—
32	C₂³×D₄	—	251	60	no	—
33	C₂⁴×C₄	D₂₅₁	1	1	yes	characteristic, complemented
34	C₂⁵	—	502	32	no	—
35	C₂⁵	—	251	60	no	—
36	C₂⁵	C₂×D₂₅₁	1	1	yes	characteristic
37	C₂²×D₄	—	251	560	no	—
38	C₂³×C₄	C₂×D₂₅₁	1	30	yes	complemented
39	C₂⁴	—	502	480	no	—
40	C₂⁴	—	251	280	no	—
41	C₂⁴	D₁₀₀₄	1	16	yes	direct product factor
42	C₂⁴	C₂²×D₂₅₁	1	15	yes	—
43	C₂×D₄	—	251	960	no	—
44	C₂²×C₄	C₂²×D₂₅₁	1	140	yes	complemented
45	C₂³	—	502	1120	no	—
46	C₂³	—	251	240	no	—
47	C₂³	C₂×D₁₀₀₄	1	120	yes	direct product factor
48	C₂³	C₂³×D₂₅₁	1	35	yes	—
49	D₄	—	251	256	no	—
50	C₂×C₄	C₂³×D₂₅₁	1	120	yes	complemented
51	C₂²	—	502	480	no	—
52	C₂²	—	251	32	no	—
53	C₂²	C₂²×D₁₀₀₄	1	140	yes	direct product factor
54	C₂²	C₂⁴×D₂₅₁	1	15	yes	—
55	C₄	C₂⁴×D₂₅₁	1	16	yes	complemented
56	C₂	—	502	32	no	—
57	C₂	C₂³×D₁₀₀₄	1	30	yes	direct product factor
58	C₂	C₂⁵×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×D₂₅₁
commutator, cosocle	C₂×C₂₅₁	C₂⁶
Frattini	C₂	C₂⁵×D₂₅₁
Fitting	C₂⁴×C₄×C₂₅₁	C₂
socle	C₂⁵×C₂₅₁	C₂²
hypercenter	C₂⁴×C₄	D₂₅₁
hypocenter	C₂₅₁	C₂⁴×D₄
LCS	G ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁵×D₂₅₁	C₂	semidirect
2. C₂³×D₁₀₀₄	C₂	direct (N × H)
3. C₂⁴×C₄×C₂₅₁	C₂	semidirect
5. C₂⁴×D₂₅₁	C₂²	semidirect
6. C₂²×D₁₀₀₄	C₂²	direct (N × H)
8. C₂³×C₄×C₂₅₁	C₂²	semidirect
10. C₂³×D₂₅₁	C₂³	semidirect
11. C₂×D₁₀₀₄	C₂³	direct (N × H)
12. C₂⁴×C₂₅₁	D₄	semidirect
14. C₂²×C₄×C₂₅₁	C₂³	semidirect
16. C₂²×D₂₅₁	C₂⁴	semidirect
17. D₁₀₀₄	C₂⁴	direct (N × H)
18. C₂³×C₂₅₁	C₂×D₄	semidirect
20. C₂×C₄×C₂₅₁	C₂⁴	semidirect
21. C₂²×C₂₅₁	C₂²×D₄	semidirect
24. C₂×D₂₅₁	C₂⁵	semidirect
25. C₄×C₂₅₁	C₂⁵	semidirect
26. C₂×C₂₅₁	C₂³×D₄	semidirect
29. C₂₅₁	C₂⁴×D₄	semidirect
33. C₂⁴×C₄	D₂₅₁	semidirect
38. C₂³×C₄	C₂×D₂₅₁	semidirect
41. C₂⁴	D₁₀₀₄	direct (N × H)
44. C₂²×C₄	C₂²×D₂₅₁	semidirect
47. C₂³	C₂×D₁₀₀₄	direct (N × H)
50. C₂×C₄	C₂³×D₂₅₁	semidirect
53. C₂²	C₂²×D₁₀₀₄	direct (N × H)
55. C₄	C₂⁴×D₂₅₁	semidirect
57. C₂	C₂³×D₁₀₀₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×D₂₅₁	2
C₂³×D₁₀₀₄	2
C₂⁴×C₄×C₂₅₁	2
C₂⁴×D₄	251

Subgroups — C24×D1004 #44565158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₁₀₀₄ #44565158