Subgroups — C₂⁴×C₄×D₂₅₁ #44565154

Order 32128.

10552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×D₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₄×D₂₅₁	C₂	1	60	yes	maximal, direct product factor
4	C₂⁴×C₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₂₅₁	C₄	1	32	yes	direct product factor
6	C₂⁴×D₂₅₁	C₂²	1	30	yes	—
7	C₂³×Dic₂₅₁	C₂²	1	30	yes	complemented
8	C₂²×C₄×D₂₅₁	C₂²	1	560	yes	direct product factor
9	C₂⁵×C₂₅₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₅₁	C₂²	1	30	yes	complemented
11	C₂³×D₂₅₁	C₂×C₄	1	480	yes	direct product factor
12	C₂³×D₂₅₁	C₂³	1	140	yes	—
13	C₂²×Dic₂₅₁	C₂³	1	140	yes	complemented
14	C₂×C₄×D₂₅₁	C₂³	1	960	yes	direct product factor
15	C₂⁴×C₂₅₁	C₂×C₄	1	16	yes	complemented
16	C₂⁴×C₂₅₁	C₂³	1	15	yes	—
17	C₂²×C₄×C₂₅₁	C₂³	1	140	yes	complemented
18	C₂²×D₂₅₁	C₂²×C₄	1	1120	yes	direct product factor
19	C₂²×D₂₅₁	C₂⁴	1	120	yes	—
20	C₂×Dic₂₅₁	C₂⁴	1	120	yes	complemented
21	C₄×D₂₅₁	C₂⁴	1	256	yes	direct product factor
22	C₂³×C₂₅₁	C₂²×C₄	1	120	yes	complemented
23	C₂³×C₂₅₁	C₂⁴	1	35	yes	—
24	C₂×C₄×C₂₅₁	C₂⁴	1	120	yes	complemented
25	C₂²×C₂₅₁	C₂³×C₄	1	140	yes	complemented
26	C₂²×C₂₅₁	C₂⁵	1	15	yes	—
27	C₂×D₂₅₁	C₂³×C₄	1	480	yes	direct product factor
28	C₂×D₂₅₁	C₂⁵	1	16	yes	—
29	C₄×C₂₅₁	C₂⁵	1	16	yes	complemented
30	Dic₂₅₁	C₂⁵	1	16	yes	complemented
31	C₂×C₂₅₁	C₂⁴×C₄	1	30	yes	complemented
32	C₂×C₂₅₁	C₂⁶	1	1	yes	characteristic
33	D₂₅₁	C₂⁴×C₄	1	32	yes	direct product factor
34	C₂₅₁	C₂⁵×C₄	1	1	yes	characteristic, complemented, Sylow
35	C₂⁵×C₄	—	251	1	no	maximal, Sylow
36	C₂⁶	—	251	1	no	—
37	C₂⁴×C₄	—	251	1	no	—
38	C₂⁴×C₄	—	251	60	no	—
39	C₂⁴×C₄	D₂₅₁	1	1	yes	characteristic, direct product factor
40	C₂⁵	—	251	32	no	—
41	C₂⁵	—	251	30	no	—
42	C₂⁵	C₂×D₂₅₁	1	1	yes	characteristic
43	C₂³×C₄	—	251	30	no	—
44	C₂³×C₄	—	251	560	no	—
45	C₂³×C₄	C₂×D₂₅₁	1	30	yes	direct product factor
46	C₂⁴	—	251	480	no	—
47	C₂⁴	—	251	140	no	—
48	C₂⁴	C₄×D₂₅₁	1	16	yes	direct product factor
49	C₂⁴	C₂²×D₂₅₁	1	15	yes	—
50	C₂²×C₄	—	251	140	no	—
51	C₂²×C₄	—	251	960	no	—
52	C₂²×C₄	C₂²×D₂₅₁	1	140	yes	direct product factor
53	C₂³	—	251	1120	no	—
54	C₂³	—	251	120	no	—
55	C₂³	C₂×C₄×D₂₅₁	1	120	yes	direct product factor
56	C₂³	C₂³×D₂₅₁	1	35	yes	—
57	C₂×C₄	—	251	120	no	—
58	C₂×C₄	—	251	256	no	—
59	C₂×C₄	C₂³×D₂₅₁	1	120	yes	direct product factor
60	C₂²	—	251	480	no	—
61	C₂²	—	251	16	no	—
62	C₂²	C₂²×C₄×D₂₅₁	1	140	yes	direct product factor
63	C₂²	C₂⁴×D₂₅₁	1	15	yes	—
64	C₄	—	251	16	no	—
65	C₄	C₂⁴×D₂₅₁	1	16	yes	direct product factor
66	C₂	—	251	32	no	—
67	C₂	C₂³×C₄×D₂₅₁	1	30	yes	direct product factor
68	C₂	C₂⁵×D₂₅₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂⁴×Dic₂₅₁	C₂	semidirect
3. C₂³×C₄×D₂₅₁	C₂	direct (N × H)
4. C₂⁴×C₄×C₂₅₁	C₂	semidirect
5. C₂⁴×D₂₅₁	C₄	direct (N × H)
7. C₂³×Dic₂₅₁	C₂²	semidirect
8. C₂²×C₄×D₂₅₁	C₂²	direct (N × H)
10. C₂³×C₄×C₂₅₁	C₂²	semidirect
11. C₂³×D₂₅₁	C₂×C₄	direct (N × H)
13. C₂²×Dic₂₅₁	C₂³	semidirect
14. C₂×C₄×D₂₅₁	C₂³	direct (N × H)
15. C₂⁴×C₂₅₁	C₂×C₄	semidirect
17. C₂²×C₄×C₂₅₁	C₂³	semidirect
18. C₂²×D₂₅₁	C₂²×C₄	direct (N × H)
20. C₂×Dic₂₅₁	C₂⁴	semidirect
21. C₄×D₂₅₁	C₂⁴	direct (N × H)
22. C₂³×C₂₅₁	C₂²×C₄	semidirect
24. C₂×C₄×C₂₅₁	C₂⁴	semidirect
25. C₂²×C₂₅₁	C₂³×C₄	semidirect
27. C₂×D₂₅₁	C₂³×C₄	direct (N × H)
29. C₄×C₂₅₁	C₂⁵	semidirect
30. Dic₂₅₁	C₂⁵	semidirect
31. C₂×C₂₅₁	C₂⁴×C₄	semidirect
33. D₂₅₁	C₂⁴×C₄	direct (N × H)
34. C₂₅₁	C₂⁵×C₄	semidirect
39. C₂⁴×C₄	D₂₅₁	direct (N × H)
45. C₂³×C₄	C₂×D₂₅₁	direct (N × H)
48. C₂⁴	C₄×D₂₅₁	direct (N × H)
52. C₂²×C₄	C₂²×D₂₅₁	direct (N × H)
55. C₂³	C₂×C₄×D₂₅₁	direct (N × H)
59. C₂×C₄	C₂³×D₂₅₁	direct (N × H)
62. C₂²	C₂²×C₄×D₂₅₁	direct (N × H)
65. C₄	C₂⁴×D₂₅₁	direct (N × H)
67. C₂	C₂³×C₄×D₂₅₁	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
C₂⁴×Dic₂₅₁	2
C₂³×C₄×D₂₅₁	2
C₂⁴×C₄×C₂₅₁	2
C₂⁵×C₄	251

Subgroups — C24×C4×D251 #44565154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₄×D₂₅₁ #44565154