Subgroups — C₂²×#127×C₂₅₁ #44548027

Order 32128.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₂₅₁	C₂	1	6	yes	maximal, complemented
2	C₂×#127×C₂₅₁	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄²×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁵×C₂₅₁	C₂²	1	4	yes	—
5	C₂²×D₄×C₂₅₁	—	2	48	no	—
6	C₂²×D₄×C₂₅₁	C₂²	1	72	yes	complemented
7	C₂³×C₄×C₂₅₁	C₂²	1	3	yes	—
8	#127×C₂₅₁	C₂²	1	64	yes	direct product factor
9	C₂×C₄²×C₂₅₁	C₂²	1	12	yes	complemented
10	C₂⁴×C₂₅₁	—	2	48	no	—
11	C₂⁴×C₂₅₁	C₂³	1	24	yes	—
12	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
13	C₂×D₄×C₂₅₁	—	2	288	no	—
14	C₂×D₄×C₂₅₁	C₂³	1	96	yes	complemented
15	C₂²×C₄×C₂₅₁	D₄	1	24	yes	complemented
16	C₂²×C₄×C₂₅₁	C₂³	1	18	yes	—
17	C₄²×C₂₅₁	C₂³	1	16	yes	complemented
18	C₂³×C₂₅₁	—	2	144	no	—
19	C₂³×C₂₅₁	—	4	64	no	—
20	C₂³×C₂₅₁	C₂×D₄	1	12	yes	—
21	C₂³×C₂₅₁	C₂⁴	1	16	yes	—
22	C₂³×C₂₅₁	C₂⁴	1	3	yes	—
23	D₄×C₂₅₁	—	2	192	no	—
24	C₂×C₄×C₂₅₁	C₂×D₄	1	72	yes	complemented
25	C₂×C₄×C₂₅₁	C₂⁴	1	12	yes	—
26	C₂²×C₂₅₁	—	2	48	no	—
27	C₂²×C₂₅₁	—	4	96	no	—
28	C₂²×C₂₅₁	#127	1	16	yes	direct product factor
29	C₂²×C₂₅₁	C₂²×D₄	1	18	yes	—
30	C₂²×C₂₅₁	C₂⁵	1	1	yes	characteristic
31	C₄×C₂₅₁	C₂²×D₄	1	24	yes	complemented
32	C₂×C₂₅₁	—	4	16	no	—
33	C₂×C₂₅₁	C₂×#127	1	12	yes	direct product factor
34	C₂×C₂₅₁	C₂³×D₄	1	3	yes	—
35	C₂₅₁	C₂²×#127	1	1	yes	characteristic, direct product factor, Sylow
36	C₂²×#127	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂³×D₄	C₂×C₂₅₁	1	6	yes	complemented
38	C₂×#127	C₂×C₂₅₁	1	24	yes	direct product factor
39	C₂²×C₄²	C₂×C₂₅₁	1	1	yes	characteristic, complemented
40	C₂⁵	C₂²×C₂₅₁	1	4	yes	—
41	C₂²×D₄	—	2	48	no	—
42	C₂²×D₄	C₂²×C₂₅₁	1	72	yes	complemented
43	C₂³×C₄	C₂²×C₂₅₁	1	3	yes	—
44	#127	C₂²×C₂₅₁	1	64	yes	direct product factor
45	C₂×C₄²	C₂²×C₂₅₁	1	12	yes	complemented
46	C₂⁴	—	2	48	no	—
47	C₂⁴	C₂³×C₂₅₁	1	24	yes	—
48	C₂⁴	C₂³×C₂₅₁	1	1	yes	characteristic
49	C₂×D₄	—	2	288	no	—
50	C₂×D₄	C₂³×C₂₅₁	1	96	yes	complemented
51	C₂²×C₄	D₄×C₂₅₁	1	24	yes	complemented
52	C₂²×C₄	C₂³×C₂₅₁	1	18	yes	—
53	C₄²	C₂³×C₂₅₁	1	16	yes	complemented
54	C₂³	—	2	144	no	—
55	C₂³	—	4	64	no	—
56	C₂³	C₂×D₄×C₂₅₁	1	12	yes	—
57	C₂³	C₂⁴×C₂₅₁	1	16	yes	—
58	C₂³	C₂⁴×C₂₅₁	1	3	yes	—
59	D₄	—	2	192	no	—
60	C₂×C₄	C₂×D₄×C₂₅₁	1	72	yes	complemented
61	C₂×C₄	C₂⁴×C₂₅₁	1	12	yes	—
62	C₂²	—	2	48	no	—
63	C₂²	—	4	96	no	—
64	C₂²	#127×C₂₅₁	1	16	yes	direct product factor
65	C₂²	C₂²×D₄×C₂₅₁	1	18	yes	—
66	C₂²	C₂⁵×C₂₅₁	1	1	yes	characteristic
67	C₄	C₂²×D₄×C₂₅₁	1	24	yes	complemented
68	C₂	—	4	16	no	—
69	C₂	C₂×#127×C₂₅₁	1	12	yes	direct product factor
70	C₂	C₂³×D₄×C₂₅₁	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₂₅₁	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₅₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×C₂₅₁	C₂	semidirect
2. C₂×#127×C₂₅₁	C₂	direct (N × H)
3. C₂²×C₄²×C₂₅₁	C₂	semidirect
6. C₂²×D₄×C₂₅₁	C₂²	semidirect
8. #127×C₂₅₁	C₂²	direct (N × H)
9. C₂×C₄²×C₂₅₁	C₂²	semidirect
14. C₂×D₄×C₂₅₁	C₂³	semidirect
15. C₂²×C₄×C₂₅₁	D₄	semidirect
17. C₄²×C₂₅₁	C₂³	semidirect
24. C₂×C₄×C₂₅₁	C₂×D₄	semidirect
28. C₂²×C₂₅₁	#127	direct (N × H)
31. C₄×C₂₅₁	C₂²×D₄	semidirect
33. C₂×C₂₅₁	C₂×#127	direct (N × H)
35. C₂₅₁	C₂²×#127	direct (N × H)
36. C₂²×#127	C₂₅₁	direct (N × H)
37. C₂³×D₄	C₂×C₂₅₁	semidirect
38. C₂×#127	C₂×C₂₅₁	direct (N × H)
39. C₂²×C₄²	C₂×C₂₅₁	semidirect
42. C₂²×D₄	C₂²×C₂₅₁	semidirect
44. #127	C₂²×C₂₅₁	direct (N × H)
45. C₂×C₄²	C₂²×C₂₅₁	semidirect
50. C₂×D₄	C₂³×C₂₅₁	semidirect
51. C₂²×C₄	D₄×C₂₅₁	semidirect
53. C₄²	C₂³×C₂₅₁	semidirect
60. C₂×C₄	C₂×D₄×C₂₅₁	semidirect
64. C₂²	#127×C₂₅₁	direct (N × H)
67. C₄	C₂²×D₄×C₂₅₁	semidirect
69. C₂	C₂×#127×C₂₅₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₂₅₁	2
C₂×#127×C₂₅₁	2
C₂²×C₄²×C₂₅₁	2
C₂²×#127	251

Subgroups — C22×#127×C251 #44548027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#127×C₂₅₁ #44548027