Subgroups — C₂²×C₄×C₂₅₁⋊C₈ #44558766

Order 32128.

760 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂₅₁⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₂₅₁⋊C₈	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄²×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₂₅₁⋊C₈	C₄	1	16	yes	direct product factor
5	C₂²×C₂₅₁⋊C₈	C₂²	1	12	yes	—
6	C₄×C₂₅₁⋊C₈	C₂²	1	16	yes	direct product factor
7	C₂³×C₄×C₂₅₁	C₄	1	2	yes	—
8	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₂₅₁	C₄	1	6	yes	—
10	C₂×C₄²×C₂₅₁	C₂²	1	6	yes	—
11	C₂×C₂₅₁⋊C₈	C₂×C₄	1	48	yes	direct product factor
12	C₂×C₂₅₁⋊C₈	C₂³	1	8	yes	—
13	C₂⁴×C₂₅₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₅₁	C₈	1	16	yes	complemented
15	C₂²×C₄×C₂₅₁	C₂×C₄	1	8	yes	—
16	C₂²×C₄×C₂₅₁	C₂×C₄	1	3	yes	—
17	C₂²×C₄×C₂₅₁	C₂×C₄	1	12	yes	—
18	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
19	C₄²×C₂₅₁	C₂×C₄	1	12	yes	—
20	C₄²×C₂₅₁	C₂³	1	4	yes	—
21	C₂₅₁⋊C₈	C₂²×C₄	1	16	yes	direct product factor
22	C₂³×C₂₅₁	C₄²	1	4	yes	—
23	C₂³×C₂₅₁	C₂×C₈	1	8	yes	—
24	C₂³×C₂₅₁	C₂²×C₄	1	3	yes	—
25	C₂×C₄×C₂₅₁	C₄²	1	12	yes	—
26	C₂×C₄×C₂₅₁	C₂×C₈	1	48	yes	complemented
27	C₂×C₄×C₂₅₁	C₂²×C₄	1	12	yes	—
28	C₂×C₄×C₂₅₁	C₂²×C₄	1	3	yes	—
29	C₂×C₄×C₂₅₁	C₂²×C₄	1	8	yes	—
30	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
31	C₂²×C₂₅₁	C₄×C₈	1	16	yes	complemented
32	C₂²×C₂₅₁	C₂×C₄²	1	6	yes	—
33	C₂²×C₂₅₁	C₂²×C₈	1	12	yes	—
34	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
35	C₄×C₂₅₁	C₂×C₄²	1	6	yes	—
36	C₄×C₂₅₁	C₂²×C₈	1	16	yes	complemented
37	C₄×C₂₅₁	C₂³×C₄	1	2	yes	—
38	C₂×C₂₅₁	C₂×C₄×C₈	1	12	yes	complemented
39	C₂×C₂₅₁	C₂²×C₄²	1	1	yes	characteristic
40	C₂×C₂₅₁	C₂³×C₈	1	2	yes	—
41	C₂₅₁	C₂²×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
42	C₂²×C₄×C₈	—	251	1	no	maximal, Sylow
43	C₂³×C₈	—	251	2	no	—
44	C₂²×C₄²	D₂₅₁	1	1	yes	characteristic
45	C₂×C₄×C₈	—	251	12	no	—
46	C₂³×C₄	Dic₂₅₁	1	2	yes	—
47	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
48	C₂²×C₈	—	251	16	no	—
49	C₂²×C₈	—	251	12	no	—
50	C₂×C₄²	Dic₂₅₁	1	6	yes	—
51	C₂×C₄²	C₂×D₂₅₁	1	6	yes	—
52	C₄×C₈	—	251	16	no	—
53	C₂⁴	C₂×Dic₂₅₁	1	1	yes	characteristic
54	C₂²×C₄	C₂₅₁⋊C₈	1	16	yes	direct product factor
55	C₂²×C₄	C₄×D₂₅₁	1	8	yes	—
56	C₂²×C₄	C₂×Dic₂₅₁	1	3	yes	—
57	C₂²×C₄	C₂×Dic₂₅₁	1	12	yes	—
58	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
59	C₂×C₈	—	251	48	no	—
60	C₂×C₈	—	251	8	no	—
61	C₄²	C₂×Dic₂₅₁	1	12	yes	—
62	C₄²	C₂²×D₂₅₁	1	4	yes	—
63	C₂³	C₄×Dic₂₅₁	1	4	yes	—
64	C₂³	C₂×C₂₅₁⋊C₈	1	8	yes	—
65	C₂³	C₂²×Dic₂₅₁	1	3	yes	—
66	C₂×C₄	C₄×Dic₂₅₁	1	12	yes	—
67	C₂×C₄	C₂×C₂₅₁⋊C₈	1	48	yes	direct product factor
68	C₂×C₄	C₂×C₄×D₂₅₁	1	12	yes	—
69	C₂×C₄	C₂²×Dic₂₅₁	1	3	yes	—
70	C₂×C₄	C₂²×Dic₂₅₁	1	8	yes	—
71	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
72	C₈	—	251	16	no	—
73	C₂²	C₄×C₂₅₁⋊C₈	1	16	yes	direct product factor
74	C₂²	C₂×C₄×Dic₂₅₁	1	6	yes	—
75	C₂²	C₂²×C₂₅₁⋊C₈	1	12	yes	—
76	C₂²	C₂³×Dic₂₅₁	1	1	yes	characteristic
77	C₄	C₂×C₄×Dic₂₅₁	1	6	yes	—
78	C₄	C₂²×C₂₅₁⋊C₈	1	16	yes	direct product factor
79	C₄	C₂²×C₄×D₂₅₁	1	2	yes	—
80	C₂	C₂×C₄×C₂₅₁⋊C₈	1	12	yes	direct product factor
81	C₂	C₂²×C₄×Dic₂₅₁	1	1	yes	characteristic
82	C₂	C₂³×C₂₅₁⋊C₈	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	D₂₅₁
commutator, hypocenter	C₂₅₁	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₂²×C₄²×C₂₅₁	C₂
socle, prime radical	C₂⁴×C₂₅₁	C₂×C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
LCS	G ▷ C₂₅₁
UCS	C₁ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₂₅₁⋊C₈	C₂	direct (N × H)
4. C₂²×C₂₅₁⋊C₈	C₄	direct (N × H)
6. C₄×C₂₅₁⋊C₈	C₂²	direct (N × H)
11. C₂×C₂₅₁⋊C₈	C₂×C₄	direct (N × H)
14. C₂²×C₄×C₂₅₁	C₈	semidirect
21. C₂₅₁⋊C₈	C₂²×C₄	direct (N × H)
26. C₂×C₄×C₂₅₁	C₂×C₈	semidirect
31. C₂²×C₂₅₁	C₄×C₈	semidirect
36. C₄×C₂₅₁	C₂²×C₈	semidirect
38. C₂×C₂₅₁	C₂×C₄×C₈	semidirect
41. C₂₅₁	C₂²×C₄×C₈	semidirect
54. C₂²×C₄	C₂₅₁⋊C₈	direct (N × H)
67. C₂×C₄	C₂×C₂₅₁⋊C₈	direct (N × H)
73. C₂²	C₄×C₂₅₁⋊C₈	direct (N × H)
78. C₄	C₂²×C₂₅₁⋊C₈	direct (N × H)
80. C₂	C₂×C₄×C₂₅₁⋊C₈	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×C₂₅₁⋊C₈	2
C₂×C₄×C₂₅₁⋊C₈	2
C₂²×C₄²×C₂₅₁	2
C₂²×C₄×C₈	251

Subgroups — C22×C4×C251⋊C8 #44558766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₂₅₁⋊C₈ #44558766