Subgroups — C₂×C₈×C₂₅₁⋊C₈ #44548638

Order 32128.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂₅₁⋊C₈	C₂	1	2	yes	maximal
2	C₈×C₂₅₁⋊C₈	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₂₅₁⋊C₈	C₄	1	4	yes	—
5	C₄×C₂₅₁⋊C₈	C₄	1	4	yes	—
6	C₄×C₂₅₁⋊C₈	C₂²	1	4	yes	—
7	C₂²×C₈×C₂₅₁	C₄	1	2	yes	—
8	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₂₅₁	C₄	1	2	yes	—
10	C₄×C₈×C₂₅₁	C₂²	1	2	yes	—
11	C₂×C₂₅₁⋊C₈	C₈	1	16	yes	direct product factor
12	C₂×C₂₅₁⋊C₈	C₂×C₄	1	8	yes	—
13	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₂₅₁	C₈	1	8	yes	complemented
16	C₂×C₈×C₂₅₁	C₂×C₄	1	4	yes	—
17	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
18	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
20	C₂₅₁⋊C₈	C₂×C₈	1	16	yes	direct product factor
21	C₂³×C₂₅₁	C₄²	1	1	yes	characteristic
22	C₂×C₄×C₂₅₁	C₄²	1	2	yes	—
23	C₂×C₄×C₂₅₁	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₂₅₁	C₂×C₈	1	8	yes	—
25	C₂×C₄×C₂₅₁	C₂×C₈	1	4	yes	—
26	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
27	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
28	C₈×C₂₅₁	C₂×C₈	1	8	yes	complemented
29	C₂²×C₂₅₁	C₄×C₈	1	4	yes	—
30	C₂²×C₂₅₁	C₄×C₈	1	2	yes	—
31	C₂²×C₂₅₁	C₂×C₄²	1	1	yes	characteristic
32	C₄×C₂₅₁	C₄×C₈	1	4	yes	—
33	C₄×C₂₅₁	C₄×C₈	1	2	yes	—
34	C₄×C₂₅₁	C₂²×C₈	1	4	yes	—
35	C₄×C₂₅₁	C₂²×C₈	1	2	yes	—
36	C₂×C₂₅₁	C₈²	1	4	yes	complemented
37	C₂×C₂₅₁	C₂×C₄×C₈	1	2	yes	—
38	C₂×C₂₅₁	C₂×C₄×C₈	1	1	yes	characteristic
39	C₂₅₁	C₂×C₈²	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₈²	—	251	1	no	maximal, Sylow
41	C₂×C₄×C₈	—	251	2	no	—
42	C₂×C₄×C₈	D₂₅₁	1	1	yes	characteristic
43	C₈²	—	251	4	no	—
44	C₂²×C₈	—	251	4	no	—
45	C₂²×C₈	Dic₂₅₁	1	2	yes	—
46	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
47	C₄×C₈	—	251	4	no	—
48	C₄×C₈	—	251	4	no	—
49	C₄×C₈	Dic₂₅₁	1	2	yes	—
50	C₄×C₈	C₂×D₂₅₁	1	2	yes	—
51	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
52	C₂²×C₄	C₂×Dic₂₅₁	1	1	yes	characteristic
53	C₂×C₈	—	251	16	no	—
54	C₂×C₈	—	251	8	no	—
55	C₂×C₈	C₂₅₁⋊C₈	1	8	yes	direct product factor
56	C₂×C₈	C₂×Dic₂₅₁	1	4	yes	—
57	C₄²	C₄×D₂₅₁	1	2	yes	—
58	C₄²	C₂×Dic₂₅₁	1	1	yes	characteristic
59	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
60	C₂³	C₄×Dic₂₅₁	1	1	yes	characteristic
61	C₂×C₄	C₄×Dic₂₅₁	1	2	yes	—
62	C₂×C₄	C₄×Dic₂₅₁	1	1	yes	characteristic
63	C₂×C₄	C₈×D₂₅₁	1	8	yes	—
64	C₂×C₄	C₂×C₂₅₁⋊C₈	1	4	yes	—
65	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
66	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
67	C₈	—	251	16	no	—
68	C₈	C₂×C₂₅₁⋊C₈	1	8	yes	direct product factor
69	C₂²	C₈×Dic₂₅₁	1	4	yes	—
70	C₂²	C₄×C₂₅₁⋊C₈	1	2	yes	—
71	C₂²	C₂×C₄×Dic₂₅₁	1	1	yes	characteristic
72	C₄	C₈×Dic₂₅₁	1	4	yes	—
73	C₄	C₄×C₂₅₁⋊C₈	1	2	yes	—
74	C₄	C₂×C₈×D₂₅₁	1	4	yes	—
75	C₄	C₂²×C₂₅₁⋊C₈	1	2	yes	—
76	C₂	C₈×C₂₅₁⋊C₈	1	4	yes	direct product factor
77	C₂	C₂×C₈×Dic₂₅₁	1	2	yes	—
78	C₂	C₂×C₄×C₂₅₁⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₈	D₂₅₁
commutator, hypocenter	C₂₅₁	C₂×C₈²
Frattini	C₄²	C₂²×D₂₅₁
Fitting	C₂×C₄×C₈×C₂₅₁	C₂
socle, prime radical	C₂³×C₂₅₁	C₄²
cosocle	C₄²×C₂₅₁	C₂³
LCS	G ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₈×C₂₅₁⋊C₈	C₂	direct (N × H)
11. C₂×C₂₅₁⋊C₈	C₈	direct (N × H)
15. C₂×C₈×C₂₅₁	C₈	semidirect
20. C₂₅₁⋊C₈	C₂×C₈	direct (N × H)
28. C₈×C₂₅₁	C₂×C₈	semidirect
36. C₂×C₂₅₁	C₈²	semidirect
39. C₂₅₁	C₂×C₈²	semidirect
55. C₂×C₈	C₂₅₁⋊C₈	direct (N × H)
68. C₈	C₂×C₂₅₁⋊C₈	direct (N × H)
76. C₂	C₈×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
C₂×C₄×C₂₅₁⋊C₈	2
C₈×C₂₅₁⋊C₈	2
C₂×C₄×C₈×C₂₅₁	2
C₂×C₈²	251

Subgroups — C2×C8×C251⋊C8 #44548638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×C₂₅₁⋊C₈ #44548638