Subgroups — #1290×C₂₅₁ #44545877

Order 32128.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#432×C₂₅₁	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×C₂₅₁	C₄	1	2	yes	—
4	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
5	C₄⋊C₈×C₂₅₁	—	2	2	no	—
6	C₄⋊C₈×C₂₅₁	—	2	2	no	—
7	C₂²⋊C₈×C₂₅₁	—	2	2	no	—
8	#97×C₂₅₁	C₄	1	4	yes	—
9	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
11	C₂²×C₄×C₂₅₁	Q₈	1	1	yes	characteristic
12	C₂×C₈×C₂₅₁	—	2	4	no	—
13	C₂×C₈×C₂₅₁	—	2	2	no	—
14	C₂×C₈×C₂₅₁	—	4	2	no	—
15	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
16	C₄²×C₂₅₁	D₄	1	2	yes	—
17	C₂³×C₂₅₁	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₅₁	—	2	2	no	—
19	C₂×C₄×C₂₅₁	—	2	1	no	—
20	C₂×C₄×C₂₅₁	—	2	1	no	—
21	C₂×C₄×C₂₅₁	—	4	1	no	—
22	C₂×C₄×C₂₅₁	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
24	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₅₁	C₄⋊C₄	1	2	yes	—
26	C₈×C₂₅₁	—	4	4	no	—
27	C₈×C₂₅₁	—	4	2	no	—
28	C₂²×C₂₅₁	—	2	1	no	—
29	C₂²×C₂₅₁	—	2	1	no	—
30	C₂²×C₂₅₁	#95	1	1	yes	characteristic
31	C₂²×C₂₅₁	#108	1	2	yes	—
32	C₄×C₂₅₁	—	2	2	no	—
33	C₄×C₂₅₁	—	2	2	no	—
34	C₄×C₂₅₁	—	2	1	no	—
35	C₄×C₂₅₁	#100	1	1	yes	characteristic
36	C₄×C₂₅₁	#101	1	1	yes	characteristic
37	C₂×C₂₅₁	—	2	2	no	—
38	C₂×C₂₅₁	H(3,ℤ/4)	1	1	yes	characteristic
39	C₂×C₂₅₁	#341	1	1	yes	characteristic
40	C₂×C₂₅₁	#343	1	1	yes	characteristic
41	C₂₅₁	#1290	1	1	yes	characteristic, direct product factor, Sylow
42	#1290	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	#432	C₂×C₂₅₁	1	2	yes	—
44	C₂×C₄⋊C₈	C₂×C₂₅₁	1	1	yes	characteristic
45	C₂²×C₈	C₄×C₂₅₁	1	2	yes	—
46	C₂×C₄²	C₂²×C₂₅₁	1	1	yes	characteristic
47	C₄⋊C₈	—	2	2	no	—
48	C₄⋊C₈	—	2	2	no	—
49	C₂²⋊C₈	—	2	2	no	—
50	#97	C₄×C₂₅₁	1	4	yes	—
51	C₂²×C₄	C₂×C₄×C₂₅₁	1	1	yes	characteristic
52	C₂²×C₄	D₄×C₂₅₁	1	1	yes	characteristic
53	C₂²×C₄	Q₈×C₂₅₁	1	1	yes	characteristic
54	C₂×C₈	—	2	4	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	4	2	no	—
57	C₄²	C₂×C₄×C₂₅₁	1	2	yes	—
58	C₄²	D₄×C₂₅₁	1	2	yes	—
59	C₂³	C₄⋊C₄×C₂₅₁	1	1	yes	characteristic
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	C₄²×C₂₅₁	1	1	yes	characteristic
65	C₂×C₄	C₂²⋊C₄×C₂₅₁	1	2	yes	—
66	C₂×C₄	C₂²⋊C₄×C₂₅₁	1	1	yes	characteristic
67	C₂×C₄	C₄⋊C₄×C₂₅₁	1	2	yes	—
68	C₈	—	4	4	no	—
69	C₈	—	4	2	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	2	1	no	—
72	C₂²	#95×C₂₅₁	1	1	yes	characteristic
73	C₂²	#108×C₂₅₁	1	2	yes	—
74	C₄	—	2	2	no	—
75	C₄	—	2	2	no	—
76	C₄	—	2	1	no	—
77	C₄	#100×C₂₅₁	1	1	yes	characteristic
78	C₄	#101×C₂₅₁	1	1	yes	characteristic
79	C₂	—	2	2	no	—
80	C₂	H(3,ℤ/4)×C₂₅₁	1	1	yes	characteristic
81	C₂	#341×C₂₅₁	1	1	yes	characteristic
82	C₂	#343×C₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₅₁	#95
commutator	C₂×C₄	C₄²×C₂₅₁
Frattini, cosocle	C₂×C₄²	C₂²×C₂₅₁
prime radical	C₂³×C₂₅₁	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₅₁ ◁ C₂×C₄²×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
41. C₂₅₁	#1290	direct (N × H)
42. #1290	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
#432×C₂₅₁	2
C₂×C₄⋊C₈×C₂₅₁	2
#1290	251

Subgroups — #1290×C251 #44545877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1290×C₂₅₁ #44545877