Subgroups — C₂×C₃₇×2₋¹⁺⁶ #22684355

Order 9472.

59186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×2₋¹⁺⁶	C₂	1	64	yes	maximal, direct product factor
2	C₂×C₃₇×#585	C₂	1	36	yes	maximal, complemented
3	C₂²×2₋¹⁺⁴×C₃₇	C₂	1	27	yes	maximal, complemented
4	C₃₇×#585	C₂²	1	1152	yes	complemented
5	C₂×2₋¹⁺⁴×C₃₇	C₂²	1	216	yes	complemented
6	C₂×2₋¹⁺⁴×C₃₇	C₂²	1	864	yes	complemented
7	C₂×H(5,𝔽₂)×C₃₇	C₂²	1	120	yes	complemented
8	C₂²×#41×C₃₇	C₂²	1	270	yes	complemented
9	C₂³×Q₈×C₃₇	C₂²	1	45	yes	complemented
10	2₋¹⁺⁴×C₃₇	C₂³	1	3456	yes	complemented
11	H(5,𝔽₂)×C₃₇	C₂³	1	1920	yes	complemented
12	C₂×#41×C₃₇	C₂³	1	720	yes	—
13	C₂×#41×C₃₇	C₂³	1	4320	yes	complemented
14	C₂²×Q₈×C₃₇	C₂³	1	270	yes	—
15	C₂²×Q₈×C₃₇	C₂³	1	720	yes	complemented
16	C₂²×D₄×C₃₇	C₂³	1	270	yes	—
17	C₂³×C₄×C₃₇	C₂³	1	135	yes	—
18	C₂⁴×C₃₇	C₂⁴	1	45	yes	—
19	#41×C₃₇	C₂⁴	1	5760	yes	—
20	C₂×Q₈×C₃₇	C₂⁴	1	120	yes	—
21	C₂×Q₈×C₃₇	C₂⁴	1	2160	yes	—
22	C₂×D₄×C₃₇	C₂⁴	1	216	yes	—
23	C₂×D₄×C₃₇	C₂⁴	1	2160	yes	—
24	C₂²×C₄×C₃₇	C₂⁴	1	270	yes	—
25	C₂²×C₄×C₃₇	C₂⁴	1	1080	yes	—
26	C₂³×C₃₇	—	4	90	no	—
27	C₂³×C₃₇	C₂⁵	1	27	yes	—
28	C₂³×C₃₇	C₂⁵	1	180	yes	—
29	Q₈×C₃₇	C₂⁵	1	480	yes	—
30	D₄×C₃₇	C₂⁵	1	864	yes	—
31	C₂×C₄×C₃₇	C₂⁵	1	1080	yes	—
32	C₂×C₄×C₃₇	C₂⁵	1	36	yes	—
33	C₂×2₋¹⁺⁶	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₃₇	—	2	54	no	—
35	C₂²×C₃₇	—	4	180	no	—
36	C₂²×C₃₇	C₂⁶	1	54	yes	—
37	C₂²×C₃₇	C₂⁶	1	1	yes	characteristic
38	C₄×C₃₇	C₂⁶	1	72	yes	—
39	2₋¹⁺⁶	C₂×C₃₇	1	64	yes	direct product factor
40	C₂×#585	C₂×C₃₇	1	36	yes	complemented
41	C₂²×2₋¹⁺⁴	C₂×C₃₇	1	27	yes	complemented
42	C₂×C₃₇	—	2	54	no	—
43	C₂×C₃₇	2₋¹⁺⁶	1	2	yes	direct product factor
44	C₂×C₃₇	C₂⁷	1	1	yes	characteristic
45	#585	C₂²×C₃₇	1	1152	yes	complemented
46	C₂×2₋¹⁺⁴	C₂²×C₃₇	1	216	yes	complemented
47	C₂×2₋¹⁺⁴	C₂²×C₃₇	1	864	yes	complemented
48	C₂×H(5,𝔽₂)	C₂²×C₃₇	1	120	yes	complemented
49	C₂²×#41	C₂²×C₃₇	1	270	yes	complemented
50	C₂³×Q₈	C₂²×C₃₇	1	45	yes	complemented
51	C₃₇	C₂×2₋¹⁺⁶	1	1	yes	characteristic, direct product factor, Sylow
52	2₋¹⁺⁴	C₂³×C₃₇	1	3456	yes	complemented
53	H(5,𝔽₂)	C₂³×C₃₇	1	1920	yes	complemented
54	C₂×#41	C₂³×C₃₇	1	720	yes	—
55	C₂×#41	C₂³×C₃₇	1	4320	yes	complemented
56	C₂²×Q₈	C₂³×C₃₇	1	270	yes	—
57	C₂²×Q₈	C₂³×C₃₇	1	720	yes	complemented
58	C₂²×D₄	C₂³×C₃₇	1	270	yes	—
59	C₂³×C₄	C₂³×C₃₇	1	135	yes	—
60	C₂⁴	C₂⁴×C₃₇	1	45	yes	—
61	#41	C₂⁴×C₃₇	1	5760	yes	—
62	C₂×Q₈	C₂⁴×C₃₇	1	120	yes	—
63	C₂×Q₈	C₂⁴×C₃₇	1	2160	yes	—
64	C₂×D₄	C₂⁴×C₃₇	1	216	yes	—
65	C₂×D₄	C₂⁴×C₃₇	1	2160	yes	—
66	C₂²×C₄	C₂⁴×C₃₇	1	270	yes	—
67	C₂²×C₄	C₂⁴×C₃₇	1	1080	yes	—
68	C₂³	—	4	90	no	—
69	C₂³	C₂⁵×C₃₇	1	27	yes	—
70	C₂³	C₂⁵×C₃₇	1	180	yes	—
71	Q₈	C₂⁵×C₃₇	1	480	yes	—
72	D₄	C₂⁵×C₃₇	1	864	yes	—
73	C₂×C₄	C₂⁵×C₃₇	1	36	yes	—
74	C₂×C₄	C₂⁵×C₃₇	1	1080	yes	—
75	C₂²	—	2	54	no	—
76	C₂²	—	4	180	no	—
77	C₂²	C₂⁶×C₃₇	1	54	yes	—
78	C₂²	C₂⁶×C₃₇	1	1	yes	characteristic
79	C₄	C₂⁶×C₃₇	1	72	yes	—
80	C₂	—	2	54	no	—
81	C₂	C₃₇×2₋¹⁺⁶	1	2	yes	direct product factor
82	C₂	C₂⁷×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂⁶
commutator, Frattini, cosocle	C₂	C₂⁷×C₃₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×2₋¹⁺⁶	C₂	direct (N × H)
2. C₂×C₃₇×#585	C₂	semidirect
3. C₂²×2₋¹⁺⁴×C₃₇	C₂	semidirect
4. C₃₇×#585	C₂²	semidirect
5. C₂×2₋¹⁺⁴×C₃₇	C₂²	semidirect
6. C₂×2₋¹⁺⁴×C₃₇	C₂²	semidirect
7. C₂×H(5,𝔽₂)×C₃₇	C₂²	semidirect
8. C₂²×#41×C₃₇	C₂²	semidirect
9. C₂³×Q₈×C₃₇	C₂²	semidirect
10. 2₋¹⁺⁴×C₃₇	C₂³	semidirect
11. H(5,𝔽₂)×C₃₇	C₂³	semidirect
13. C₂×#41×C₃₇	C₂³	semidirect
15. C₂²×Q₈×C₃₇	C₂³	semidirect
33. C₂×2₋¹⁺⁶	C₃₇	direct (N × H)
39. 2₋¹⁺⁶	C₂×C₃₇	direct (N × H)
40. C₂×#585	C₂×C₃₇	semidirect
41. C₂²×2₋¹⁺⁴	C₂×C₃₇	semidirect
43. C₂×C₃₇	2₋¹⁺⁶	direct (N × H)
45. #585	C₂²×C₃₇	semidirect
46. C₂×2₋¹⁺⁴	C₂²×C₃₇	semidirect
47. C₂×2₋¹⁺⁴	C₂²×C₃₇	semidirect
48. C₂×H(5,𝔽₂)	C₂²×C₃₇	semidirect
49. C₂²×#41	C₂²×C₃₇	semidirect
50. C₂³×Q₈	C₂²×C₃₇	semidirect
51. C₃₇	C₂×2₋¹⁺⁶	direct (N × H)
52. 2₋¹⁺⁴	C₂³×C₃₇	semidirect
53. H(5,𝔽₂)	C₂³×C₃₇	semidirect
55. C₂×#41	C₂³×C₃₇	semidirect
57. C₂²×Q₈	C₂³×C₃₇	semidirect
81. C₂	C₃₇×2₋¹⁺⁶	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×2₋¹⁺⁶	2
C₂×C₃₇×#585	2
C₂²×2₋¹⁺⁴×C₃₇	2
C₂×2₋¹⁺⁶	37

Subgroups — C2×C37×2−1+6 #22684355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×2₋¹⁺⁶ #22684355