Subgroups — C₂×#585×C₁₇₃ #36351378

Order 22144.

5956 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#585×C₁₇₃	C₂	1	32	yes	maximal, direct product factor
2	C₂×2₋¹⁺⁴×C₁₇₃	C₂	1	6	yes	maximal, complemented
3	C₂×H(5,𝔽₂)×C₁₇₃	C₂	1	10	yes	maximal, complemented
4	C₂²×#41×C₁₇₃	C₂	1	15	yes	maximal, complemented
5	2₋¹⁺⁴×C₁₇₃	C₂²	1	96	yes	complemented
6	H(5,𝔽₂)×C₁₇₃	C₂²	1	160	yes	complemented
7	C₂×#41×C₁₇₃	C₂²	1	20	yes	complemented
8	C₂×#41×C₁₇₃	C₂²	1	60	yes	complemented
9	C₂×#41×C₁₇₃	C₂²	1	240	yes	complemented
10	C₂²×Q₈×C₁₇₃	C₂²	1	15	yes	complemented
11	C₂²×D₄×C₁₇₃	C₂²	1	45	yes	complemented
12	C₂³×C₄×C₁₇₃	C₂²	1	15	yes	complemented
13	C₂⁴×C₁₇₃	C₂³	1	15	yes	—
14	#41×C₁₇₃	C₂³	1	160	yes	complemented
15	#41×C₁₇₃	C₂³	1	480	yes	complemented
16	C₂×Q₈×C₁₇₃	C₂³	1	20	yes	—
17	C₂×Q₈×C₁₇₃	C₂³	1	120	yes	complemented
18	C₂×D₄×C₁₇₃	C₂³	1	60	yes	—
19	C₂×D₄×C₁₇₃	C₂³	1	360	yes	complemented
20	C₂²×C₄×C₁₇₃	C₂³	1	45	yes	—
21	C₂²×C₄×C₁₇₃	C₂³	1	15	yes	—
22	C₂²×C₄×C₁₇₃	C₂³	1	120	yes	complemented
23	C₂³×C₁₇₃	—	4	30	no	—
24	C₂³×C₁₇₃	C₂⁴	1	15	yes	—
25	C₂³×C₁₇₃	C₂⁴	1	60	yes	—
26	Q₈×C₁₇₃	C₂⁴	1	80	yes	—
27	D₄×C₁₇₃	C₂⁴	1	240	yes	—
28	C₂×C₄×C₁₇₃	C₂⁴	1	15	yes	—
29	C₂×C₄×C₁₇₃	C₂⁴	1	60	yes	—
30	C₂×C₄×C₁₇₃	C₂⁴	1	180	yes	—
31	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
32	C₂²×C₁₇₃	—	2	30	no	—
33	C₂²×C₁₇₃	—	4	60	no	—
34	C₂²×C₁₇₃	C₂⁵	1	30	yes	—
35	C₂²×C₁₇₃	C₂⁵	1	1	yes	characteristic
36	C₄×C₁₇₃	C₂⁵	1	2	yes	—
37	C₄×C₁₇₃	C₂⁵	1	30	yes	—
38	C₂×C₁₇₃	—	2	30	no	—
39	C₂×C₁₇₃	#585	1	2	yes	direct product factor
40	C₂×C₁₇₃	C₂⁶	1	1	yes	characteristic
41	C₁₇₃	C₂×#585	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×#585	C₁₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	#585	C₂×C₁₇₃	1	32	yes	direct product factor
44	C₂×2₋¹⁺⁴	C₂×C₁₇₃	1	6	yes	complemented
45	C₂×H(5,𝔽₂)	C₂×C₁₇₃	1	10	yes	complemented
46	C₂²×#41	C₂×C₁₇₃	1	15	yes	complemented
47	2₋¹⁺⁴	C₂²×C₁₇₃	1	96	yes	complemented
48	H(5,𝔽₂)	C₂²×C₁₇₃	1	160	yes	complemented
49	C₂×#41	C₂²×C₁₇₃	1	20	yes	complemented
50	C₂×#41	C₂²×C₁₇₃	1	60	yes	complemented
51	C₂×#41	C₂²×C₁₇₃	1	240	yes	complemented
52	C₂²×Q₈	C₂²×C₁₇₃	1	15	yes	complemented
53	C₂²×D₄	C₂²×C₁₇₃	1	45	yes	complemented
54	C₂³×C₄	C₂²×C₁₇₃	1	15	yes	complemented
55	C₂⁴	C₂³×C₁₇₃	1	15	yes	—
56	#41	C₂³×C₁₇₃	1	160	yes	complemented
57	#41	C₂³×C₁₇₃	1	480	yes	complemented
58	C₂×Q₈	C₂³×C₁₇₃	1	20	yes	—
59	C₂×Q₈	C₂³×C₁₇₃	1	120	yes	complemented
60	C₂×D₄	C₂³×C₁₇₃	1	60	yes	—
61	C₂×D₄	C₂³×C₁₇₃	1	360	yes	complemented
62	C₂²×C₄	C₂³×C₁₇₃	1	15	yes	—
63	C₂²×C₄	C₂³×C₁₇₃	1	45	yes	—
64	C₂²×C₄	C₂³×C₁₇₃	1	120	yes	complemented
65	C₂³	—	4	30	no	—
66	C₂³	C₂⁴×C₁₇₃	1	15	yes	—
67	C₂³	C₂⁴×C₁₇₃	1	60	yes	—
68	Q₈	C₂⁴×C₁₇₃	1	80	yes	—
69	D₄	C₂⁴×C₁₇₃	1	240	yes	—
70	C₂×C₄	C₂⁴×C₁₇₃	1	15	yes	—
71	C₂×C₄	C₂⁴×C₁₇₃	1	60	yes	—
72	C₂×C₄	C₂⁴×C₁₇₃	1	180	yes	—
73	C₂×C₄	C₂⁴×C₁₇₃	1	1	yes	characteristic
74	C₂²	—	2	30	no	—
75	C₂²	—	4	60	no	—
76	C₂²	C₂⁵×C₁₇₃	1	30	yes	—
77	C₂²	C₂⁵×C₁₇₃	1	1	yes	characteristic
78	C₄	C₂⁵×C₁₇₃	1	2	yes	—
79	C₄	C₂⁵×C₁₇₃	1	30	yes	—
80	C₂	—	2	30	no	—
81	C₂	#585×C₁₇₃	1	2	yes	direct product factor
82	C₂	C₂⁶×C₁₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₇₃	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁶×C₁₇₃
socle	C₂²×C₁₇₃	C₂⁵
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #585×C₁₇₃	C₂	direct (N × H)
2. C₂×2₋¹⁺⁴×C₁₇₃	C₂	semidirect
3. C₂×H(5,𝔽₂)×C₁₇₃	C₂	semidirect
4. C₂²×#41×C₁₇₃	C₂	semidirect
5. 2₋¹⁺⁴×C₁₇₃	C₂²	semidirect
6. H(5,𝔽₂)×C₁₇₃	C₂²	semidirect
7. C₂×#41×C₁₇₃	C₂²	semidirect
8. C₂×#41×C₁₇₃	C₂²	semidirect
9. C₂×#41×C₁₇₃	C₂²	semidirect
10. C₂²×Q₈×C₁₇₃	C₂²	semidirect
11. C₂²×D₄×C₁₇₃	C₂²	semidirect
12. C₂³×C₄×C₁₇₃	C₂²	semidirect
14. #41×C₁₇₃	C₂³	semidirect
15. #41×C₁₇₃	C₂³	semidirect
17. C₂×Q₈×C₁₇₃	C₂³	semidirect
19. C₂×D₄×C₁₇₃	C₂³	semidirect
22. C₂²×C₄×C₁₇₃	C₂³	semidirect
39. C₂×C₁₇₃	#585	direct (N × H)
41. C₁₇₃	C₂×#585	direct (N × H)
42. C₂×#585	C₁₇₃	direct (N × H)
43. #585	C₂×C₁₇₃	direct (N × H)
44. C₂×2₋¹⁺⁴	C₂×C₁₇₃	semidirect
45. C₂×H(5,𝔽₂)	C₂×C₁₇₃	semidirect
46. C₂²×#41	C₂×C₁₇₃	semidirect
47. 2₋¹⁺⁴	C₂²×C₁₇₃	semidirect
48. H(5,𝔽₂)	C₂²×C₁₇₃	semidirect
49. C₂×#41	C₂²×C₁₇₃	semidirect
50. C₂×#41	C₂²×C₁₇₃	semidirect
51. C₂×#41	C₂²×C₁₇₃	semidirect
52. C₂²×Q₈	C₂²×C₁₇₃	semidirect
53. C₂²×D₄	C₂²×C₁₇₃	semidirect
54. C₂³×C₄	C₂²×C₁₇₃	semidirect
56. #41	C₂³×C₁₇₃	semidirect
57. #41	C₂³×C₁₇₃	semidirect
59. C₂×Q₈	C₂³×C₁₇₃	semidirect
61. C₂×D₄	C₂³×C₁₇₃	semidirect
64. C₂²×C₄	C₂³×C₁₇₃	semidirect
81. C₂	#585×C₁₇₃	direct (N × H)

Maximal subgroups

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

subgroup	index
#585×C₁₇₃	2
C₂×2₋¹⁺⁴×C₁₇₃	2
C₂×H(5,𝔽₂)×C₁₇₃	2
C₂²×#41×C₁₇₃	2
C₂×#585	173

Subgroups — C2×#585×C173 #36351378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#585×C₁₇₃ #36351378