Subgroups — D₃×H(5,𝔽₂) #5965

Order 192.

898 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	#1009	C₂	1	6	yes	maximal, complemented
3	D₃×#41	C₂	1	6	yes	maximal, complemented
4	#1004	C₂	1	9	yes	maximal, complemented
5	C₂×D₃×D₄	C₂	1	9	yes	maximal, complemented
6	C₂×H(5,𝔽₂)	—	3	1	no	maximal, Sylow
7	C₂³×D₃	C₂²	1	6	yes	complemented
8	C₃×#41	C₂²	1	6	yes	complemented
9	C₂×C₃×D₄	C₂²	1	9	yes	complemented
10	C₂×#67	C₂²	1	18	yes	complemented
11	#239	C₂²	1	6	yes	complemented
12	D₃×Q₈	C₂²	1	2	yes	complemented
13	#237	C₂²	1	18	yes	complemented
14	D₃×D₄	C₂²	1	18	yes	complemented
15	D₃×D₄	C₂²	1	36	yes	complemented
16	#235	C₂²	1	18	yes	complemented
17	C₂×D₁₂	C₂²	1	9	yes	complemented
18	C₂×C₄×D₃	C₂²	1	9	yes	complemented
19	H(5,𝔽₂)	—	3	6	no	—
20	H(5,𝔽₂)	—	3	9	no	—
21	H(5,𝔽₂)	D₃	1	1	yes	characteristic, direct product factor
22	C₂×#41	—	3	6	no	—
23	C₂²×D₄	—	3	9	no	—
24	C₂³×C₃	C₂³	1	6	yes	complemented
25	C₂²×D₃	—	4	12	no	—
26	C₂²×D₃	C₂³	1	9	yes	—
27	C₂²×D₃	C₂³	1	18	yes	complemented
28	C₃×Q₈	C₂³	1	2	yes	complemented
29	C₃×D₄	C₂³	1	18	yes	complemented
30	C₂×C₃×C₄	C₂³	1	9	yes	complemented
31	#67	C₂³	1	36	yes	complemented
32	C₂×Dic₃	C₂³	1	9	yes	complemented
33	D₁₂	C₂³	1	18	yes	complemented
34	C₄×D₃	C₂³	1	6	yes	—
35	C₄×D₃	C₂³	1	18	yes	complemented
36	Dic₆	C₂³	1	6	yes	complemented
37	C₂⁴	—	3	6	no	—
38	#41	—	3	6	no	—
39	#41	—	3	18	no	—
40	#41	—	3	18	no	—
41	#41	C₂×D₃	1	6	yes	complemented
42	C₂×Q₈	—	3	2	no	—
43	C₂×D₄	—	3	18	no	—
44	C₂×D₄	—	3	18	no	—
45	C₂×D₄	—	3	36	no	—
46	C₂×D₄	—	3	9	no	—
47	C₂×D₄	C₂×D₃	1	9	yes	complemented
48	C₂²×C₄	—	3	9	no	—
49	C₂²×C₃	—	4	6	no	—
50	C₂²×C₃	C₂⁴	1	9	yes	—
51	C₂×D₃	—	2	18	no	—
52	C₂×D₃	—	4	18	no	—
53	C₂×D₃	C₂⁴	1	9	yes	—
54	C₂×D₃	C₂⁴	1	1	yes	characteristic
55	C₃×C₄	C₂⁴	1	6	yes	—
56	Dic₃	C₂⁴	1	6	yes	—
57	C₂³	—	3	9	no	—
58	C₂³	—	3	18	no	—
59	C₂³	—	12	12	no	—
60	C₂³	C₂²×D₃	1	6	yes	complemented
61	Q₈	—	3	6	no	—
62	Q₈	C₂²×D₃	1	2	yes	complemented
63	D₄	—	3	36	no	—
64	D₄	—	3	18	no	—
65	D₄	C₂²×D₃	1	18	yes	complemented
66	C₂×C₄	—	3	9	no	—
67	C₂×C₄	—	3	6	no	—
68	C₂×C₄	—	3	18	no	—
69	C₂×C₄	C₂²×D₃	1	9	yes	complemented
70	C₂×C₃	—	2	9	no	—
71	C₂×C₃	C₂⁵	1	1	yes	characteristic
72	D₃	—	2	9	no	—
73	D₃	H(5,𝔽₂)	1	2	yes	direct product factor
74	C₂²	—	3	1	no	—
75	C₂²	—	3	9	no	—
76	C₂²	—	4	6	no	—
77	C₂²	—	6	18	no	—
78	C₂²	—	12	18	no	—
79	C₂²	C₂³×D₃	1	9	yes	—
80	C₄	—	3	6	no	—
81	C₄	C₂³×D₃	1	6	yes	—
82	C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
83	C₂	—	2	9	no	—
84	C₂	—	3	2	no	—
85	C₂	—	6	9	no	—
86	C₂	C₂⁴×D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₃
commutator, socle, cosocle	C₂×C₃	C₂⁵
Fitting	C₃×H(5,𝔽₂)	C₂
hypercenter	H(5,𝔽₂)	D₃
hypocenter	C₃	C₂×H(5,𝔽₂)
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₃×H(5,𝔽₂)	C₂	semidirect
2. #1009	C₂	semidirect
3. D₃×#41	C₂	semidirect
4. #1004	C₂	semidirect
5. C₂×D₃×D₄	C₂	semidirect
7. C₂³×D₃	C₂²	semidirect
8. C₃×#41	C₂²	semidirect
9. C₂×C₃×D₄	C₂²	semidirect
10. C₂×#67	C₂²	semidirect
11. #239	C₂²	semidirect
12. D₃×Q₈	C₂²	semidirect
13. #237	C₂²	semidirect
14. D₃×D₄	C₂²	semidirect
15. D₃×D₄	C₂²	semidirect
16. #235	C₂²	semidirect
17. C₂×D₁₂	C₂²	semidirect
18. C₂×C₄×D₃	C₂²	semidirect
21. H(5,𝔽₂)	D₃	direct (N × H)
24. C₂³×C₃	C₂³	semidirect
27. C₂²×D₃	C₂³	semidirect
28. C₃×Q₈	C₂³	semidirect
29. C₃×D₄	C₂³	semidirect
30. C₂×C₃×C₄	C₂³	semidirect
31. #67	C₂³	semidirect
32. C₂×Dic₃	C₂³	semidirect
33. D₁₂	C₂³	semidirect
35. C₄×D₃	C₂³	semidirect
36. Dic₆	C₂³	semidirect
41. #41	C₂×D₃	semidirect
47. C₂×D₄	C₂×D₃	semidirect
60. C₂³	C₂²×D₃	semidirect
62. Q₈	C₂²×D₃	semidirect
65. D₄	C₂²×D₃	semidirect
69. C₂×C₄	C₂²×D₃	semidirect
73. D₃	H(5,𝔽₂)	direct (N × H)
82. C₃	C₂×H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₃×H(5,𝔽₂)	2
#1009	2
D₃×#41	2
#1004	2
C₂×D₃×D₄	2
C₂×H(5,𝔽₂)	3

Subgroups — D3×H(5,𝔽2) #5965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×H(5,𝔽₂) #5965