Subgroups — C₃×C₅×H(3,𝔽₄) #11756650

Order 960.

520 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#124	C₂	1	9	yes	maximal, complemented
2	C₃×C₅×#120	C₂	1	6	yes	maximal, complemented
3	C₅×H(3,𝔽₄)	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂⁴×C₃×C₅	C₂²	1	2	yes	complemented
5	C₂×C₃×C₅×Q₈	C₂²	1	3	yes	complemented
6	C₂×C₃×C₅×D₄	C₂²	1	9	yes	—
7	C₃×C₅×C₂²⋊C₄	C₂²	1	18	yes	complemented
8	C₃×C₄²×C₅	C₂²	1	3	yes	complemented
9	C₃×H(3,𝔽₄)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₅×#124	C₂×C₃	1	9	yes	complemented
11	C₅×#120	C₂×C₃	1	6	yes	complemented
12	C₂³×C₃×C₅	—	4	6	no	—
13	C₂³×C₃×C₅	C₂³	1	6	yes	—
14	C₃×C₅×Q₈	—	4	3	no	—
15	C₃×C₅×D₄	—	4	9	no	—
16	C₂×C₃×C₄×C₅	C₂³	1	9	yes	—
17	C₃×#124	C₂×C₅	1	9	yes	complemented
18	C₃×#120	C₂×C₅	1	6	yes	complemented
19	C₂⁴×C₅	C₂²×C₃	1	2	yes	complemented
20	C₂×C₅×Q₈	C₂²×C₃	1	3	yes	complemented
21	C₂×C₅×D₄	C₂²×C₃	1	9	yes	—
22	C₅×C₂²⋊C₄	C₂²×C₃	1	18	yes	complemented
23	C₄²×C₅	C₂²×C₃	1	3	yes	complemented
24	H(3,𝔽₄)	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
25	C₂²×C₃×C₅	—	2	18	no	—
26	C₂²×C₃×C₅	—	4	8	no	—
27	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
28	C₃×C₄×C₅	—	2	9	no	—
29	C₂⁴×C₃	C₂²×C₅	1	2	yes	complemented
30	C₂×C₃×Q₈	C₂²×C₅	1	3	yes	complemented
31	C₂×C₃×D₄	C₂²×C₅	1	9	yes	—
32	C₃×C₂²⋊C₄	C₂²×C₅	1	18	yes	complemented
33	C₃×C₄²	C₂²×C₅	1	3	yes	complemented
34	C₂³×C₅	—	4	6	no	—
35	C₂³×C₅	C₂³×C₃	1	6	yes	—
36	C₅×Q₈	—	4	3	no	—
37	C₅×D₄	—	4	9	no	—
38	C₂×C₄×C₅	C₂³×C₃	1	9	yes	—
39	#124	C₂×C₃×C₅	1	9	yes	complemented
40	#120	C₂×C₃×C₅	1	6	yes	complemented
41	C₂×C₃×C₅	—	4	6	no	—
42	C₂×C₃×C₅	H(5,𝔽₂)	1	3	yes	—
43	C₂³×C₃	—	4	6	no	—
44	C₂³×C₃	C₂³×C₅	1	6	yes	—
45	C₃×Q₈	—	4	3	no	—
46	C₃×D₄	—	4	9	no	—
47	C₂×C₃×C₄	C₂³×C₅	1	9	yes	—
48	C₂²×C₅	—	2	18	no	—
49	C₂²×C₅	—	4	8	no	—
50	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
51	C₄×C₅	—	2	9	no	—
52	C₂⁴	C₂²×C₃×C₅	1	2	yes	complemented
53	C₂×Q₈	C₂²×C₃×C₅	1	3	yes	complemented
54	C₂×D₄	C₂²×C₃×C₅	1	9	yes	—
55	C₂²⋊C₄	C₂²×C₃×C₅	1	18	yes	complemented
56	C₄²	C₂²×C₃×C₅	1	3	yes	complemented
57	C₃×C₅	H(3,𝔽₄)	1	1	yes	characteristic, direct product factor, Hall
58	C₂²×C₃	—	2	18	no	—
59	C₂²×C₃	—	4	8	no	—
60	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
61	C₃×C₄	—	2	9	no	—
62	C₂×C₅	—	4	6	no	—
63	C₂×C₅	C₃×H(5,𝔽₂)	1	3	yes	—
64	C₂³	—	4	6	no	—
65	C₂³	C₂³×C₃×C₅	1	6	yes	—
66	Q₈	—	4	3	no	—
67	D₄	—	4	9	no	—
68	C₂×C₄	C₂³×C₃×C₅	1	9	yes	—
69	C₂×C₃	—	4	6	no	—
70	C₂×C₃	C₅×H(5,𝔽₂)	1	3	yes	—
71	C₅	C₃×H(3,𝔽₄)	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²	—	2	18	no	—
73	C₂²	—	4	8	no	—
74	C₂²	C₂⁴×C₃×C₅	1	1	yes	characteristic
75	C₄	—	2	9	no	—
76	C₃	C₅×H(3,𝔽₄)	1	1	yes	characteristic, direct product factor, Sylow
77	C₂	—	4	6	no	—
78	C₂	C₃×C₅×H(5,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#124	C₂	semidirect
2. C₃×C₅×#120	C₂	semidirect
3. C₅×H(3,𝔽₄)	C₃	direct (N × H)
4. C₂⁴×C₃×C₅	C₂²	semidirect
5. C₂×C₃×C₅×Q₈	C₂²	semidirect
7. C₃×C₅×C₂²⋊C₄	C₂²	semidirect
8. C₃×C₄²×C₅	C₂²	semidirect
9. C₃×H(3,𝔽₄)	C₅	direct (N × H)
10. C₅×#124	C₂×C₃	semidirect
11. C₅×#120	C₂×C₃	semidirect
17. C₃×#124	C₂×C₅	semidirect
18. C₃×#120	C₂×C₅	semidirect
19. C₂⁴×C₅	C₂²×C₃	semidirect
20. C₂×C₅×Q₈	C₂²×C₃	semidirect
22. C₅×C₂²⋊C₄	C₂²×C₃	semidirect
23. C₄²×C₅	C₂²×C₃	semidirect
24. H(3,𝔽₄)	C₃×C₅	direct (N × H)
29. C₂⁴×C₃	C₂²×C₅	semidirect
30. C₂×C₃×Q₈	C₂²×C₅	semidirect
32. C₃×C₂²⋊C₄	C₂²×C₅	semidirect
33. C₃×C₄²	C₂²×C₅	semidirect
39. #124	C₂×C₃×C₅	semidirect
40. #120	C₂×C₃×C₅	semidirect
52. C₂⁴	C₂²×C₃×C₅	semidirect
53. C₂×Q₈	C₂²×C₃×C₅	semidirect
55. C₂²⋊C₄	C₂²×C₃×C₅	semidirect
56. C₄²	C₂²×C₃×C₅	semidirect
57. C₃×C₅	H(3,𝔽₄)	direct (N × H)
71. C₅	C₃×H(3,𝔽₄)	direct (N × H)
76. C₃	C₅×H(3,𝔽₄)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#124	2
C₃×C₅×#120	2
C₅×H(3,𝔽₄)	3
C₃×H(3,𝔽₄)	5

Subgroups — C3×C5×H(3,𝔽4) #11756650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×H(3,𝔽₄) #11756650