Subgroups — C₃×C₅×#997 #13089442

Order 1440.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic, complemented
2	C₅×#997	C₃	1	3	yes	maximal, direct product factor
3	C₂×C₃×C₅×A₄	—	4	3	no	maximal
4	C₃×C₅×SL(2,𝔽₃)	—	4	2	no	maximal
5	C₃×#997	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₃×C₅×#41	—	3	2	no	—
7	C₂×C₃×C₅×D₄	—	3	3	no	—
8	C₃×C₅×A₄	—	4	3	no	—
9	C₅×H(5,𝔽₂)	C₃²	1	1	yes	characteristic, complemented, Hall
10	C₂³×C₃×C₅	—	3	1	no	—
11	C₂³×C₃×C₅	A₄	1	3	yes	complemented
12	C₂×C₅×A₄	—	4	9	no	—
13	C₃×C₅×Q₈	A₄	1	2	yes	complemented
14	C₃×C₅×D₄	—	3	6	no	—
15	C₂×C₃×C₄×C₅	—	3	3	no	—
16	C₅×SL(2,𝔽₃)	—	4	6	no	—
17	C₃×H(5,𝔽₂)	C₃×C₅	1	1	yes	characteristic, complemented
18	#997	C₃×C₅	1	3	yes	direct product factor
19	C₂×C₃²×C₅	—	16	1	no	—
20	C₅×#41	—	3	2	no	—
21	C₂×C₅×D₄	—	3	3	no	—
22	C₂×C₃×A₄	—	4	3	no	—
23	C₃×SL(2,𝔽₃)	—	4	2	no	—
24	C₂²×C₃×C₅	—	3	3	no	—
25	C₂²×C₃×C₅	—	4	3	no	—
26	C₂²×C₃×C₅	—	12	1	no	—
27	C₅×A₄	—	4	9	no	—
28	C₃×C₄×C₅	—	3	2	no	—
29	C₃×#41	—	3	2	no	—
30	C₂×C₃×D₄	—	3	3	no	—
31	C₃²×C₅	—	16	1	no	Hall
32	C₂³×C₅	—	3	1	no	—
33	C₂³×C₅	C₃×A₄	1	3	yes	complemented
34	C₅×Q₈	C₃×A₄	1	2	yes	complemented
35	C₅×D₄	—	3	6	no	—
36	C₂×C₄×C₅	—	3	3	no	—
37	C₃×A₄	—	4	3	no	—
38	H(5,𝔽₂)	C₃²×C₅	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₃×C₅	—	6	3	no	—
40	C₂×C₃×C₅	—	16	3	no	—
41	C₂×C₃×C₅	C₂²⋊A₄	1	1	yes	characteristic
42	C₂³×C₃	—	3	1	no	—
43	C₂³×C₃	C₅×A₄	1	3	yes	complemented
44	C₂×A₄	—	4	9	no	—
45	C₃×Q₈	C₅×A₄	1	2	yes	complemented
46	C₃×D₄	—	3	6	no	—
47	C₂×C₃×C₄	—	3	3	no	—
48	SL(2,𝔽₃)	—	4	6	no	—
49	C₂²×C₅	—	3	3	no	—
50	C₂²×C₅	—	4	3	no	—
51	C₂²×C₅	—	12	1	no	—
52	C₄×C₅	—	3	2	no	—
53	C₂×C₃²	—	16	1	no	—
54	#41	—	3	2	no	—
55	C₂×D₄	—	3	3	no	—
56	C₃×C₅	—	16	3	no	—
57	C₃×C₅	#997	1	1	yes	characteristic, direct product factor
58	C₂²×C₃	—	3	3	no	—
59	C₂²×C₃	—	4	3	no	—
60	C₂²×C₃	—	12	1	no	—
61	A₄	—	4	9	no	—
62	C₃×C₄	—	3	2	no	—
63	C₂×C₅	—	6	3	no	—
64	C₂×C₅	C₃×C₂²⋊A₄	1	1	yes	characteristic
65	C₃²	—	16	1	no	Sylow
66	C₂³	—	3	1	no	—
67	C₂³	C₃×C₅×A₄	1	3	yes	complemented
68	Q₈	C₃×C₅×A₄	1	2	yes	complemented
69	D₄	—	3	6	no	—
70	C₂×C₄	—	3	3	no	—
71	C₂×C₃	—	6	3	no	—
72	C₂×C₃	—	16	3	no	—
73	C₂×C₃	C₅×C₂²⋊A₄	1	1	yes	characteristic
74	C₅	C₃×#997	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²	—	3	3	no	—
76	C₂²	—	4	3	no	—
77	C₂²	—	12	1	no	—
78	C₄	—	3	2	no	—
79	C₃	—	16	3	no	—
80	C₃	C₅×#997	1	1	yes	characteristic, direct product factor
81	C₂	—	6	3	no	—
82	C₂	C₃×C₅×C₂²⋊A₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂×C₃×C₅	C₂²⋊A₄
commutator, cosocle, hypocenter	H(5,𝔽₂)	C₃²×C₅
Frattini	C₂	C₃×C₅×C₂²⋊A₄
Fitting	C₃×C₅×H(5,𝔽₂)	C₃
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×H(5,𝔽₂)	C₃	semidirect
2. C₅×#997	C₃	direct (N × H)
5. C₃×#997	C₅	direct (N × H)
9. C₅×H(5,𝔽₂)	C₃²	semidirect
11. C₂³×C₃×C₅	A₄	semidirect
13. C₃×C₅×Q₈	A₄	semidirect
17. C₃×H(5,𝔽₂)	C₃×C₅	semidirect
18. #997	C₃×C₅	direct (N × H)
33. C₂³×C₅	C₃×A₄	semidirect
34. C₅×Q₈	C₃×A₄	semidirect
38. H(5,𝔽₂)	C₃²×C₅	semidirect
43. C₂³×C₃	C₅×A₄	semidirect
45. C₃×Q₈	C₅×A₄	semidirect
57. C₃×C₅	#997	direct (N × H)
67. C₂³	C₃×C₅×A₄	semidirect
68. Q₈	C₃×C₅×A₄	semidirect
74. C₅	C₃×#997	direct (N × H)
80. C₃	C₅×#997	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×H(5,𝔽₂)	3
C₅×#997	3
C₂×C₃×C₅×A₄	4
C₃×C₅×SL(2,𝔽₃)	4
C₃×#997	5

Subgroups — C3×C5×#997 #13089442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#997 #13089442