Subgroups — C₂×C₅×#239 #92082

Order 480.

328 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#239	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₅×D₁₂	C₂	1	3	yes	maximal, complemented
4	C₂×C₄×C₅×D₃	C₂	1	3	yes	maximal, complemented
5	C₂×C₅×#41	—	3	1	no	maximal, Hall
6	C₂²×C₅×D₃	C₂²	1	3	yes	—
7	C₃×C₅×Q₈	C₂²	1	4	yes	complemented
8	C₂×C₃×C₄×C₅	C₂²	1	3	yes	—
9	C₂×C₅×Dic₃	C₂²	1	1	yes	characteristic
10	C₅×D₁₂	C₂²	1	12	yes	complemented
11	C₄×C₅×D₃	C₂²	1	12	yes	complemented
12	C₂×#239	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₅×#41	—	3	8	no	—
14	C₂×C₅×Q₈	D₃	1	1	yes	characteristic, complemented
15	C₂×C₅×D₄	—	3	3	no	—
16	C₂²×C₄×C₅	—	3	3	no	—
17	C₂²×C₃×C₅	C₂³	1	1	yes	characteristic
18	C₂×C₅×D₃	—	2	6	no	—
19	C₂×C₅×D₃	C₂³	1	6	yes	—
20	C₃×C₄×C₅	C₂³	1	6	yes	—
21	C₅×Dic₃	C₂³	1	2	yes	—
22	C₂×C₃×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
23	#239	C₂×C₅	1	8	yes	direct product factor
24	C₂×D₁₂	C₂×C₅	1	3	yes	complemented
25	C₂×C₄×D₃	C₂×C₅	1	3	yes	complemented
26	C₂³×C₅	—	3	3	no	—
27	C₅×Q₈	C₂×D₃	1	4	yes	complemented
28	C₅×D₄	—	3	12	no	—
29	C₂×C₄×C₅	—	3	1	no	—
30	C₂×C₄×C₅	—	3	12	no	—
31	C₂×C₄×C₅	C₂×D₃	1	3	yes	—
32	C₂×#41	—	3	1	no	Sylow
33	C₂×C₃×C₅	#41	1	2	yes	complemented
34	C₂×C₃×C₅	C₂⁴	1	1	yes	characteristic
35	C₅×D₃	—	2	6	no	—
36	C₂²×D₃	C₂²×C₅	1	3	yes	—
37	C₃×Q₈	C₂²×C₅	1	4	yes	complemented
38	C₂×C₃×C₄	C₂²×C₅	1	3	yes	—
39	C₂×Dic₃	C₂²×C₅	1	1	yes	characteristic
40	D₁₂	C₂²×C₅	1	12	yes	complemented
41	C₄×D₃	C₂²×C₅	1	12	yes	complemented
42	C₂²×C₅	—	3	6	no	—
43	C₂²×C₅	—	6	6	no	—
44	C₂²×C₅	C₂²×D₃	1	1	yes	characteristic
45	C₄×C₅	—	3	2	no	—
46	C₄×C₅	C₂²×D₃	1	6	yes	—
47	#41	—	3	8	no	—
48	C₂×Q₈	C₅×D₃	1	1	yes	characteristic, complemented
49	C₂×D₄	—	3	3	no	—
50	C₂²×C₄	—	3	3	no	—
51	C₃×C₅	C₂×#41	1	1	yes	characteristic, complemented, Hall
52	C₂²×C₃	C₂³×C₅	1	1	yes	characteristic
53	C₂×D₃	—	2	6	no	—
54	C₂×D₃	C₂³×C₅	1	6	yes	—
55	C₃×C₄	C₂³×C₅	1	6	yes	—
56	Dic₃	C₂³×C₅	1	2	yes	—
57	C₂×C₅	—	6	6	no	—
58	C₂×C₅	#239	1	2	yes	direct product factor
59	C₂×C₅	C₂³×D₃	1	1	yes	characteristic
60	C₂³	—	3	3	no	—
61	Q₈	C₂×C₅×D₃	1	4	yes	complemented
62	D₄	—	3	12	no	—
63	C₂×C₄	—	3	1	no	—
64	C₂×C₄	—	3	12	no	—
65	C₂×C₄	C₂×C₅×D₃	1	3	yes	—
66	C₂×C₃	C₅×#41	1	2	yes	complemented
67	C₂×C₃	C₂⁴×C₅	1	1	yes	characteristic
68	D₃	—	2	6	no	—
69	C₅	C₂×#239	1	1	yes	characteristic, direct product factor, Sylow
70	C₂²	—	3	6	no	—
71	C₂²	—	6	6	no	—
72	C₂²	C₂²×C₅×D₃	1	1	yes	characteristic
73	C₄	—	3	2	no	—
74	C₄	C₂²×C₅×D₃	1	6	yes	—
75	C₃	C₂×C₅×#41	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	6	6	no	—
77	C₂	C₅×#239	1	2	yes	direct product factor
78	C₂	C₂³×C₅×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂⁴×C₅
Frattini	C₂	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂³
hypercenter	C₂×C₅×Q₈	D₃
hypocenter	C₃	C₂×C₅×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×Q₈	C₂	semidirect
2. C₅×#239	C₂	direct (N × H)
3. C₂×C₅×D₁₂	C₂	semidirect
4. C₂×C₄×C₅×D₃	C₂	semidirect
7. C₃×C₅×Q₈	C₂²	semidirect
10. C₅×D₁₂	C₂²	semidirect
11. C₄×C₅×D₃	C₂²	semidirect
12. C₂×#239	C₅	direct (N × H)
14. C₂×C₅×Q₈	D₃	semidirect
22. C₂×C₃×Q₈	C₂×C₅	semidirect
23. #239	C₂×C₅	direct (N × H)
24. C₂×D₁₂	C₂×C₅	semidirect
25. C₂×C₄×D₃	C₂×C₅	semidirect
27. C₅×Q₈	C₂×D₃	semidirect
33. C₂×C₃×C₅	#41	semidirect
37. C₃×Q₈	C₂²×C₅	semidirect
40. D₁₂	C₂²×C₅	semidirect
41. C₄×D₃	C₂²×C₅	semidirect
48. C₂×Q₈	C₅×D₃	semidirect
51. C₃×C₅	C₂×#41	semidirect
58. C₂×C₅	#239	direct (N × H)
61. Q₈	C₂×C₅×D₃	semidirect
66. C₂×C₃	C₅×#41	semidirect
69. C₅	C₂×#239	direct (N × H)
75. C₃	C₂×C₅×#41	semidirect
77. C₂	C₅×#239	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×Q₈	2
C₅×#239	2
C₂×C₅×D₁₂	2
C₂×C₄×C₅×D₃	2
C₂×C₅×#41	3
C₂×#239	5

Subgroups — C2×C5×#239 #92082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#239 #92082