Subgroups — C₂×#6821 #92097

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	#6821	C₂	1	8	yes	maximal, direct product factor
3	C₂×D₆₀	C₂	1	3	yes	maximal, complemented
4	C₂×C₄×D₁₅	C₂	1	3	yes	maximal, complemented
5	C₂×#713	—	3	1	no	maximal, Hall
6	C₂²×D₁₅	C₂²	1	3	yes	—
7	C₃×C₅×Q₈	C₂²	1	4	yes	complemented
8	C₂×C₃×C₄×C₅	C₂²	1	3	yes	—
9	C₂×Dic₁₅	C₂²	1	1	yes	characteristic
10	D₆₀	C₂²	1	12	yes	complemented
11	C₄×D₁₅	C₂²	1	12	yes	complemented
12	C₂×#239	—	5	1	no	maximal, Hall
13	C₂×C₅×Q₈	D₃	1	1	yes	characteristic, complemented
14	#713	—	3	8	no	—
15	C₂×D₂₀	—	3	3	no	—
16	C₂×C₄×D₅	—	3	3	no	—
17	C₂²×C₃×C₅	C₂³	1	1	yes	characteristic
18	C₂×D₁₅	—	2	6	no	—
19	C₂×D₁₅	C₂³	1	6	yes	—
20	C₃×C₄×C₅	C₂³	1	6	yes	—
21	Dic₁₅	C₂³	1	2	yes	—
22	C₂×C₃×Q₈	D₅	1	1	yes	characteristic, complemented
23	#239	—	5	8	no	—
24	C₂×D₁₂	—	5	3	no	—
25	C₂×C₄×D₃	—	5	3	no	—
26	C₂²×D₅	—	3	3	no	—
27	C₅×Q₈	C₂×D₃	1	4	yes	complemented
28	C₂×C₄×C₅	C₂×D₃	1	3	yes	—
29	C₂×Dic₅	—	3	1	no	—
30	D₂₀	—	3	12	no	—
31	C₄×D₅	—	3	12	no	—
32	C₂×#41	—	15	1	no	Sylow
33	C₂×C₃×C₅	#41	1	2	yes	complemented
34	C₂×C₃×C₅	C₂⁴	1	1	yes	characteristic
35	D₁₅	—	2	6	no	—
36	C₂²×D₃	—	5	3	no	—
37	C₃×Q₈	C₂×D₅	1	4	yes	complemented
38	C₂×C₃×C₄	C₂×D₅	1	3	yes	—
39	C₂×Dic₃	—	5	1	no	—
40	D₁₂	—	5	12	no	—
41	C₄×D₃	—	5	12	no	—
42	C₂²×C₅	C₂²×D₃	1	1	yes	characteristic
43	C₂×D₅	—	3	6	no	—
44	C₂×D₅	—	6	6	no	—
45	C₄×C₅	C₂²×D₃	1	6	yes	—
46	Dic₅	—	3	2	no	—
47	#41	—	15	8	no	—
48	C₂×Q₈	D₁₅	1	1	yes	characteristic, complemented
49	C₂×D₄	—	15	3	no	—
50	C₂²×C₄	—	15	3	no	—
51	C₃×C₅	C₂×#41	1	1	yes	characteristic, complemented, Hall
52	C₂²×C₃	C₂²×D₅	1	1	yes	characteristic
53	C₂×D₃	—	5	6	no	—
54	C₂×D₃	—	10	6	no	—
55	C₃×C₄	C₂²×D₅	1	6	yes	—
56	Dic₃	—	5	2	no	—
57	C₂×C₅	#239	1	2	yes	complemented
58	C₂×C₅	C₂³×D₃	1	1	yes	characteristic
59	D₅	—	6	6	no	—
60	C₂³	—	15	3	no	—
61	Q₈	C₂×D₁₅	1	4	yes	complemented
62	D₄	—	15	12	no	—
63	C₂×C₄	—	15	1	no	—
64	C₂×C₄	—	15	12	no	—
65	C₂×C₄	C₂×D₁₅	1	3	yes	—
66	C₂×C₃	#713	1	2	yes	complemented
67	C₂×C₃	C₂³×D₅	1	1	yes	characteristic
68	D₃	—	10	6	no	—
69	C₅	C₂×#239	1	1	yes	characteristic, complemented, Sylow
70	C₂²	—	15	6	no	—
71	C₂²	—	30	6	no	—
72	C₂²	C₂²×D₁₅	1	1	yes	characteristic
73	C₄	—	15	2	no	—
74	C₄	C₂²×D₁₅	1	6	yes	—
75	C₃	C₂×#713	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	30	6	no	—
77	C₂	#6821	1	2	yes	direct product factor
78	C₂	C₂³×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₅
commutator, cosocle	C₂×C₃×C₅	C₂⁴
Frattini	C₂	C₂³×D₁₅
Fitting	C₂×C₃×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂³
hypercenter	C₂×Q₈	D₁₅
hypocenter	C₃×C₅	C₂×#41
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×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. #6821	C₂	direct (N × H)
3. C₂×D₆₀	C₂	semidirect
4. C₂×C₄×D₁₅	C₂	semidirect
7. C₃×C₅×Q₈	C₂²	semidirect
10. D₆₀	C₂²	semidirect
11. C₄×D₁₅	C₂²	semidirect
13. C₂×C₅×Q₈	D₃	semidirect
22. C₂×C₃×Q₈	D₅	semidirect
27. C₅×Q₈	C₂×D₃	semidirect
33. C₂×C₃×C₅	#41	semidirect
37. C₃×Q₈	C₂×D₅	semidirect
48. C₂×Q₈	D₁₅	semidirect
51. C₃×C₅	C₂×#41	semidirect
57. C₂×C₅	#239	semidirect
61. Q₈	C₂×D₁₅	semidirect
66. C₂×C₃	#713	semidirect
69. C₅	C₂×#239	semidirect
75. C₃	C₂×#713	semidirect
77. C₂	#6821	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
#6821	2
C₂×D₆₀	2
C₂×C₄×D₁₅	2
C₂×#713	3
C₂×#239	5

Subgroups — C2×#6821 #92097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6821 #92097