Subgroups — C₂×C₃×#676 #91617

Order 480.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#676	C₂	1	4	yes	maximal, direct product factor
5	C₂×#676	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×D₅	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×Dic₅	C₄	1	1	yes	characteristic
9	C₃×C₄×D₅	C₄	1	2	yes	—
10	C₃×C₄×D₅	C₂²	1	2	yes	—
11	C₃×C₅×C₈	C₂²	1	2	yes	complemented
12	C₃×#168	C₂²	1	2	yes	complemented
13	C₂×C₃×M₁₆	—	5	1	no	maximal, Hall
14	C₂×C₄×D₅	C₂×C₃	1	1	yes	characteristic
15	C₂×C₅×C₈	C₂×C₃	1	1	yes	characteristic, complemented
16	C₂×#168	C₂×C₃	1	1	yes	characteristic, complemented
17	#676	C₂×C₃	1	4	yes	direct product factor
18	C₂²×C₃×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×D₅	—	2	2	no	—
20	C₂×C₃×D₅	C₂×C₄	1	2	yes	—
21	C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₃×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₃×Dic₅	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄	—	5	1	no	—
25	C₃×M₁₆	—	5	4	no	—
26	C₂×C₃×C₈	—	5	1	no	—
27	C₂×C₃×C₈	D₅	1	1	yes	characteristic, complemented
28	C₂²×D₅	C₃×C₄	1	1	yes	characteristic
29	C₂×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
30	C₂×Dic₅	C₃×C₄	1	1	yes	characteristic
31	C₄×D₅	C₃×C₄	1	2	yes	—
32	C₄×D₅	C₂²×C₃	1	2	yes	—
33	C₅×C₈	C₂²×C₃	1	2	yes	complemented
34	#168	C₂²×C₃	1	2	yes	complemented
35	C₂×M₁₆	—	5	1	no	Sylow
36	C₂×C₃×C₅	M₁₆	1	2	yes	complemented
37	C₂×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
38	C₃×D₅	—	2	2	no	—
39	C₂³×C₃	—	5	1	no	—
40	C₂×C₃×C₄	—	5	1	no	—
41	C₂×C₃×C₄	—	5	2	no	—
42	C₂×C₃×C₄	—	5	2	no	—
43	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
44	C₃×C₈	—	5	2	no	—
45	C₃×C₈	C₂×D₅	1	2	yes	complemented
46	C₂²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
47	C₂×D₅	—	2	2	no	—
48	C₂×D₅	C₂×C₃×C₄	1	2	yes	—
49	C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
50	C₄×C₅	C₂³×C₃	1	1	yes	characteristic
51	Dic₅	C₂×C₃×C₄	1	2	yes	—
52	C₂²×C₄	—	5	1	no	—
53	M₁₆	—	5	4	no	—
54	C₂×C₈	—	5	1	no	—
55	C₂×C₈	C₃×D₅	1	1	yes	characteristic, complemented
56	C₃×C₅	C₂×M₁₆	1	1	yes	characteristic, complemented, Hall
57	C₂²×C₃	—	5	2	no	—
58	C₂²×C₃	—	10	2	no	—
59	C₂²×C₃	C₄×D₅	1	1	yes	characteristic
60	C₃×C₄	—	5	2	no	—
61	C₃×C₄	C₄×D₅	1	1	yes	characteristic
62	C₃×C₄	C₂²×D₅	1	1	yes	characteristic
63	C₂×C₅	C₃×M₁₆	1	2	yes	complemented
64	C₂×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
65	D₅	—	2	2	no	—
66	C₂³	—	5	1	no	—
67	C₂×C₄	—	5	1	no	—
68	C₂×C₄	—	5	2	no	—
69	C₂×C₄	—	5	2	no	—
70	C₂×C₄	C₂×C₃×D₅	1	1	yes	characteristic
71	C₈	—	5	2	no	—
72	C₈	C₂×C₃×D₅	1	2	yes	complemented
73	C₂×C₃	—	10	2	no	—
74	C₂×C₃	#676	1	2	yes	direct product factor
75	C₂×C₃	C₂×C₄×D₅	1	1	yes	characteristic
76	C₅	C₂×C₃×M₁₆	1	1	yes	characteristic, complemented, Sylow
77	C₂²	—	5	2	no	—
78	C₂²	—	10	2	no	—
79	C₂²	C₃×C₄×D₅	1	1	yes	characteristic
80	C₄	—	5	2	no	—
81	C₄	C₃×C₄×D₅	1	1	yes	characteristic
82	C₄	C₂²×C₃×D₅	1	1	yes	characteristic
83	C₃	C₂×#676	1	1	yes	characteristic, direct product factor, Sylow
84	C₂	—	10	2	no	—
85	C₂	C₃×#676	1	2	yes	direct product factor
86	C₂	C₂×C₃×C₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₅
commutator	C₂×C₅	C₂²×C₃×C₄
Frattini	C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₂×C₄
prime radical	C₂²×C₃×D₅	C₄
cosocle	C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₈	D₅
hypocenter	C₅	C₂×C₃×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×C₈	C₂	semidirect
3. C₂×C₃×#168	C₂	semidirect
4. C₃×#676	C₂	direct (N × H)
5. C₂×#676	C₃	direct (N × H)
11. C₃×C₅×C₈	C₂²	semidirect
12. C₃×#168	C₂²	semidirect
15. C₂×C₅×C₈	C₂×C₃	semidirect
16. C₂×#168	C₂×C₃	semidirect
17. #676	C₂×C₃	direct (N × H)
27. C₂×C₃×C₈	D₅	semidirect
33. C₅×C₈	C₂²×C₃	semidirect
34. #168	C₂²×C₃	semidirect
36. C₂×C₃×C₅	M₁₆	semidirect
45. C₃×C₈	C₂×D₅	semidirect
55. C₂×C₈	C₃×D₅	semidirect
56. C₃×C₅	C₂×M₁₆	semidirect
63. C₂×C₅	C₃×M₁₆	semidirect
72. C₈	C₂×C₃×D₅	semidirect
74. C₂×C₃	#676	direct (N × H)
76. C₅	C₂×C₃×M₁₆	semidirect
83. C₃	C₂×#676	direct (N × H)
85. C₂	C₃×#676	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×C₄×D₅	2
C₂×C₃×C₅×C₈	2
C₂×C₃×#168	2
C₃×#676	2
C₂×#676	3
C₂×C₃×M₁₆	5

Subgroups — C2×C3×#676 #91617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#676 #91617