Subgroups — C₂×C₃×#681 #91638

Order 480.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₃×#681	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃×#168	C₂	1	2	yes	maximal, complemented
4	C₂×#681	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₅	C₄	1	1	yes	characteristic
6	C₂×C₃×C₄×C₅	C₄	1	2	yes	—
7	C₂×C₃×C₄×C₅	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄×C₅	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₃×#168	C₂²	1	4	yes	complemented
11	C₂×C₃×M₁₆	—	5	1	no	maximal, Hall
12	C₂²×C₄×C₅	C₂×C₃	1	1	yes	characteristic
13	#681	C₂×C₃	1	4	yes	direct product factor
14	C₂×#168	C₂×C₃	1	2	yes	complemented
15	C₂²×C₃×C₅	—	2	2	no	—
16	C₂²×C₃×C₅	C₂×C₄	1	2	yes	—
17	C₂²×C₃×C₅	C₂×C₄	1	1	yes	characteristic
18	C₃×C₄×C₅	C₂×C₄	1	2	yes	—
19	C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₃×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂²×C₃×C₄	D₅	1	1	yes	characteristic
22	C₃×M₁₆	—	5	4	no	—
23	C₂×C₃×C₈	—	5	2	no	—
24	C₂³×C₅	C₃×C₄	1	1	yes	characteristic
25	C₂×C₄×C₅	C₃×C₄	1	2	yes	—
26	C₂×C₄×C₅	C₃×C₄	1	1	yes	characteristic
27	C₂×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
28	C₂×C₄×C₅	C₂²×C₃	1	2	yes	—
29	#168	C₂²×C₃	1	4	yes	complemented
30	C₂×M₁₆	—	5	1	no	Sylow
31	C₂×C₃×C₅	—	2	2	no	—
32	C₂×C₃×C₅	M₁₆	1	2	yes	complemented
33	C₂×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
34	C₂³×C₃	Dic₅	1	1	yes	characteristic
35	C₂×C₃×C₄	Dic₅	1	1	yes	characteristic
36	C₂×C₃×C₄	Dic₅	1	2	yes	—
37	C₂×C₃×C₄	C₂×D₅	1	2	yes	—
38	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
39	C₃×C₈	—	5	4	no	—
40	C₂²×C₅	—	2	2	no	—
41	C₂²×C₅	C₂×C₃×C₄	1	2	yes	—
42	C₂²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
43	C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
44	C₄×C₅	C₂×C₃×C₄	1	2	yes	—
45	C₄×C₅	C₂³×C₃	1	1	yes	characteristic
46	C₂²×C₄	C₃×D₅	1	1	yes	characteristic
47	M₁₆	—	5	4	no	—
48	C₂×C₈	—	5	2	no	—
49	C₃×C₅	C₂×M₁₆	1	1	yes	characteristic, complemented, Hall
50	C₂²×C₃	—	2	2	no	—
51	C₂²×C₃	C₂×Dic₅	1	2	yes	—
52	C₂²×C₃	C₂×Dic₅	1	1	yes	characteristic
53	C₃×C₄	C₂×Dic₅	1	2	yes	—
54	C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
55	C₃×C₄	C₂²×D₅	1	1	yes	characteristic
56	C₂×C₅	—	2	2	no	—
57	C₂×C₅	C₃×M₁₆	1	2	yes	complemented
58	C₂×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
59	C₂³	C₃×Dic₅	1	1	yes	characteristic
60	C₂×C₄	C₃×Dic₅	1	2	yes	—
61	C₂×C₄	C₃×Dic₅	1	1	yes	characteristic
62	C₂×C₄	C₂×C₃×D₅	1	1	yes	characteristic
63	C₂×C₄	C₂×C₃×D₅	1	2	yes	—
64	C₈	—	5	4	no	—
65	C₂×C₃	—	2	2	no	—
66	C₂×C₃	#681	1	2	yes	direct product factor
67	C₂×C₃	C₂²×Dic₅	1	1	yes	characteristic
68	C₅	C₂×C₃×M₁₆	1	1	yes	characteristic, complemented, Sylow
69	C₂²	—	2	2	no	—
70	C₂²	C₂×C₃×Dic₅	1	2	yes	—
71	C₂²	C₂×C₃×Dic₅	1	1	yes	characteristic
72	C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
73	C₄	C₂×C₃×Dic₅	1	2	yes	—
74	C₄	C₂²×C₃×D₅	1	1	yes	characteristic
75	C₃	C₂×#681	1	1	yes	characteristic, direct product factor, Sylow
76	C₂	—	2	2	no	—
77	C₂	C₃×#681	1	2	yes	direct product factor
78	C₂	C₂²×C₃×Dic₅	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	C₂²×C₃×C₄×C₅	C₂
socle	C₂²×C₃×C₅	C₂×C₄
prime radical	C₂³×C₃×C₅	C₄
cosocle	C₄×C₅	C₂³×C₃
hypercenter	C₂²×C₃×C₄	D₅
hypocenter	C₅	C₂×C₃×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₃×#681	C₂	direct (N × H)
3. C₂×C₃×#168	C₂	semidirect
4. C₂×#681	C₃	direct (N × H)
10. C₃×#168	C₂²	semidirect
13. #681	C₂×C₃	direct (N × H)
14. C₂×#168	C₂×C₃	semidirect
29. #168	C₂²×C₃	semidirect
32. C₂×C₃×C₅	M₁₆	semidirect
49. C₃×C₅	C₂×M₁₆	semidirect
57. C₂×C₅	C₃×M₁₆	semidirect
66. C₂×C₃	#681	direct (N × H)
68. C₅	C₂×C₃×M₁₆	semidirect
75. C₃	C₂×#681	direct (N × H)
77. C₂	C₃×#681	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×C₅	2
C₃×#681	2
C₂×C₃×#168	2
C₂×#681	3
C₂×C₃×M₁₆	5

Subgroups — C2×C3×#681 #91638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#681 #91638