Subgroups — #91116

Order 480.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×D₄	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#3964	—	3	1	no	maximal, Hall
5	C₂³×C₃×C₅	—	2	1	no	—
6	C₃×C₅×D₄	—	2	1	no	—
7	C₃×C₅×D₄	C₄	1	2	yes	complemented
8	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₅	—	2	1	no	—
10	C₃⋊#168	—	2	1	no	—
11	#832	—	5	1	no	maximal, Hall
12	C₂×C₅×D₄	D₃	1	1	yes	characteristic
13	C₄⋊Dic₅	—	3	1	no	—
14	C₂×#168	—	3	1	no	—
15	C₂²×C₃×C₅	—	2	2	no	—
16	C₂²×C₃×C₅	—	4	1	no	—
17	C₂²×C₃×C₅	—	4	1	no	—
18	C₂²×C₃×C₅	D₄	1	1	yes	characteristic
19	C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₃×C₄×C₅	D₄	1	1	yes	characteristic
21	Dic₁₅	—	4	1	no	—
22	C₂×C₃×D₄	D₅	1	1	yes	characteristic
23	C₄⋊Dic₃	—	5	1	no	—
24	C₂×C₃⋊C₈	—	5	1	no	—
25	C₂³×C₅	—	2	1	no	—
26	C₅×D₄	—	2	1	no	—
27	C₅×D₄	Dic₃	1	2	yes	complemented
28	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
29	C₂×Dic₅	—	6	1	no	—
30	#168	—	6	1	no	—
31	#102	—	15	1	no	Sylow
32	C₂×C₃×C₅	—	4	2	no	—
33	C₂×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₃×C₅	D₈	1	1	yes	characteristic
35	C₂×C₃×C₅	SD₁₆	1	1	yes	characteristic
36	C₂³×C₃	—	2	1	no	—
37	C₃×D₄	—	2	1	no	—
38	C₃×D₄	Dic₅	1	2	yes	complemented
39	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
40	C₂×Dic₃	—	10	1	no	—
41	C₃⋊C₈	—	10	1	no	—
42	C₂²×C₅	—	2	2	no	—
43	C₂²×C₅	—	4	1	no	—
44	C₂²×C₅	—	4	1	no	—
45	C₂²×C₅	#67	1	1	yes	characteristic
46	C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
47	C₄×C₅	#67	1	1	yes	characteristic
48	Dic₅	—	12	1	no	—
49	C₂×D₄	D₁₅	1	1	yes	characteristic
50	C₂×C₈	—	15	1	no	—
51	C₄⋊C₄	—	15	1	no	—
52	C₃×C₅	#102	1	1	yes	characteristic, complemented, Hall
53	C₂²×C₃	—	2	2	no	—
54	C₂²×C₃	—	4	1	no	—
55	C₂²×C₃	—	4	1	no	—
56	C₂²×C₃	C₂²⋊D₅	1	1	yes	characteristic
57	C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
58	C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
59	Dic₃	—	20	1	no	—
60	C₂×C₅	—	4	2	no	—
61	C₂×C₅	#213	1	1	yes	characteristic
62	C₂×C₅	#214	1	1	yes	characteristic
63	C₂×C₅	#217	1	1	yes	characteristic
64	C₂³	—	2	1	no	—
65	D₄	—	2	1	no	—
66	D₄	Dic₁₅	1	2	yes	complemented
67	C₂×C₄	—	30	1	no	—
68	C₂×C₄	C₂×D₁₅	1	1	yes	characteristic
69	C₈	—	30	1	no	—
70	C₂×C₃	—	4	2	no	—
71	C₂×C₃	#686	1	1	yes	characteristic
72	C₂×C₃	#687	1	1	yes	characteristic
73	C₂×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
74	C₅	#832	1	1	yes	characteristic, complemented, Sylow
75	C₂²	—	2	2	no	—
76	C₂²	—	4	1	no	—
77	C₂²	—	4	1	no	—
78	C₂²	#1220	1	1	yes	characteristic
79	C₄	—	60	1	no	—
80	C₄	C₂×Dic₁₅	1	1	yes	characteristic
81	C₄	#1220	1	1	yes	characteristic
82	C₃	#3964	1	1	yes	characteristic, complemented, Sylow
83	C₂	—	4	2	no	—
84	C₂	#6715	1	1	yes	characteristic
85	C₂	#6716	1	1	yes	characteristic
86	C₂	#6719	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#1220
commutator	C₃×C₄×C₅	C₂×C₄
Frattini	C₂×C₄	C₂×D₁₅
Fitting, prime radical	C₂×C₃×C₅×D₄	C₂
socle	C₂²×C₃×C₅	D₄
cosocle	C₂×C₃×C₄×C₅	C₂²
hypercenter	C₂×D₄	D₁₅
hypocenter	C₃×C₅	#102
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
2. C₄⋊Dic₁₅	C₂	semidirect
3. C₂×C₃⋊#168	C₂	semidirect
7. C₃×C₅×D₄	C₄	semidirect
27. C₅×D₄	Dic₃	semidirect
38. C₃×D₄	Dic₅	semidirect
52. C₃×C₅	#102	semidirect
66. D₄	Dic₁₅	semidirect
74. C₅	#832	semidirect
82. C₃	#3964	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×D₄	2
C₄⋊Dic₁₅	2
C₂×C₃⋊#168	2
#3964	3
#832	5