Subgroups — #91114

Order 480.

100 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	C₂×C₃⋊#168	C₂	1	2	yes	maximal, complemented
3	C₂²⋊#168	—	3	1	no	maximal, Hall
4	C₂³×C₃×C₅	C₄	1	1	yes	characteristic
5	C₂×C₃×C₄×C₅	—	2	2	no	—
6	C₂×C₃×C₄×C₅	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₃⋊#168	—	2	2	no	—
9	#830	—	5	1	no	maximal, Hall
10	C₂²×C₄×C₅	D₃	1	1	yes	characteristic
11	C₂×#168	—	3	2	no	—
12	C₂²×C₃×C₅	—	2	1	no	—
13	C₂²×C₃×C₅	—	2	1	no	—
14	C₂²×C₃×C₅	C₈	1	2	yes	complemented
15	C₂²×C₃×C₅	C₂×C₄	1	1	yes	characteristic
16	C₃×C₄×C₅	—	2	1	no	—
17	C₃×C₄×C₅	D₄	1	2	yes	—
18	C₂²×C₃×C₄	D₅	1	1	yes	characteristic
19	C₂×C₃⋊C₈	—	5	2	no	—
20	C₂³×C₅	Dic₃	1	1	yes	characteristic
21	C₂×C₄×C₅	—	2	2	no	—
22	C₂×C₄×C₅	Dic₃	1	1	yes	characteristic
23	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
24	#168	—	6	2	no	—
25	C₂²⋊C₈	—	15	1	no	Sylow
26	C₂×C₃×C₅	—	2	2	no	—
27	C₂×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₃×C₅	C₂×C₈	1	1	yes	characteristic
29	C₂×C₃×C₅	M₁₆	1	1	yes	characteristic
30	C₂³×C₃	Dic₅	1	1	yes	characteristic
31	C₂×C₃×C₄	—	2	2	no	—
32	C₂×C₃×C₄	Dic₅	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
34	C₃⋊C₈	—	10	2	no	—
35	C₂²×C₅	—	2	1	no	—
36	C₂²×C₅	—	2	1	no	—
37	C₂²×C₅	C₃⋊C₈	1	2	yes	complemented
38	C₂²×C₅	C₂×Dic₃	1	1	yes	characteristic
39	C₄×C₅	—	2	1	no	—
40	C₄×C₅	#67	1	2	yes	—
41	C₂²×C₄	D₁₅	1	1	yes	characteristic
42	C₂×C₈	—	15	2	no	—
43	C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic, complemented, Hall
44	C₂²×C₃	—	2	1	no	—
45	C₂²×C₃	—	2	1	no	—
46	C₂²×C₃	#168	1	2	yes	complemented
47	C₂²×C₃	C₂×Dic₅	1	1	yes	characteristic
48	C₃×C₄	—	2	1	no	—
49	C₃×C₄	C₂²⋊D₅	1	2	yes	—
50	C₂×C₅	—	2	2	no	—
51	C₂×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
52	C₂×C₅	#208	1	1	yes	characteristic
53	C₂×C₅	#217	1	1	yes	characteristic
54	C₂³	Dic₁₅	1	1	yes	characteristic
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	Dic₁₅	1	1	yes	characteristic
57	C₂×C₄	C₂×D₁₅	1	1	yes	characteristic
58	C₈	—	30	2	no	—
59	C₂×C₃	—	2	2	no	—
60	C₂×C₃	C₂×#168	1	1	yes	characteristic
61	C₂×C₃	#681	1	1	yes	characteristic
62	C₂×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
63	C₅	#830	1	1	yes	characteristic, complemented, Sylow
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	C₃⋊#168	1	2	yes	complemented
67	C₂²	C₂×Dic₁₅	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	#1220	1	2	yes	—
70	C₃	C₂²⋊#168	1	1	yes	characteristic, complemented, Sylow
71	C₂	—	2	2	no	—
72	C₂	C₂×C₃⋊#168	1	1	yes	characteristic
73	C₂	#6710	1	1	yes	characteristic
74	C₂	#6719	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₁₅
commutator	C₂×C₃×C₅	C₂×C₈
Fitting	C₂²×C₃×C₄×C₅	C₂
socle	C₂²×C₃×C₅	C₂×C₄
prime radical	C₂³×C₃×C₅	C₄
cosocle	C₂×C₃×C₄×C₅	C₂²
hypercenter	C₂²×C₄	D₁₅
hypocenter	C₃×C₅	C₂²⋊C₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃⋊#168	C₂	semidirect
14. C₂²×C₃×C₅	C₈	semidirect
37. C₂²×C₅	C₃⋊C₈	semidirect
43. C₃×C₅	C₂²⋊C₈	semidirect
46. C₂²×C₃	#168	semidirect
63. C₅	#830	semidirect
66. C₂²	C₃⋊#168	semidirect
70. C₃	C₂²⋊#168	semidirect

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
C₂×C₃⋊#168	2
C₂²⋊#168	3
#830	5