Subgroups — #91375

Order 480.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₆₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#6667	C₂	1	4	yes	maximal, complemented
5	#4093	—	3	1	no	maximal, Hall
6	C₂²×D₁₅	C₂²	1	2	yes	—
7	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	D₆₀	—	2	2	no	—
9	C₂×C₅×Dic₃	C₂²	1	2	yes	—
10	C₅×Dic₆	—	2	2	no	—
11	C₂×C₃×Dic₅	C₂²	1	2	yes	—
12	#875	—	5	1	no	maximal, Hall
13	C₂×C₅×Q₈	—	3	1	no	—
14	C₂×D₂₀	—	3	1	no	—
15	#685	—	3	4	no	—
16	C₄×Dic₅	D₃	1	1	yes	characteristic, complemented
17	C₂²×C₃×C₅	C₂³	1	1	yes	characteristic
18	C₂×D₁₅	—	2	2	no	—
19	C₂×D₁₅	—	2	4	no	—
20	C₃×C₄×C₅	D₄	1	2	yes	—
21	C₃×Dic₅	—	2	2	no	—
22	C₅×Dic₃	—	2	2	no	—
23	C₂×D₁₂	—	5	1	no	—
24	C₂×Dic₆	D₅	1	1	yes	characteristic, complemented
25	C₃×C₄²	—	5	1	no	—
26	(C₂×D₃)⋊C₄	—	5	4	no	—
27	C₂²×D₅	—	3	2	no	—
28	C₅×Q₈	—	6	2	no	—
29	C₂×C₄×C₅	—	3	2	no	—
30	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
31	C₂×Dic₅	C₂×D₃	1	2	yes	—
32	D₂₀	—	6	2	no	—
33	#124	—	15	1	no	Sylow
34	C₂×C₃×C₅	C₂×D₄	1	1	yes	characteristic
35	C₂×C₃×C₅	#41	1	2	yes	—
36	D₁₅	—	4	2	no	—
37	C₂²×D₃	—	5	2	no	—
38	C₂×C₃×C₄	—	5	2	no	—
39	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
40	C₂×Dic₃	C₂×D₅	1	2	yes	—
41	D₁₂	—	10	2	no	—
42	Dic₆	—	2	2	no	—
43	C₂²×C₅	C₂²×D₃	1	1	yes	characteristic
44	C₂×D₅	—	6	2	no	—
45	C₂×D₅	—	6	4	no	—
46	C₄×C₅	—	6	2	no	—
47	C₄×C₅	D₁₂	1	2	yes	—
48	Dic₅	—	2	2	no	—
49	C₂×Q₈	—	3	1	no	—
50	C₂×D₄	—	15	1	no	—
51	C₂²⋊C₄	—	15	4	no	—
52	C₄²	—	5	1	no	—
53	C₃×C₅	#124	1	1	yes	characteristic, complemented, Hall
54	C₂²×C₃	C₂²×D₅	1	1	yes	characteristic
55	C₂×D₃	—	10	4	no	—
56	C₂×D₃	—	10	2	no	—
57	C₃×C₄	—	10	2	no	—
58	C₃×C₄	C₂²⋊D₅	1	2	yes	—
59	Dic₃	—	2	2	no	—
60	C₂×C₅	C₂×D₁₂	1	1	yes	characteristic
61	C₂×C₅	#235	1	2	yes	—
62	D₅	—	12	2	no	—
63	C₂³	—	15	2	no	—
64	Q₈	—	6	2	no	—
65	D₄	—	30	2	no	—
66	C₂×C₄	—	3	2	no	—
67	C₂×C₄	—	5	2	no	—
68	C₂×C₄	D₃×D₅	1	1	yes	characteristic
69	C₂×C₃	#713	1	2	yes	—
70	C₂×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
71	D₃	—	20	2	no	—
72	C₅	#875	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	30	4	no	—
74	C₂²	—	30	2	no	—
75	C₂²	C₂×D₃×D₅	1	1	yes	characteristic
76	C₄	—	6	2	no	—
77	C₄	—	10	2	no	—
78	C₄	#1203	1	2	yes	—
79	C₃	#4093	1	1	yes	characteristic, complemented, Sylow
80	C₂	—	60	2	no	—
81	C₂	#6773	1	2	yes	—
82	C₂	C₂×#1203	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂³
Fitting	C₂×C₃×C₄×C₅	C₂²
prime radical	C₂×D₆₀	C₂
hypercenter	C₂×C₄	D₃×D₅
hypocenter	C₃×C₅	#124
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×Dic₆	C₂	semidirect
3. C₃×C₄×Dic₅	C₂	semidirect
4. #6667	C₂	semidirect
16. C₄×Dic₅	D₃	semidirect
24. C₂×Dic₆	D₅	semidirect
53. C₃×C₅	#124	semidirect
72. C₅	#875	semidirect
79. C₃	#4093	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₆₀	2
C₂×C₅×Dic₆	2
C₃×C₄×Dic₅	2
#6667	2
#4093	3
#875	5