Subgroups — #91115

Order 480.

152 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	C₂²×Dic₁₅	C₂	1	2	yes	maximal
3	#3963	—	3	1	no	maximal, Hall
4	C₂³×C₃×C₅	C₂²	1	1	yes	characteristic
5	C₂×C₃×C₄×C₅	—	2	2	no	—
6	C₂×C₃×C₄×C₅	C₄	1	2	yes	complemented
7	C₂×Dic₁₅	—	2	2	no	—
8	C₂×Dic₁₅	—	2	2	no	—
9	C₂×Dic₁₅	C₄	1	4	yes	complemented
10	C₃⋊#95	—	5	1	no	maximal, Hall
11	C₂²×C₄×C₅	D₃	1	1	yes	characteristic
12	C₂²×Dic₅	—	3	2	no	—
13	C₂²×C₃×C₅	C₂×C₄	1	2	yes	—
14	C₂²×C₃×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₃×C₅	D₄	1	1	yes	characteristic
16	C₂²×C₃×C₅	D₄	1	2	yes	—
17	C₂²×C₃×C₅	Q₈	1	1	yes	characteristic
18	C₃×C₄×C₅	—	2	2	no	—
19	Dic₁₅	—	2	4	no	—
20	C₂²×C₃×C₄	D₅	1	1	yes	characteristic
21	C₂²×Dic₃	—	5	2	no	—
22	C₂³×C₅	C₂×D₃	1	1	yes	characteristic
23	C₂×C₄×C₅	—	2	2	no	—
24	C₂×C₄×C₅	Dic₃	1	2	yes	complemented
25	C₂×Dic₅	—	3	4	no	—
26	C₂×Dic₅	—	6	2	no	—
27	C₂×Dic₅	—	6	2	no	—
28	#95	—	15	1	no	Sylow
29	C₂×C₃×C₅	C₄²	1	1	yes	characteristic
30	C₂×C₃×C₅	C₂²⋊C₄	1	2	yes	—
31	C₂×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₃×C₅	C₄⋊C₄	1	2	yes	—
33	C₂×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
34	C₂³×C₃	C₂×D₅	1	1	yes	characteristic
35	C₂×C₃×C₄	—	2	2	no	—
36	C₂×C₃×C₄	Dic₅	1	2	yes	complemented
37	C₂×Dic₃	—	5	4	no	—
38	C₂×Dic₃	—	10	2	no	—
39	C₂×Dic₃	—	10	2	no	—
40	C₂²×C₅	Dic₆	1	1	yes	characteristic
41	C₂²×C₅	C₄×D₃	1	2	yes	—
42	C₂²×C₅	D₁₂	1	1	yes	characteristic
43	C₂²×C₅	C₂×Dic₃	1	1	yes	characteristic
44	C₂²×C₅	#67	1	2	yes	—
45	C₄×C₅	—	2	2	no	—
46	Dic₅	—	6	4	no	—
47	C₂²×C₄	—	15	2	no	—
48	C₂²×C₄	D₁₅	1	1	yes	characteristic
49	C₃×C₅	#95	1	1	yes	characteristic, complemented, Hall
50	C₂²×C₃	Dic₁₀	1	1	yes	characteristic
51	C₂²×C₃	C₄×D₅	1	2	yes	—
52	C₂²×C₃	D₂₀	1	1	yes	characteristic
53	C₂²×C₃	C₂×Dic₅	1	1	yes	characteristic
54	C₂²×C₃	C₂²⋊D₅	1	2	yes	—
55	C₃×C₄	—	2	2	no	—
56	Dic₃	—	10	4	no	—
57	C₂×C₅	C₄×Dic₃	1	1	yes	characteristic
58	C₂×C₅	Dic₃⋊C₄	1	2	yes	—
59	C₂×C₅	C₄⋊Dic₃	1	1	yes	characteristic
60	C₂×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
61	C₂×C₅	#217	1	1	yes	characteristic
62	C₂³	C₂×D₁₅	1	1	yes	characteristic
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	15	4	no	—
65	C₂×C₄	—	30	2	no	—
66	C₂×C₄	—	30	2	no	—
67	C₂×C₄	Dic₁₅	1	2	yes	complemented
68	C₂×C₃	C₄×Dic₅	1	1	yes	characteristic
69	C₂×C₃	#683	1	2	yes	—
70	C₂×C₃	C₄⋊Dic₅	1	1	yes	characteristic
71	C₂×C₃	#685	1	2	yes	—
72	C₂×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
73	C₅	C₃⋊#95	1	1	yes	characteristic, complemented, Sylow
74	C₂²	Dic₃₀	1	1	yes	characteristic
75	C₂²	C₄×D₁₅	1	2	yes	—
76	C₂²	D₆₀	1	1	yes	characteristic
77	C₂²	C₂×Dic₁₅	1	1	yes	characteristic
78	C₂²	#1220	1	2	yes	—
79	C₄	—	2	2	no	—
80	C₄	—	30	4	no	—
81	C₃	#3963	1	1	yes	characteristic, complemented, Sylow
82	C₂	C₄×Dic₁₅	1	1	yes	characteristic
83	C₂	#6712	1	2	yes	—
84	C₂	C₄⋊Dic₁₅	1	1	yes	characteristic
85	C₂	#6714	1	2	yes	—
86	C₂	#6719	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂×D₁₅
commutator	C₂×C₃×C₅	C₄²
Fitting	C₂²×C₃×C₄×C₅	C₂
socle, prime radical, cosocle	C₂³×C₃×C₅	C₂²
hypercenter	C₂²×C₄	D₁₅
hypocenter	C₃×C₅	#95
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
6. C₂×C₃×C₄×C₅	C₄	semidirect
9. C₂×Dic₁₅	C₄	semidirect
24. C₂×C₄×C₅	Dic₃	semidirect
36. C₂×C₃×C₄	Dic₅	semidirect
49. C₃×C₅	#95	semidirect
67. C₂×C₄	Dic₁₅	semidirect
73. C₅	C₃⋊#95	semidirect
81. C₃	#3963	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
C₂²×Dic₁₅	2
#3963	3
C₃⋊#95	5