Subgroups — #92098

Order 480.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#6821	C₂	1	4	yes	maximal, complemented
3	Q₈×D₁₅	C₂	1	4	yes	maximal, complemented
4	#6817	C₂	1	6	yes	maximal, complemented
5	#4147	—	3	1	no	maximal, Hall
6	C₃×C₅×Q₈	C₂²	1	4	yes	—
7	C₂×C₃×C₄×C₅	C₂²	1	3	yes	—
8	#1220	C₂²	1	4	yes	—
9	D₆₀	C₂²	1	6	yes	—
10	C₄×D₁₅	C₂²	1	12	yes	—
11	Dic₃₀	C₂²	1	6	yes	—
12	#1007	—	5	1	no	maximal, Hall
13	C₂×C₅×Q₈	D₃	1	1	yes	characteristic, complemented
14	#713	—	3	4	no	—
15	Q₈×D₅	—	3	4	no	—
16	#709	—	3	6	no	—
17	C₂²×C₃×C₅	C₂³	1	1	yes	characteristic
18	C₂×D₁₅	C₂³	1	4	yes	—
19	C₃×C₄×C₅	C₂³	1	6	yes	—
20	Dic₁₅	C₂³	1	4	yes	—
21	C₂×C₃×Q₈	D₅	1	1	yes	characteristic, complemented
22	#239	—	5	4	no	—
23	D₃×Q₈	—	5	4	no	—
24	#235	—	5	6	no	—
25	C₅×Q₈	C₂×D₃	1	4	yes	—
26	C₂×C₄×C₅	C₂×D₃	1	3	yes	—
27	C₂²⋊D₅	—	3	4	no	—
28	D₂₀	—	3	6	no	—
29	C₄×D₅	—	3	12	no	—
30	Dic₁₀	—	3	6	no	—
31	2₋¹⁺⁴	—	15	1	no	Sylow
32	C₂×C₃×C₅	—	2	1	no	—
33	C₂×C₃×C₅	C₂⁴	1	1	yes	characteristic
34	D₁₅	—	2	4	no	—
35	C₃×Q₈	C₂×D₅	1	4	yes	—
36	C₂×C₃×C₄	C₂×D₅	1	3	yes	—
37	#67	—	5	4	no	—
38	D₁₂	—	5	6	no	—
39	C₄×D₃	—	5	12	no	—
40	Dic₆	—	5	6	no	—
41	C₂²×C₅	C₂²×D₃	1	1	yes	characteristic
42	C₂×D₅	—	3	4	no	—
43	C₄×C₅	C₂²×D₃	1	6	yes	—
44	Dic₅	—	3	4	no	—
45	#41	—	15	4	no	—
46	#41	—	15	6	no	—
47	C₂×Q₈	—	15	4	no	—
48	C₂×Q₈	D₁₅	1	1	yes	characteristic, complemented
49	C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Hall
50	C₂²×C₃	C₂²×D₅	1	1	yes	characteristic
51	C₂×D₃	—	5	4	no	—
52	C₃×C₄	C₂²×D₅	1	6	yes	—
53	Dic₃	—	5	4	no	—
54	C₂×C₅	—	2	1	no	—
55	C₂×C₅	C₂³×D₃	1	1	yes	characteristic
56	D₅	—	6	4	no	—
57	Q₈	—	15	6	no	—
58	Q₈	C₂×D₁₅	1	4	yes	—
59	D₄	—	15	4	no	—
60	D₄	—	15	6	no	—
61	C₂×C₄	—	15	12	no	—
62	C₂×C₄	C₂×D₁₅	1	3	yes	—
63	C₂×C₃	—	2	1	no	—
64	C₂×C₃	C₂³×D₅	1	1	yes	characteristic
65	D₃	—	10	4	no	—
66	C₅	#1007	1	1	yes	characteristic, complemented, Sylow
67	C₂²	—	15	4	no	—
68	C₂²	C₂²×D₁₅	1	1	yes	characteristic
69	C₄	—	15	4	no	—
70	C₄	C₂²×D₁₅	1	6	yes	—
71	C₃	#4147	1	1	yes	characteristic, complemented, Sylow
72	C₂	—	2	1	no	—
73	C₂	—	30	4	no	—
74	C₂	C₂³×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₁₅
commutator, socle, cosocle	C₂×C₃×C₅	C₂⁴
Fitting	C₂×C₃×C₅×Q₈	C₂
hypercenter	C₂×Q₈	D₁₅
hypocenter	C₃×C₅	2₋¹⁺⁴
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×Q₈	C₂	semidirect
2. #6821	C₂	semidirect
3. Q₈×D₁₅	C₂	semidirect
4. #6817	C₂	semidirect
13. C₂×C₅×Q₈	D₃	semidirect
21. C₂×C₃×Q₈	D₅	semidirect
48. C₂×Q₈	D₁₅	semidirect
49. C₃×C₅	2₋¹⁺⁴	semidirect
66. C₅	#1007	semidirect
71. C₃	#4147	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×Q₈	2
#6821	2
Q₈×D₁₅	2
#6817	2
#4147	3
#1007	5