Subgroups — #91104

Order 480.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#3952	—	3	1	no	maximal, Hall
5	C₂²×D₁₅	C₄	1	1	yes	characteristic
6	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₅	C₄	1	1	yes	characteristic
8	C₄×D₁₅	—	2	1	no	—
9	C₄×D₁₅	—	2	1	no	—
10	C₃×C₅×C₈	—	2	1	no	—
11	C₃⋊#168	—	2	1	no	—
12	(C₂×D₃)⋊C₈	—	5	1	no	maximal, Hall
13	C₂×C₄×D₅	—	3	1	no	—
14	C₂×C₅×C₈	D₃	1	1	yes	characteristic, complemented
15	C₂×#168	—	3	1	no	—
16	C₂²×C₃×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×D₁₅	—	2	1	no	—
18	C₂×D₁₅	—	2	1	no	—
19	C₂×D₁₅	C₈	1	2	yes	complemented
20	C₃×C₄×C₅	D₄	1	1	yes	characteristic
21	C₃×C₄×C₅	D₄	1	1	yes	characteristic
22	Dic₁₅	—	2	1	no	—
23	C₂×C₄×D₃	—	5	1	no	—
24	C₂×C₃×C₈	D₅	1	1	yes	characteristic, complemented
25	C₂×C₃⋊C₈	—	5	1	no	—
26	C₂²×D₅	—	3	1	no	—
27	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
28	C₂×Dic₅	—	3	1	no	—
29	C₄×D₅	—	6	1	no	—
30	C₄×D₅	—	6	1	no	—
31	C₅×C₈	—	2	1	no	—
32	#168	—	6	1	no	—
33	C₂²⋊C₈	—	15	1	no	Sylow
34	C₂×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₃×C₅	C₂×C₈	1	1	yes	characteristic
36	C₂×C₃×C₅	M₁₆	1	1	yes	characteristic
37	D₁₅	—	2	2	no	—
38	C₂²×D₃	—	5	1	no	—
39	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
40	C₂×Dic₃	—	5	1	no	—
41	C₄×D₃	—	10	1	no	—
42	C₄×D₃	—	10	1	no	—
43	C₃×C₈	—	2	1	no	—
44	C₃⋊C₈	—	10	1	no	—
45	C₂²×C₅	C₄×D₃	1	1	yes	characteristic
46	C₂×D₅	—	3	2	no	—
47	C₂×D₅	—	6	1	no	—
48	C₂×D₅	—	6	1	no	—
49	C₄×C₅	D₁₂	1	1	yes	characteristic
50	C₄×C₅	#67	1	1	yes	characteristic
51	Dic₅	—	6	1	no	—
52	C₂²×C₄	—	15	1	no	—
53	C₂×C₈	—	15	1	no	—
54	C₂×C₈	D₁₅	1	1	yes	characteristic, complemented
55	C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic, complemented, Hall
56	C₂²×C₃	C₄×D₅	1	1	yes	characteristic
57	C₂×D₃	—	5	2	no	—
58	C₂×D₃	—	10	1	no	—
59	C₂×D₃	—	10	1	no	—
60	C₃×C₄	D₂₀	1	1	yes	characteristic
61	C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
62	Dic₃	—	10	1	no	—
63	C₂×C₅	D₃×C₈	1	1	yes	characteristic
64	C₂×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
65	C₂×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
66	D₅	—	6	2	no	—
67	C₂³	—	15	1	no	—
68	C₂×C₄	—	15	1	no	—
69	C₂×C₄	—	30	1	no	—
70	C₂×C₄	—	30	1	no	—
71	C₂×C₄	C₂×D₁₅	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	30	1	no	—
74	C₂×C₃	C₈×D₅	1	1	yes	characteristic
75	C₂×C₃	#676	1	1	yes	characteristic
76	C₂×C₃	#685	1	1	yes	characteristic
77	D₃	—	10	2	no	—
78	C₅	(C₂×D₃)⋊C₈	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	15	2	no	—
80	C₂²	—	30	1	no	—
81	C₂²	—	30	1	no	—
82	C₂²	C₄×D₁₅	1	1	yes	characteristic
83	C₄	—	30	1	no	—
84	C₄	D₆₀	1	1	yes	characteristic
85	C₄	#1220	1	1	yes	characteristic
86	C₃	#3952	1	1	yes	characteristic, complemented, Sylow
87	C₂	—	30	2	no	—
88	C₂	C₈×D₁₅	1	1	yes	characteristic
89	C₂	#6705	1	1	yes	characteristic
90	C₂	#6714	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×C₈	C₂	semidirect
3. C₂×C₃⋊#168	C₂	semidirect
14. C₂×C₅×C₈	D₃	semidirect
19. C₂×D₁₅	C₈	semidirect
24. C₂×C₃×C₈	D₅	semidirect
54. C₂×C₈	D₁₅	semidirect
55. C₃×C₅	C₂²⋊C₈	semidirect
78. C₅	(C₂×D₃)⋊C₈	semidirect
86. C₃	#3952	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₅	2
C₂×C₃×C₅×C₈	2
C₂×C₃⋊#168	2
#3952	3
(C₂×D₃)⋊C₈	5