Subgroups — C₅×#888 #91691

Order 480.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×Dic₆	C₂	1	2	yes	maximal
2	C₃×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₅×Dic₃⋊C₄	C₂	1	2	yes	maximal
5	C₄×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
6	C₅×C₄⋊Q₈	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄×C₅	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×Dic₃	C₂²	1	2	yes	—
10	C₂×C₅×Dic₃	C₂²	1	1	yes	characteristic
11	C₂×C₅×Dic₃	C₂²	1	1	yes	characteristic
12	C₅×Dic₆	—	2	4	no	—
13	#888	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₅×Q₈	—	3	2	no	—
15	C₅×C₄⋊C₄	—	3	2	no	—
16	C₅×C₄⋊C₄	—	3	1	no	—
17	C₅×C₄⋊C₄	D₃	1	1	yes	characteristic
18	C₄²×C₅	—	3	1	no	—
19	C₂²×C₃×C₅	C₂³	1	1	yes	characteristic
20	C₃×C₄×C₅	—	2	2	no	—
21	C₃×C₄×C₅	Q₈	1	2	yes	complemented
22	C₅×Dic₃	—	2	2	no	—
23	C₅×Dic₃	D₄	1	2	yes	—
24	C₅×Dic₃	Q₈	1	2	yes	complemented
25	C₂×Dic₆	C₂×C₅	1	2	yes	—
26	C₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
27	C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic
28	Dic₃⋊C₄	C₂×C₅	1	2	yes	—
29	C₄×Dic₃	C₂×C₅	1	1	yes	characteristic
30	C₅×Q₈	—	6	4	no	—
31	C₂×C₄×C₅	—	3	1	no	—
32	C₂×C₄×C₅	—	3	1	no	—
33	C₂×C₄×C₅	—	3	2	no	—
34	C₂×C₄×C₅	C₂×D₃	1	2	yes	—
35	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
36	C₄⋊Q₈	—	3	1	no	Sylow
37	C₂×C₃×C₅	C₂×D₄	1	1	yes	characteristic
38	C₂×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₃×C₄	C₂²×C₅	1	2	yes	—
41	C₂×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
42	C₂×Dic₃	C₂²×C₅	1	2	yes	—
43	C₂×Dic₃	C₂²×C₅	1	1	yes	characteristic
44	C₂×Dic₃	C₂²×C₅	1	1	yes	characteristic
45	Dic₆	—	2	4	no	—
46	C₂²×C₅	C₂²×D₃	1	1	yes	characteristic
47	C₄×C₅	—	2	2	no	—
48	C₄×C₅	—	3	2	no	—
49	C₄×C₅	—	3	2	no	—
50	C₄×C₅	—	6	2	no	—
51	C₄×C₅	Dic₆	1	2	yes	complemented
52	C₂×Q₈	—	3	2	no	—
53	C₄⋊C₄	—	3	2	no	—
54	C₄⋊C₄	—	3	1	no	—
55	C₄⋊C₄	C₅×D₃	1	1	yes	characteristic
56	C₄²	—	3	1	no	—
57	C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic, complemented, Hall
58	C₂²×C₃	C₂³×C₅	1	1	yes	characteristic
59	C₃×C₄	—	2	2	no	—
60	C₃×C₄	C₅×Q₈	1	2	yes	complemented
61	Dic₃	—	2	2	no	—
62	Dic₃	C₅×D₄	1	2	yes	—
63	Dic₃	C₅×Q₈	1	2	yes	complemented
64	C₂×C₅	C₂×Dic₆	1	1	yes	characteristic
65	C₂×C₅	D₃×D₄	1	1	yes	characteristic
66	C₂×C₅	D₃×Q₈	1	1	yes	characteristic
67	Q₈	—	6	4	no	—
68	C₂×C₄	—	3	1	no	—
69	C₂×C₄	—	3	1	no	—
70	C₂×C₄	—	3	2	no	—
71	C₂×C₄	C₂×C₅×D₃	1	2	yes	—
72	C₂×C₄	C₂×C₅×D₃	1	1	yes	characteristic
73	C₂×C₃	C₂×C₅×D₄	1	1	yes	characteristic
74	C₂×C₃	C₂×C₅×Q₈	1	1	yes	characteristic
75	C₂×C₃	C₂×C₅×Q₈	1	1	yes	characteristic
76	C₅	#888	1	1	yes	characteristic, direct product factor, Sylow
77	C₂²	C₂²×C₅×D₃	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	3	2	no	—
80	C₄	—	3	2	no	—
81	C₄	—	6	2	no	—
82	C₄	C₅×Dic₆	1	2	yes	complemented
83	C₃	C₅×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
84	C₂	C₂×C₅×Dic₆	1	1	yes	characteristic
85	C₂	C₅×D₃×D₄	1	1	yes	characteristic
86	C₂	C₅×D₃×Q₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂³×C₅
Frattini	C₂²	C₂²×C₅×D₃
Fitting	C₃×C₅×C₄⋊C₄	C₂
socle, prime radical	C₂²×C₃×C₅	C₂³
hypercenter	C₅×C₄⋊C₄	D₃
hypocenter	C₃	C₅×C₄⋊Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
13. #888	C₅	direct (N × H)
21. C₃×C₄×C₅	Q₈	semidirect
24. C₅×Dic₃	Q₈	semidirect
51. C₄×C₅	Dic₆	semidirect
57. C₃×C₅	C₄⋊Q₈	semidirect
60. C₃×C₄	C₅×Q₈	semidirect
63. Dic₃	C₅×Q₈	semidirect
76. C₅	#888	direct (N × H)
82. C₄	C₅×Dic₆	semidirect
83. C₃	C₅×C₄⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×Dic₆	2
C₃×C₅×C₄⋊C₄	2
C₅×C₄⋊Dic₃	2
C₅×Dic₃⋊C₄	2
C₄×C₅×Dic₃	2
C₅×C₄⋊Q₈	3
#888	5

Subgroups — C5×#888 #91691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#888 #91691