Subgroups — C₂³×C₅×#168 #13106934

Order 1600.

912 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₅²	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#168	C₂	1	14	yes	maximal, direct product factor
3	C₂⁴×C₅²	C₄	1	1	yes	characteristic
4	C₂²×C₄×C₅²	C₄	1	7	yes	—
5	C₂²×C₄×C₅²	C₂²	1	7	yes	—
6	C₂×C₅×#168	C₂²	1	28	yes	direct product factor
7	C₂³×C₅×C₈	—	5	1	no	maximal
8	C₂³×#168	C₅	1	1	yes	maximal, characteristic, direct product factor
9	C₂³×C₅²	C₈	1	8	yes	complemented
10	C₂³×C₅²	C₂×C₄	1	7	yes	—
11	C₂×C₄×C₅²	C₂×C₄	1	21	yes	—
12	C₂×C₄×C₅²	C₂³	1	7	yes	—
13	C₅×#168	C₂³	1	8	yes	direct product factor
14	C₂³×C₄×C₅	—	2	2	no	—
15	C₂³×C₄×C₅	D₅	1	1	yes	characteristic
16	C₂³×C₄×C₅	C₂×C₅	1	1	yes	characteristic
17	C₂²×C₅×C₈	—	5	14	no	—
18	C₂²×#168	C₂×C₅	1	14	yes	direct product factor
19	C₂²×C₅²	C₂×C₈	1	28	yes	complemented
20	C₂²×C₅²	C₂²×C₄	1	7	yes	—
21	C₄×C₅²	C₂²×C₄	1	7	yes	—
22	C₄×C₅²	C₂⁴	1	1	yes	characteristic
23	C₂⁴×C₅	—	2	2	no	—
24	C₂⁴×C₅	Dic₅	1	1	yes	characteristic
25	C₂⁴×C₅	C₄×C₅	1	1	yes	characteristic
26	C₂²×C₄×C₅	—	2	14	no	—
27	C₂²×C₄×C₅	—	2	14	no	—
28	C₂²×C₄×C₅	Dic₅	1	7	yes	—
29	C₂²×C₄×C₅	C₄×C₅	1	7	yes	—
30	C₂²×C₄×C₅	C₂×D₅	1	7	yes	—
31	C₂²×C₄×C₅	C₂²×C₅	1	7	yes	—
32	C₂×C₅×C₈	—	5	28	no	—
33	C₂×#168	C₂²×C₅	1	28	yes	direct product factor
34	C₂³×C₈	—	5	1	no	Sylow
35	C₂×C₅²	C₂²×C₈	1	14	yes	complemented
36	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
37	C₂³×C₅	—	2	16	no	—
38	C₂³×C₅	—	2	14	no	—
39	C₂³×C₅	#168	1	8	yes	direct product factor
40	C₂³×C₅	C₅×C₈	1	8	yes	complemented
41	C₂³×C₅	C₂×Dic₅	1	7	yes	—
42	C₂³×C₅	C₂×C₄×C₅	1	7	yes	—
43	C₂×C₄×C₅	—	2	42	no	—
44	C₂×C₄×C₅	—	2	14	no	—
45	C₂×C₄×C₅	C₂×Dic₅	1	21	yes	—
46	C₂×C₄×C₅	C₂×C₄×C₅	1	21	yes	—
47	C₂×C₄×C₅	C₂²×D₅	1	7	yes	—
48	C₂×C₄×C₅	C₂³×C₅	1	7	yes	—
49	C₅×C₈	—	5	8	no	—
50	#168	C₂³×C₅	1	8	yes	direct product factor
51	C₂³×C₄	C₅×D₅	1	1	yes	characteristic
52	C₂²×C₈	—	5	14	no	—
53	C₅²	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₅	—	2	14	no	—
55	C₂²×C₅	—	2	56	no	—
56	C₂²×C₅	C₂×#168	1	28	yes	direct product factor
57	C₂²×C₅	C₂×C₅×C₈	1	28	yes	complemented
58	C₂²×C₅	C₂²×Dic₅	1	7	yes	—
59	C₂²×C₅	C₂²×C₄×C₅	1	7	yes	—
60	C₄×C₅	—	2	14	no	—
61	C₄×C₅	—	2	2	no	—
62	C₄×C₅	C₂²×Dic₅	1	7	yes	—
63	C₄×C₅	C₂²×C₄×C₅	1	7	yes	—
64	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
65	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
66	C₂⁴	C₅×Dic₅	1	1	yes	characteristic
67	C₂²×C₄	C₅×Dic₅	1	7	yes	—
68	C₂²×C₄	C₂×C₅×D₅	1	7	yes	—
69	C₂×C₈	—	5	28	no	—
70	C₂×C₅	—	2	28	no	—
71	C₂×C₅	—	2	2	no	—
72	C₂×C₅	C₂²×#168	1	14	yes	direct product factor
73	C₂×C₅	C₂²×C₅×C₈	1	14	yes	complemented
74	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
75	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
76	C₂³	C₅×#168	1	8	yes	direct product factor
77	C₂³	C₂×C₅×Dic₅	1	7	yes	—
78	C₂×C₄	C₂×C₅×Dic₅	1	21	yes	—
79	C₂×C₄	C₂²×C₅×D₅	1	7	yes	—
80	C₈	—	5	8	no	—
81	C₅	—	2	2	no	—
82	C₅	C₂³×#168	1	1	yes	characteristic, direct product factor
83	C₅	C₂³×C₅×C₈	1	1	yes	characteristic, complemented
84	C₂²	C₂×C₅×#168	1	28	yes	direct product factor
85	C₂²	C₂²×C₅×Dic₅	1	7	yes	—
86	C₄	C₂²×C₅×Dic₅	1	7	yes	—
87	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
88	C₂	C₂²×C₅×#168	1	14	yes	direct product factor
89	C₂	C₂³×C₅×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄×C₅	D₅
commutator, hypocenter	C₅	C₂³×C₅×C₈
Frattini	C₄	C₂³×C₅×D₅
Fitting	C₂³×C₄×C₅²	C₂
socle, prime radical	C₂⁴×C₅²	C₄
cosocle	C₄×C₅	C₂⁴×C₅
LCS	G ▷ C₅
UCS	C₁ ◁ C₂³×C₄×C₅

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×#168	C₂	direct (N × H)
6. C₂×C₅×#168	C₂²	direct (N × H)
8. C₂³×#168	C₅	direct (N × H)
9. C₂³×C₅²	C₈	semidirect
13. C₅×#168	C₂³	direct (N × H)
18. C₂²×#168	C₂×C₅	direct (N × H)
19. C₂²×C₅²	C₂×C₈	semidirect
33. C₂×#168	C₂²×C₅	direct (N × H)
35. C₂×C₅²	C₂²×C₈	semidirect
39. C₂³×C₅	#168	direct (N × H)
40. C₂³×C₅	C₅×C₈	semidirect
50. #168	C₂³×C₅	direct (N × H)
53. C₅²	C₂³×C₈	semidirect
56. C₂²×C₅	C₂×#168	direct (N × H)
57. C₂²×C₅	C₂×C₅×C₈	semidirect
72. C₂×C₅	C₂²×#168	direct (N × H)
73. C₂×C₅	C₂²×C₅×C₈	semidirect
76. C₂³	C₅×#168	direct (N × H)
82. C₅	C₂³×#168	direct (N × H)
83. C₅	C₂³×C₅×C₈	semidirect
84. C₂²	C₂×C₅×#168	direct (N × H)
88. C₂	C₂²×C₅×#168	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₅²	2
C₂²×C₅×#168	2
C₂³×C₅×C₈	5
C₂³×#168	5

Subgroups — C23×C5×#168 #13106934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₅×#168 #13106934