Subgroups — C₂³×#704 #10619418

Order 640.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×#704	C₂	1	28	yes	maximal, direct product factor
3	C₂³×#170	C₂	1	2	yes	maximal, complemented
4	C₂⁵×C₅	C₄	1	1	yes	characteristic
5	C₂³×Dic₅	C₄	1	14	yes	—
6	C₂³×Dic₅	C₄	1	1	yes	characteristic
7	C₂³×Dic₅	C₂²	1	1	yes	characteristic
8	C₂³×Dic₅	C₂²	1	14	yes	—
9	C₂×#704	C₂²	1	112	yes	direct product factor
10	C₂²×#170	C₂²	1	28	yes	complemented
11	C₂³×M₁₆	—	5	1	no	maximal, Sylow
12	C₂⁴×C₅	—	2	8	no	—
13	C₂⁴×C₅	C₂×C₄	1	14	yes	—
14	C₂⁴×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₅	C₂×C₄	1	84	yes	—
16	C₂²×Dic₅	C₂×C₄	1	7	yes	—
17	C₂²×Dic₅	C₂×C₄	1	14	yes	—
18	C₂²×Dic₅	C₂³	1	7	yes	—
19	C₂²×Dic₅	C₂³	1	28	yes	—
20	#704	C₂³	1	64	yes	direct product factor
21	C₂×#170	C₂³	1	56	yes	complemented
22	C₂⁴×C₄	—	5	1	no	—
23	C₂²×M₁₆	—	5	28	no	—
24	C₂³×C₈	—	5	2	no	—
25	C₂³×C₅	—	2	56	no	—
26	C₂³×C₅	M₁₆	1	8	yes	complemented
27	C₂³×C₅	C₂²×C₄	1	28	yes	—
28	C₂³×C₅	C₂²×C₄	1	7	yes	—
29	C₂×Dic₅	C₂²×C₄	1	56	yes	—
30	C₂×Dic₅	C₂²×C₄	1	21	yes	—
31	C₂×Dic₅	C₂²×C₄	1	28	yes	—
32	C₂×Dic₅	C₂⁴	1	7	yes	—
33	C₂×Dic₅	C₂⁴	1	8	yes	—
34	#170	C₂⁴	1	16	yes	complemented
35	C₂⁵	AGL(1,𝔽₅)	1	1	yes	characteristic
36	C₂³×C₄	—	5	14	no	—
37	C₂³×C₄	—	5	14	no	—
38	C₂³×C₄	—	5	1	no	—
39	C₂³×C₄	—	5	1	no	—
40	C₂×M₁₆	—	5	112	no	—
41	C₂²×C₈	—	5	28	no	—
42	C₂²×C₅	—	2	56	no	—
43	C₂²×C₅	C₂×M₁₆	1	28	yes	complemented
44	C₂²×C₅	C₂³×C₄	1	8	yes	—
45	C₂²×C₅	C₂³×C₄	1	7	yes	—
46	Dic₅	C₂³×C₄	1	7	yes	—
47	Dic₅	C₂³×C₄	1	8	yes	—
48	Dic₅	C₂⁵	1	1	yes	characteristic
49	C₂⁴	—	2	8	no	—
50	C₂⁴	C₂×AGL(1,𝔽₅)	1	14	yes	—
51	C₂⁴	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₂²×C₄	—	5	7	no	—
53	C₂²×C₄	—	5	28	no	—
54	C₂²×C₄	—	5	84	no	—
55	C₂²×C₄	—	5	14	no	—
56	C₂²×C₄	—	5	7	no	—
57	M₁₆	—	5	64	no	—
58	C₂×C₈	—	5	56	no	—
59	C₂×C₅	—	2	8	no	—
60	C₂×C₅	C₂²×M₁₆	1	14	yes	complemented
61	C₂×C₅	C₂⁴×C₄	1	1	yes	characteristic
62	C₂³	—	2	56	no	—
63	C₂³	#704	1	8	yes	direct product factor
64	C₂³	C₂²×AGL(1,𝔽₅)	1	28	yes	—
65	C₂³	C₂²×AGL(1,𝔽₅)	1	7	yes	—
66	C₂×C₄	—	5	7	no	—
67	C₂×C₄	—	5	8	no	—
68	C₂×C₄	—	5	56	no	—
69	C₂×C₄	—	5	21	no	—
70	C₂×C₄	—	5	28	no	—
71	C₈	—	5	16	no	—
72	C₅	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	2	56	no	—
74	C₂²	C₂×#704	1	28	yes	direct product factor
75	C₂²	C₂³×AGL(1,𝔽₅)	1	8	yes	—
76	C₂²	C₂³×AGL(1,𝔽₅)	1	7	yes	—
77	C₄	—	5	1	no	—
78	C₄	—	5	7	no	—
79	C₄	—	5	8	no	—
80	C₂	—	2	8	no	—
81	C₂	C₂²×#704	1	14	yes	direct product factor
82	C₂	C₂⁴×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂⁴×C₄
Frattini	C₂	C₂⁴×AGL(1,𝔽₅)
Fitting, prime radical	C₂⁵×C₅	C₄
socle	C₂⁴×C₅	C₂×C₄
cosocle	Dic₅	C₂⁵
hypercenter	C₂⁵	AGL(1,𝔽₅)
hypocenter	C₅	C₂³×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
2. C₂²×#704	C₂	direct (N × H)
3. C₂³×#170	C₂	semidirect
9. C₂×#704	C₂²	direct (N × H)
10. C₂²×#170	C₂²	semidirect
20. #704	C₂³	direct (N × H)
21. C₂×#170	C₂³	semidirect
26. C₂³×C₅	M₁₆	semidirect
34. #170	C₂⁴	semidirect
43. C₂²×C₅	C₂×M₁₆	semidirect
60. C₂×C₅	C₂²×M₁₆	semidirect
63. C₂³	#704	direct (N × H)
72. C₅	C₂³×M₁₆	semidirect
74. C₂²	C₂×#704	direct (N × H)
81. C₂	C₂²×#704	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₅	2
C₂²×#704	2
C₂³×#170	2
C₂³×M₁₆	5

Subgroups — C23×#704 #10619418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#704 #10619418