Subgroups — C₂³×#6034 #13107776

Order 1600.

912 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#1039	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6034	C₂	1	14	yes	maximal, direct product factor
3	C₂⁴×C₅²	C₄	1	1	yes	characteristic
4	C₂²×#1039	C₄	1	7	yes	—
5	C₂²×#1039	C₂²	1	7	yes	—
6	C₂×#6034	C₂²	1	28	yes	direct product factor
7	C₂³×#170	—	5	2	no	maximal
8	C₂³×C₅²	C₈	1	8	yes	complemented
9	C₂³×C₅²	C₂×C₄	1	7	yes	—
10	C₂×#1039	C₂×C₄	1	21	yes	—
11	C₂×#1039	C₂³	1	7	yes	—
12	#6034	C₂³	1	8	yes	direct product factor
13	C₂³×Dic₅	—	5	2	no	—
14	C₂³×Dic₅	—	10	2	no	—
15	C₂²×#170	—	5	28	no	—
16	C₂²×C₅²	C₂×C₈	1	28	yes	complemented
17	C₂²×C₅²	C₂²×C₄	1	7	yes	—
18	#1039	C₂²×C₄	1	7	yes	—
19	#1039	C₂⁴	1	1	yes	characteristic
20	C₂⁴×C₅	—	2	2	no	—
21	C₂⁴×C₅	AGL(1,𝔽₅)	1	2	yes	—
22	C₂²×Dic₅	—	5	14	no	—
23	C₂²×Dic₅	—	5	14	no	—
24	C₂²×Dic₅	—	10	14	no	—
25	C₂²×Dic₅	—	10	14	no	—
26	C₂×#170	—	5	56	no	—
27	C₂³×C₈	—	25	1	no	Sylow
28	C₂×C₅²	C₂²×C₈	1	14	yes	complemented
29	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
30	C₂³×C₅	—	2	16	no	—
31	C₂³×C₅	—	2	14	no	—
32	C₂³×C₅	#170	1	16	yes	complemented
33	C₂³×C₅	C₂×AGL(1,𝔽₅)	1	14	yes	—
34	C₂×Dic₅	—	5	42	no	—
35	C₂×Dic₅	—	5	14	no	—
36	C₂×Dic₅	—	10	42	no	—
37	C₂×Dic₅	—	10	14	no	—
38	#170	—	5	16	no	—
39	C₂³×C₄	—	25	1	no	—
40	C₂²×C₈	—	25	14	no	—
41	C₅²	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
42	C₂²×C₅	—	2	56	no	—
43	C₂²×C₅	—	2	14	no	—
44	C₂²×C₅	C₂×#170	1	56	yes	complemented
45	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	14	yes	—
46	Dic₅	—	5	14	no	—
47	Dic₅	—	5	2	no	—
48	Dic₅	—	10	14	no	—
49	Dic₅	—	10	2	no	—
50	C₂⁴	#1044	1	1	yes	characteristic
51	C₂²×C₄	—	25	7	no	—
52	C₂²×C₄	—	25	7	no	—
53	C₂×C₈	—	25	28	no	—
54	C₂×C₅	—	2	28	no	—
55	C₂×C₅	—	2	2	no	—
56	C₂×C₅	C₂²×#170	1	28	yes	complemented
57	C₂×C₅	C₂³×AGL(1,𝔽₅)	1	2	yes	—
58	C₂³	#6034	1	8	yes	direct product factor
59	C₂³	C₂×#1044	1	7	yes	—
60	C₂×C₄	—	25	7	no	—
61	C₂×C₄	—	25	21	no	—
62	C₈	—	25	8	no	—
63	C₅	—	2	2	no	—
64	C₅	C₂³×#170	1	2	yes	complemented
65	C₂²	C₂×#6034	1	28	yes	direct product factor
66	C₂²	C₂²×#1044	1	7	yes	—
67	C₄	—	25	1	no	—
68	C₄	—	25	7	no	—
69	C₂	C₂²×#6034	1	14	yes	direct product factor
70	C₂	C₂³×#1044	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴	#1044
commutator, hypocenter	C₅²	C₂³×C₈
Frattini	C₂	C₂³×#1044
Fitting, socle, prime radical	C₂⁴×C₅²	C₄
cosocle	#1039	C₂⁴
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
2. C₂²×#6034	C₂	direct (N × H)
6. C₂×#6034	C₂²	direct (N × H)
8. C₂³×C₅²	C₈	semidirect
12. #6034	C₂³	direct (N × H)
16. C₂²×C₅²	C₂×C₈	semidirect
28. C₂×C₅²	C₂²×C₈	semidirect
32. C₂³×C₅	#170	semidirect
41. C₅²	C₂³×C₈	semidirect
44. C₂²×C₅	C₂×#170	semidirect
56. C₂×C₅	C₂²×#170	semidirect
58. C₂³	#6034	direct (N × H)
64. C₅	C₂³×#170	semidirect
65. C₂²	C₂×#6034	direct (N × H)
69. C₂	C₂²×#6034	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×#1039	2
C₂²×#6034	2
C₂³×#170	5

Subgroups — C23×#6034 #13107776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6034 #13107776