Subgroups — C₄²×#674 #13025391

Order 1280.

558 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#674	C₂	1	6	yes	maximal
2	C₄²×#168	C₂	1	1	yes	maximal, characteristic
3	C₂²×#674	C₂²	1	4	yes	—
4	C₄³×C₅	C₄	1	1	yes	characteristic
5	C₂×C₄×#168	C₄	1	3	yes	—
6	C₂×C₄×#168	C₂²	1	3	yes	—
7	C₄×#674	C₄	1	24	yes	direct product factor
8	C₄²×C₁₆	—	5	1	no	maximal, Sylow
9	C₂×C₄²×C₅	C₈	1	4	yes	—
10	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
11	C₂²×#168	C₂×C₄	1	3	yes	—
12	C₂²×#168	C₂³	1	1	yes	characteristic
13	C₂×#674	C₂×C₄	1	24	yes	—
14	C₄×#168	C₈	1	12	yes	—
15	C₄×#168	C₂×C₄	1	12	yes	—
16	C₂×C₄×C₁₆	—	5	6	no	—
17	C₄²×C₈	—	5	1	no	—
18	C₂²×C₄×C₅	C₂×C₈	1	6	yes	—
19	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
20	C₄²×C₅	C₁₆	1	16	yes	complemented
21	C₄²×C₅	C₄²	1	6	yes	—
22	C₄²×C₅	C₂×C₈	1	6	yes	—
23	C₂×#168	C₄²	1	6	yes	—
24	C₂×#168	C₂×C₈	1	12	yes	—
25	C₂×#168	C₂²×C₄	1	6	yes	—
26	#674	C₄²	1	16	yes	direct product factor
27	C₂²×C₁₆	—	5	4	no	—
28	C₂×C₄×C₈	—	5	3	no	—
29	C₂×C₄×C₈	—	5	3	no	—
30	C₄³	AGL(1,𝔽₅)	1	1	yes	characteristic
31	C₄×C₁₆	—	5	24	no	—
32	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
33	C₂×C₄×C₅	C₄×C₈	1	12	yes	—
34	C₂×C₄×C₅	C₂×C₁₆	1	24	yes	—
35	C₂×C₄×C₅	C₂×C₄²	1	3	yes	—
36	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
37	#168	C₄×C₈	1	12	yes	—
38	#168	C₂×C₄²	1	4	yes	—
39	C₂²×C₈	—	5	3	no	—
40	C₂²×C₈	—	5	1	no	—
41	C₂×C₄²	#170	1	4	yes	—
42	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	3	yes	—
43	C₂×C₁₆	—	5	24	no	—
44	C₄×C₈	—	5	12	no	—
45	C₄×C₈	—	5	12	no	—
46	C₂²×C₅	C₂×C₄×C₈	1	3	yes	—
47	C₂²×C₅	C₂²×C₁₆	1	4	yes	—
48	C₄×C₅	C₄×C₁₆	1	24	yes	complemented
49	C₄×C₅	C₄³	1	1	yes	characteristic
50	C₄×C₅	C₂×C₄×C₈	1	3	yes	—
51	C₂²×C₄	C₂×#170	1	6	yes	—
52	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₂×C₈	—	5	12	no	—
54	C₂×C₈	—	5	6	no	—
55	C₂×C₈	—	5	6	no	—
56	C₄²	#674	1	16	yes	direct product factor
57	C₄²	D₅⋊C₈	1	6	yes	—
58	C₄²	C₄×AGL(1,𝔽₅)	1	6	yes	—
59	C₁₆	—	5	16	no	—
60	C₂×C₅	C₄²×C₈	1	1	yes	characteristic
61	C₂×C₅	C₂×C₄×C₁₆	1	6	yes	—
62	C₂³	C₂²×#170	1	1	yes	characteristic
63	C₂×C₄	C₂×#674	1	24	yes	—
64	C₂×C₄	C₄×#170	1	12	yes	—
65	C₂×C₄	C₂×D₅⋊C₈	1	3	yes	—
66	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	3	yes	—
67	C₈	—	5	12	no	—
68	C₈	—	5	4	no	—
69	C₅	C₄²×C₁₆	1	1	yes	characteristic, complemented, Sylow
70	C₂²	C₂²×#674	1	4	yes	—
71	C₂²	C₂×C₄×#170	1	3	yes	—
72	C₄	C₄×#674	1	24	yes	direct product factor
73	C₄	C₄×D₅⋊C₈	1	3	yes	—
74	C₄	C₄²×AGL(1,𝔽₅)	1	1	yes	characteristic
75	C₂	C₂×C₄×#674	1	6	yes	—
76	C₂	C₄²×#170	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄³	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄²×C₁₆
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₄³×C₅	C₄
socle, prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂²×#168	C₂³
LCS	G ▷ C₅
UCS	C₁ ◁ C₄³

Complements

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

N	H ≅ G/N	type
7. C₄×#674	C₄	direct (N × H)
20. C₄²×C₅	C₁₆	semidirect
26. #674	C₄²	direct (N × H)
48. C₄×C₅	C₄×C₁₆	semidirect
56. C₄²	#674	direct (N × H)
69. C₅	C₄²×C₁₆	semidirect
72. C₄	C₄×#674	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#674	2
C₄²×#168	2
C₄²×C₁₆	5

Subgroups — C42×#674 #13025391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#674 #13025391