Subgroups — C₂×#4268 #66734

Order 336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#4268	C₂	1	4	yes	maximal, direct product factor
4	C₂×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊D₇	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₇	—	2	2	no	—
7	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₇	C₂²	1	2	yes	complemented
9	C₂×D₂₁	—	2	2	no	—
10	C₂×D₂₁	C₂²	1	2	yes	complemented
11	Dic₂₁	C₂²	1	2	yes	complemented
12	C₂³×C₇	D₃	1	1	yes	characteristic, complemented
13	C₂²×D₇	—	3	1	no	—
14	C₂²⋊D₇	—	3	4	no	—
15	C₂×Dic₇	—	3	1	no	—
16	C₂×#67	—	7	1	no	maximal, Hall
17	C₂×C₃×C₇	—	2	2	no	—
18	C₂×C₃×C₇	D₄	1	2	yes	complemented
19	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
20	D₂₁	—	2	2	no	—
21	C₂²×C₇	—	2	2	no	—
22	C₂²×C₇	C₂×D₃	1	1	yes	characteristic
23	C₂²×C₇	C₂×D₃	1	2	yes	complemented
24	C₂×D₇	—	3	2	no	—
25	C₂×D₇	—	6	2	no	—
26	Dic₇	—	3	2	no	—
27	C₂³×C₃	D₇	1	1	yes	characteristic, complemented
28	C₂²×D₃	—	7	1	no	—
29	#67	—	7	4	no	—
30	C₂×Dic₃	—	7	1	no	—
31	C₃×C₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
32	C₂×D₄	—	21	1	no	Sylow
33	C₂×C₇	—	2	2	no	—
34	C₂×C₇	#67	1	2	yes	complemented
35	C₂×C₇	C₂²×D₃	1	1	yes	characteristic
36	D₇	—	6	2	no	—
37	C₂²×C₃	—	2	2	no	—
38	C₂²×C₃	C₂×D₇	1	1	yes	characteristic
39	C₂²×C₃	C₂×D₇	1	2	yes	complemented
40	C₂×D₃	—	7	2	no	—
41	C₂×D₃	—	14	2	no	—
42	Dic₃	—	7	2	no	—
43	C₂³	—	21	1	no	—
44	C₂³	D₂₁	1	1	yes	characteristic, complemented
45	D₄	—	21	4	no	—
46	C₂×C₄	—	21	1	no	—
47	C₇	C₂×#67	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₃	—	2	2	no	—
49	C₂×C₃	C₂²⋊D₇	1	2	yes	complemented
50	C₂×C₃	C₂²×D₇	1	1	yes	characteristic
51	D₃	—	14	2	no	—
52	C₂²	—	2	2	no	—
53	C₂²	—	21	2	no	—
54	C₂²	—	42	2	no	—
55	C₂²	C₂×D₂₁	1	1	yes	characteristic
56	C₂²	C₂×D₂₁	1	2	yes	complemented
57	C₄	—	21	2	no	—
58	C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic, complemented, Sylow
59	C₂	—	2	2	no	—
60	C₂	—	42	2	no	—
61	C₂	#4268	1	2	yes	direct product factor
62	C₂	C₂²×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₁
commutator, cosocle	C₂×C₃×C₇	C₂³
Frattini	C₂	C₂²×D₂₁
Fitting	C₂³×C₃×C₇	C₂
socle	C₂²×C₃×C₇	C₂²
hypercenter	C₂³	D₂₁
hypocenter	C₃×C₇	C₂×D₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₇	C₂	semidirect
2. C₂²×D₂₁	C₂	semidirect
3. #4268	C₂	direct (N × H)
4. C₂×Dic₂₁	C₂	semidirect
8. C₂²×C₃×C₇	C₂²	semidirect
10. C₂×D₂₁	C₂²	semidirect
11. Dic₂₁	C₂²	semidirect
12. C₂³×C₇	D₃	semidirect
18. C₂×C₃×C₇	D₄	semidirect
23. C₂²×C₇	C₂×D₃	semidirect
27. C₂³×C₃	D₇	semidirect
31. C₃×C₇	C₂×D₄	semidirect
34. C₂×C₇	#67	semidirect
39. C₂²×C₃	C₂×D₇	semidirect
44. C₂³	D₂₁	semidirect
47. C₇	C₂×#67	semidirect
49. C₂×C₃	C₂²⋊D₇	semidirect
56. C₂²	C₂×D₂₁	semidirect
58. C₃	C₂×C₂²⋊D₇	semidirect
61. C₂	#4268	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×C₃×C₇	2
C₂²×D₂₁	2
#4268	2
C₂×Dic₂₁	2
C₂×C₂²⋊D₇	3
C₂×#67	7

Subgroups — C2×#4268 #66734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4268 #66734