Subgroups — C₃×#1157 #66708

Order 336.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₂²⋊D₇	C₂	1	2	yes	maximal, complemented
3	C₃×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×D₇	C₂	1	2	yes	maximal, complemented
5	C₃×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#1157	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₃×D₇	C₂²	1	2	yes	—
9	C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₃×Dic₇	C₂²	1	2	yes	—
12	C₂×C₄×C₇	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂²⋊D₇	C₂×C₃	1	2	yes	complemented
14	D₂₈	C₂×C₃	1	1	yes	characteristic, complemented
15	C₄×D₇	C₂×C₃	1	2	yes	complemented
16	Dic₁₄	C₂×C₃	1	1	yes	characteristic, complemented
17	C₃×#41	—	7	1	no	maximal, Hall
18	C₂×C₃×C₇	—	2	1	no	—
19	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
20	C₃×D₇	—	2	2	no	—
21	C₂²×C₇	C₂²×C₃	1	1	yes	characteristic
22	C₂×D₇	C₂²×C₃	1	2	yes	—
23	C₄×C₇	C₂²×C₃	1	1	yes	characteristic
24	C₄×C₇	C₂²×C₃	1	1	yes	characteristic
25	Dic₇	C₂²×C₃	1	2	yes	—
26	C₃×Q₈	—	7	1	no	—
27	C₃×D₄	—	7	2	no	—
28	C₃×D₄	—	7	1	no	—
29	C₂×C₃×C₄	—	7	2	no	—
30	C₂×C₃×C₄	D₇	1	1	yes	characteristic, complemented
31	C₃×C₇	#41	1	1	yes	characteristic, complemented, Hall
32	#41	—	7	1	no	Sylow
33	C₂×C₇	—	2	1	no	—
34	C₂×C₇	C₂³×C₃	1	1	yes	characteristic
35	D₇	—	2	2	no	—
36	C₂²×C₃	—	7	2	no	—
37	C₂²×C₃	C₂×D₇	1	1	yes	characteristic
38	C₃×C₄	—	7	2	no	—
39	C₃×C₄	C₂×D₇	1	1	yes	characteristic
40	C₃×C₄	C₂×D₇	1	1	yes	characteristic
41	Q₈	—	7	1	no	—
42	D₄	—	7	2	no	—
43	D₄	—	7	1	no	—
44	C₂×C₄	—	7	2	no	—
45	C₂×C₄	C₃×D₇	1	1	yes	characteristic, complemented
46	C₇	C₃×#41	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃	—	2	1	no	—
48	C₂×C₃	—	14	2	no	—
49	C₂×C₃	C₂²×D₇	1	1	yes	characteristic
50	C₂²	—	7	2	no	—
51	C₂²	C₂×C₃×D₇	1	1	yes	characteristic
52	C₄	—	7	2	no	—
53	C₄	C₂×C₃×D₇	1	1	yes	characteristic
54	C₄	C₂×C₃×D₇	1	1	yes	characteristic
55	C₃	#1157	1	1	yes	characteristic, direct product factor, Sylow
56	C₂	—	2	1	no	—
57	C₂	—	14	2	no	—
58	C₂	C₂²×C₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂×D₇
commutator, cosocle	C₂×C₇	C₂³×C₃
Frattini	C₂	C₂²×C₃×D₇
Fitting	C₂×C₃×C₄×C₇	C₂
socle	C₂×C₃×C₇	C₂³
hypercenter	C₂×C₃×C₄	D₇
hypocenter	C₇	C₃×#41
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₇	C₂	semidirect
2. C₃×C₂²⋊D₇	C₂	semidirect
3. C₃×D₂₈	C₂	semidirect
4. C₃×C₄×D₇	C₂	semidirect
5. C₃×Dic₁₄	C₂	semidirect
6. #1157	C₃	direct (N × H)
12. C₂×C₄×C₇	C₂×C₃	semidirect
13. C₂²⋊D₇	C₂×C₃	semidirect
14. D₂₈	C₂×C₃	semidirect
15. C₄×D₇	C₂×C₃	semidirect
16. Dic₁₄	C₂×C₃	semidirect
30. C₂×C₃×C₄	D₇	semidirect
31. C₃×C₇	#41	semidirect
45. C₂×C₄	C₃×D₇	semidirect
46. C₇	C₃×#41	semidirect
55. C₃	#1157	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×C₃×C₄×C₇	2
C₃×C₂²⋊D₇	2
C₃×D₂₈	2
C₃×C₄×D₇	2
C₃×Dic₁₄	2
#1157	3
C₃×#41	7

Subgroups — C3×#1157 #66708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1157 #66708