Subgroups — C₇×#213 #66616

Order 336.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₈	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₇	—	2	1	no	—
6	C₂×D₃×C₇	—	2	1	no	—
7	C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₇×D₄	—	3	1	no	—
9	C₇×D₄	D₃	1	1	yes	characteristic, complemented
10	C₇×C₈	—	3	1	no	—
11	#213	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₃×C₇	—	4	1	no	—
13	C₂×C₃×C₇	D₄	1	1	yes	characteristic
14	D₃×C₇	—	4	1	no	—
15	C₂²×C₇	—	2	1	no	—
16	C₂²×C₇	—	6	1	no	—
17	C₄×C₇	C₂×D₃	1	1	yes	characteristic
18	C₃×D₄	C₂×C₇	1	1	yes	characteristic, complemented
19	D₁₂	C₂×C₇	1	1	yes	characteristic, complemented
20	C₃⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
21	C₃×C₇	D₈	1	1	yes	characteristic, complemented, Hall
22	D₈	—	3	1	no	Sylow
23	C₂×C₇	—	4	1	no	—
24	C₂×C₇	—	12	1	no	—
25	C₂×C₇	#67	1	1	yes	characteristic
26	C₂²×C₃	—	2	1	no	—
27	C₂×D₃	—	2	1	no	—
28	C₃×C₄	C₂²×C₇	1	1	yes	characteristic
29	D₄	—	3	1	no	—
30	D₄	D₃×C₇	1	1	yes	characteristic, complemented
31	C₈	—	3	1	no	—
32	C₇	#213	1	1	yes	characteristic, direct product factor, Sylow
33	C₂×C₃	—	4	1	no	—
34	C₂×C₃	C₇×D₄	1	1	yes	characteristic
35	D₃	—	4	1	no	—
36	C₂²	—	2	1	no	—
37	C₂²	—	6	1	no	—
38	C₄	C₂×D₃×C₇	1	1	yes	characteristic
39	C₃	C₇×D₈	1	1	yes	characteristic, complemented, Sylow
40	C₂	—	4	1	no	—
41	C₂	—	12	1	no	—
42	C₂	C₇×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇	#67
commutator, cosocle	C₃×C₄	C₂²×C₇
Frattini	C₄	C₂×D₃×C₇
Fitting	C₃×C₇×D₄	C₂
socle	C₂×C₃×C₇	D₄
hypercenter	C₇×D₄	D₃
hypocenter	C₃	C₇×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×D₄

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×D₄	C₂	semidirect
2. C₇×D₁₂	C₂	semidirect
3. C₇×C₃⋊C₈	C₂	semidirect
9. C₇×D₄	D₃	semidirect
11. #213	C₇	direct (N × H)
18. C₃×D₄	C₂×C₇	semidirect
19. D₁₂	C₂×C₇	semidirect
20. C₃⋊C₈	C₂×C₇	semidirect
21. C₃×C₇	D₈	semidirect
30. D₄	D₃×C₇	semidirect
32. C₇	#213	direct (N × H)
39. C₃	C₇×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×D₄	2
C₇×D₁₂	2
C₇×C₃⋊C₈	2
C₇×D₈	3
#213	7

Subgroups — C7×#213 #66616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#213 #66616