Subgroups — C₇×#217 #66620

Order 336.

68 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
2	C₂×C₇×Dic₃	C₂	1	2	yes	maximal, complemented
3	C₇×C₂²⋊C₄	—	3	1	no	maximal, Hall
4	C₂²×C₃×C₇	—	2	2	no	—
5	C₂²×C₃×C₇	C₄	1	2	yes	complemented
6	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
7	C₇×Dic₃	—	2	2	no	—
8	C₂³×C₇	D₃	1	1	yes	characteristic
9	C₂×C₄×C₇	—	3	2	no	—
10	#217	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₇	—	2	2	no	—
12	C₂×C₃×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃×C₇	D₄	1	2	yes	—
14	C₂²×C₇	—	2	2	no	—
15	C₂²×C₇	Dic₃	1	2	yes	complemented
16	C₂²×C₇	C₂×D₃	1	1	yes	characteristic
17	C₄×C₇	—	6	2	no	—
18	C₂³×C₃	C₂×C₇	1	1	yes	characteristic
19	C₂×Dic₃	C₂×C₇	1	2	yes	complemented
20	C₃×C₇	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
21	C₂²⋊C₄	—	3	1	no	Sylow
22	C₂×C₇	—	2	2	no	—
23	C₂×C₇	C₂×Dic₃	1	1	yes	characteristic
24	C₂×C₇	#67	1	2	yes	—
25	C₂²×C₃	—	2	2	no	—
26	C₂²×C₃	C₄×C₇	1	2	yes	complemented
27	C₂²×C₃	C₂²×C₇	1	1	yes	characteristic
28	Dic₃	—	2	2	no	—
29	C₂³	D₃×C₇	1	1	yes	characteristic
30	C₂×C₄	—	3	2	no	—
31	C₇	#217	1	1	yes	characteristic, direct product factor, Sylow
32	C₂×C₃	—	2	2	no	—
33	C₂×C₃	C₂×C₄×C₇	1	1	yes	characteristic
34	C₂×C₃	C₇×D₄	1	2	yes	—
35	C₂²	—	2	2	no	—
36	C₂²	C₇×Dic₃	1	2	yes	complemented
37	C₂²	C₂×D₃×C₇	1	1	yes	characteristic
38	C₄	—	6	2	no	—
39	C₃	C₇×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
40	C₂	—	2	2	no	—
41	C₂	C₂×C₇×Dic₃	1	1	yes	characteristic
42	C₂	C₇×#67	1	2	yes	—

	subgroup	G/H
center	C₂²×C₇	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₇
Frattini	C₂²	C₂×D₃×C₇
Fitting, prime radical	C₂³×C₃×C₇	C₂
socle	C₂²×C₃×C₇	C₂²
cosocle	C₂²×C₃	C₂²×C₇
hypercenter	C₂³×C₇	D₃
hypocenter	C₃	C₇×C₂²⋊C₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×Dic₃	C₂	semidirect
5. C₂²×C₃×C₇	C₄	semidirect
10. #217	C₇	direct (N × H)
15. C₂²×C₇	Dic₃	semidirect
19. C₂×Dic₃	C₂×C₇	semidirect
20. C₃×C₇	C₂²⋊C₄	semidirect
26. C₂²×C₃	C₄×C₇	semidirect
31. C₇	#217	direct (N × H)
36. C₂²	C₇×Dic₃	semidirect
39. C₃	C₇×C₂²⋊C₄	semidirect

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
C₂×C₇×Dic₃	2
C₇×C₂²⋊C₄	3
#217	7

Subgroups — C7×#217 #66620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#217 #66620