Subgroups — C₇×#237 #66720

Order 336.

80 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×#67	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×Dic₃	C₂	1	2	yes	maximal, complemented
4	C₄×D₃×C₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#41	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₇	C₂²	1	2	yes	—
8	C₂×D₃×C₇	C₂²	1	1	yes	characteristic
9	C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₇×Dic₃	C₂²	1	2	yes	—
11	C₇×Dic₃	C₂²	1	1	yes	characteristic
12	C₇×Q₈	—	3	1	no	—
13	C₇×D₄	—	3	2	no	—
14	C₇×D₄	D₃	1	1	yes	characteristic, complemented
15	C₂×C₄×C₇	—	3	2	no	—
16	C₂×C₄×C₇	—	3	1	no	—
17	#237	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₃×C₇	—	2	2	no	—
19	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
20	D₃×C₇	—	2	1	no	—
21	C₂²×C₇	—	3	1	no	—
22	C₂²×C₇	C₂×D₃	1	2	yes	—
23	C₄×C₇	—	3	1	no	—
24	C₄×C₇	—	3	2	no	—
25	C₄×C₇	C₂×D₃	1	1	yes	characteristic
26	C₃×D₄	C₂×C₇	1	1	yes	characteristic, complemented
27	#67	C₂×C₇	1	2	yes	complemented
28	C₂×Dic₃	C₂×C₇	1	2	yes	complemented
29	C₄×D₃	C₂×C₇	1	1	yes	characteristic, complemented
30	Dic₆	C₂×C₇	1	1	yes	characteristic, complemented
31	C₃×C₇	#41	1	1	yes	characteristic, complemented, Hall
32	#41	—	3	1	no	Sylow
33	C₂×C₇	—	2	2	no	—
34	C₂×C₇	—	6	1	no	—
35	C₂×C₇	C₂²×D₃	1	1	yes	characteristic
36	C₂²×C₃	C₂²×C₇	1	2	yes	—
37	C₂×D₃	C₂²×C₇	1	1	yes	characteristic
38	C₃×C₄	C₂²×C₇	1	1	yes	characteristic
39	Dic₃	C₂²×C₇	1	2	yes	—
40	Dic₃	C₂²×C₇	1	1	yes	characteristic
41	Q₈	—	3	1	no	—
42	D₄	—	3	2	no	—
43	D₄	D₃×C₇	1	1	yes	characteristic, complemented
44	C₂×C₄	—	3	2	no	—
45	C₂×C₄	—	3	1	no	—
46	C₇	#237	1	1	yes	characteristic, direct product factor, Sylow
47	C₂×C₃	—	2	2	no	—
48	C₂×C₃	C₂³×C₇	1	1	yes	characteristic
49	D₃	—	2	1	no	—
50	C₂²	—	3	1	no	—
51	C₂²	C₂×D₃×C₇	1	2	yes	—
52	C₄	—	3	1	no	—
53	C₄	—	3	2	no	—
54	C₄	C₂×D₃×C₇	1	1	yes	characteristic
55	C₃	C₇×#41	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	2	2	no	—
57	C₂	—	6	1	no	—
58	C₂	C₂²×D₃×C₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂³×C₇
Frattini	C₂	C₂²×D₃×C₇
Fitting	C₃×C₇×D₄	C₂
socle	C₂×C₃×C₇	C₂³
hypercenter	C₇×D₄	D₃
hypocenter	C₃	C₇×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×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×#67	C₂	semidirect
3. C₂×C₇×Dic₃	C₂	semidirect
4. C₄×D₃×C₇	C₂	semidirect
5. C₇×Dic₆	C₂	semidirect
14. C₇×D₄	D₃	semidirect
17. #237	C₇	direct (N × H)
26. C₃×D₄	C₂×C₇	semidirect
27. #67	C₂×C₇	semidirect
28. C₂×Dic₃	C₂×C₇	semidirect
29. C₄×D₃	C₂×C₇	semidirect
30. Dic₆	C₂×C₇	semidirect
31. C₃×C₇	#41	semidirect
43. D₄	D₃×C₇	semidirect
46. C₇	#237	direct (N × H)
55. C₃	C₇×#41	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×#67	2
C₂×C₇×Dic₃	2
C₄×D₃×C₇	2
C₇×Dic₆	2
C₇×#41	3
#237	7

Subgroups — C7×#237 #66720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#237 #66720