Subgroups — C₇×C₃⋊(C₃⋊#66774) #36339539

Order 21420.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₇×D₅×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₇×C₃⋊#66774	—	3	4	no	maximal
3	C₃²×C₅×C₇×C₁₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇×C₃⋊Dic₅₁	—	5	1	no	maximal, Hall
5	C₃×C₇×D₅×C₁₇	D₃	1	4	yes	—
6	C₃⋊(C₃⋊#66774)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₇×#66774	—	9	1	no	Hall
8	C₂×C₃²×C₇×C₁₇	—	5	1	no	—
9	C₃×C₅×C₇×C₁₇	Dic₃	1	4	yes	complemented
10	C₃²×D₅×C₁₇	C₂×C₇	1	1	yes	characteristic
11	C₇×Dic₅₁	—	15	4	no	—
12	C₇×C₃⋊ΓL(1,𝔽₁₆)	—	17	1	no	maximal, Hall
13	C₇×D₅×C₁₇	C₃⋊D₃	1	1	yes	characteristic
14	C₃²×C₇×C₁₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
15	C₃⋊#66774	—	3	4	no	—
16	C₃²×C₅×C₁₇	C₄×C₇	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₇×C₁₇	—	5	4	no	—
18	C₃²×C₇×D₅	D₁₇	1	1	yes	characteristic
19	C₃⋊Dic₅₁	—	5	1	no	Hall
20	C₅×C₇×C₁₇	C₃⋊Dic₃	1	1	yes	characteristic, complemented, Hall
21	C₃×D₅×C₁₇	D₃×C₇	1	4	yes	—
22	C₇×Dic₁₇	—	45	1	no	Hall
23	C₇×ΓL(1,𝔽₁₆)	—	51	4	no	—
24	C₃×C₇×C₁₇	ΓL(1,𝔽₁₆)	1	4	yes	complemented
25	#66774	—	9	1	no	Hall
26	C₃²×C₅×C₇	Dic₁₇	1	1	yes	characteristic, complemented, Hall
27	C₂×C₃²×C₁₇	—	5	1	no	—
28	C₃×C₅×C₁₇	C₇×Dic₃	1	4	yes	complemented
29	C₇×C₃⋊Dic₃	—	85	1	no	Hall
30	C₂×C₇×C₁₇	—	5	1	no	—
31	C₃×C₇×D₅	D₅₁	1	4	yes	—
32	Dic₅₁	—	15	4	no	—
33	C₃⋊ΓL(1,𝔽₁₆)	—	17	1	no	Hall
34	D₅×C₁₇	C₇×C₃⋊D₃	1	1	yes	characteristic
35	C₃²×C₁₇	C₇×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
36	C₇×AGL(1,𝔽₅)	—	153	1	no	Hall
37	C₂×C₃²×C₇	—	5	1	no	—
38	C₇×C₁₇	C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Hall
39	C₃×C₅×C₇	Dic₅₁	1	4	yes	complemented
40	C₂×C₃×C₁₇	—	5	4	no	—
41	C₃²×D₅	C₇×D₁₇	1	1	yes	characteristic
42	C₅×C₁₇	C₇×C₃⋊Dic₃	1	1	yes	characteristic, complemented, Hall
43	C₇×Dic₃	—	255	4	no	—
44	C₇×D₅	C₃⋊D₅₁	1	1	yes	characteristic
45	Dic₁₇	—	45	1	no	Hall
46	C₃²×C₇	#66774	1	1	yes	characteristic, complemented, Hall
47	ΓL(1,𝔽₁₆)	—	51	4	no	—
48	C₃×C₁₇	C₇×ΓL(1,𝔽₁₆)	1	4	yes	complemented
49	C₃²×C₅	C₇×Dic₁₇	1	1	yes	characteristic, complemented, Hall
50	C₂×C₃×C₇	—	5	4	no	—
51	C₃⋊Dic₃	—	85	1	no	Hall
52	C₅×C₇	C₃⋊Dic₅₁	1	1	yes	characteristic, complemented, Hall
53	C₂×C₁₇	—	5	1	no	—
54	C₃×D₅	C₇×D₅₁	1	4	yes	—
55	C₄×C₇	—	765	1	no	Hall
56	C₃×C₇	C₃⋊#66774	1	4	yes	complemented
57	AGL(1,𝔽₅)	—	153	1	no	Hall
58	C₂×C₃²	—	5	1	no	—
59	C₁₇	C₇×C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
60	C₃×C₅	C₇×Dic₅₁	1	4	yes	complemented
61	C₂×C₇	—	5	1	no	—
62	Dic₃	—	255	4	no	—
63	D₅	C₇×C₃⋊D₅₁	1	1	yes	characteristic
64	C₃²	C₇×#66774	1	1	yes	characteristic, complemented, Sylow
65	C₇	C₃⋊(C₃⋊#66774)	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₃	—	5	4	no	—
67	C₅	C₇×C₃⋊Dic₅₁	1	1	yes	characteristic, complemented, Sylow
68	C₄	—	765	1	no	Sylow
69	C₃	C₇×C₃⋊#66774	1	4	yes	complemented
70	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₇	C₃⋊(C₃⋊#66774)
commutator, hypocenter	C₃²×C₅×C₁₇	C₄×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇×C₁₇	C₄
prime radical	C₃²×C₇×D₅×C₁₇	C₂
cosocle	C₃²×D₅×C₁₇	C₂×C₇
LCS	G ▷ C₃²×C₅×C₁₇
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
3. C₃²×C₅×C₇×C₁₇	C₄	semidirect
6. C₃⋊(C₃⋊#66774)	C₇	direct (N × H)
9. C₃×C₅×C₇×C₁₇	Dic₃	semidirect
14. C₃²×C₇×C₁₇	AGL(1,𝔽₅)	semidirect
16. C₃²×C₅×C₁₇	C₄×C₇	semidirect
20. C₅×C₇×C₁₇	C₃⋊Dic₃	semidirect
24. C₃×C₇×C₁₇	ΓL(1,𝔽₁₆)	semidirect
26. C₃²×C₅×C₇	Dic₁₇	semidirect
28. C₃×C₅×C₁₇	C₇×Dic₃	semidirect
35. C₃²×C₁₇	C₇×AGL(1,𝔽₅)	semidirect
38. C₇×C₁₇	C₃⋊ΓL(1,𝔽₁₆)	semidirect
39. C₃×C₅×C₇	Dic₅₁	semidirect
42. C₅×C₁₇	C₇×C₃⋊Dic₃	semidirect
46. C₃²×C₇	#66774	semidirect
48. C₃×C₁₇	C₇×ΓL(1,𝔽₁₆)	semidirect
49. C₃²×C₅	C₇×Dic₁₇	semidirect
52. C₅×C₇	C₃⋊Dic₅₁	semidirect
56. C₃×C₇	C₃⋊#66774	semidirect
59. C₁₇	C₇×C₃⋊ΓL(1,𝔽₁₆)	semidirect
60. C₃×C₅	C₇×Dic₅₁	semidirect
64. C₃²	C₇×#66774	semidirect
65. C₇	C₃⋊(C₃⋊#66774)	direct (N × H)
67. C₅	C₇×C₃⋊Dic₅₁	semidirect
69. C₃	C₇×C₃⋊#66774	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₇×D₅×C₁₇	2
C₇×C₃⋊#66774	3
C₇×C₃⋊Dic₅₁	5
C₃⋊(C₃⋊#66774)	7
C₇×C₃⋊ΓL(1,𝔽₁₆)	17

Subgroups — C7×C3⋊(C3⋊#66774) #36339539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₃⋊(C₃⋊#66774) #36339539