Subgroups — C₂×C₁₇×D₇⋊(C₅²⋊C₃) #49087834

Order 35700.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×C₇⋊(C₅²⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×D₇⋊(C₅²⋊C₃)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅²×D₇×C₁₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₁₇×C₇⋊(C₅²⋊C₃)	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₅²×D₇×C₁₇	C₂×C₃	1	2	yes	complemented
6	C₂×C₅²×C₇×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂²×C₁₇×C₅²⋊C₃	—	7	1	no	maximal, Hall
8	C₅²×C₇×C₁₇	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	C₂×C₁₇×C₅²⋊C₃	—	7	2	no	—
10	C₂×C₁₇×C₅²⋊C₃	—	7	1	no	—
11	C₂×C₅×D₇×C₁₇	—	3	2	no	—
12	C₂×D₇⋊(C₅²⋊C₃)	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₅²×C₁₇	—	7	1	no	Hall
14	C₂×C₁₇×AGL(1,𝔽₇)	—	25	1	no	maximal, Hall
15	C₁₇×C₅²⋊C₃	—	7	1	no	Hall
16	C₂×C₅×C₇×C₁₇	—	3	2	no	—
17	C₅×D₇×C₁₇	—	3	4	no	—
18	C₂×C₇⋊(C₅²⋊C₃)	C₂×C₁₇	1	1	yes	characteristic, complemented
19	D₇⋊(C₅²⋊C₃)	C₂×C₁₇	1	2	yes	direct product factor
20	C₂×C₅²×C₁₇	—	7	2	no	—
21	C₂×C₅²×C₁₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
22	C₂×C₁₇×ΓL(1,𝔽₈)	—	25	1	no	—
23	C₁₇×AGL(1,𝔽₇)	—	25	2	no	—
24	C₂×C₅²×D₇	C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
25	C₅×C₇×C₁₇	—	3	2	no	—
26	C₇⋊(C₅²⋊C₃)	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
27	C₂×D₇×C₁₇	C₅²⋊C₃	1	1	yes	characteristic, complemented, Hall
28	C₅²×C₁₇	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
29	C₁₇×ΓL(1,𝔽₈)	—	25	1	no	Hall
30	C₂×C₅²×C₇	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
31	C₅²×D₇	C₂×C₃×C₁₇	1	2	yes	complemented
32	C₂²×C₅×C₁₇	—	21	2	no	—
33	C₂²×C₅²⋊C₃	—	7	1	no	Hall
34	C₂×C₇×C₁₇	C₂×C₅²⋊C₃	1	1	yes	characteristic, complemented
35	D₇×C₁₇	C₂×C₅²⋊C₃	1	2	yes	complemented
36	C₂²×C₃×C₁₇	—	175	1	no	Hall
37	C₅²×C₇	C₂²×C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
38	C₂×C₅×C₁₇	—	3	2	no	—
39	C₂×C₅×C₁₇	—	21	4	no	—
40	C₂×C₅²⋊C₃	—	7	2	no	—
41	C₂×C₅²⋊C₃	—	7	1	no	—
42	C₂×C₅×D₇	—	3	2	no	—
43	C₇×C₁₇	C₂²×C₅²⋊C₃	1	1	yes	characteristic, complemented, Hall
44	C₂×C₃×C₁₇	—	175	2	no	—
45	C₂×C₃×C₁₇	—	175	1	no	—
46	C₂²×C₅²	—	7	1	no	Hall
47	C₅×C₁₇	—	3	2	no	—
48	C₂×AGL(1,𝔽₇)	—	25	1	no	Hall
49	C₅²⋊C₃	—	7	1	no	Hall
50	C₂×C₅×C₇	—	3	2	no	—
51	C₅×D₇	—	3	4	no	—
52	C₂²×C₁₇	—	7	1	no	Hall
53	C₃×C₁₇	—	175	1	no	Hall
54	C₂×C₅²	—	7	2	no	—
55	C₂×C₅²	C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
56	C₂×ΓL(1,𝔽₈)	—	25	1	no	—
57	AGL(1,𝔽₇)	—	25	2	no	—
58	C₅×C₇	—	3	2	no	—
59	C₂×C₁₇	—	7	2	no	—
60	C₂×C₁₇	D₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, direct product factor
61	C₂×D₇	C₁₇×C₅²⋊C₃	1	1	yes	characteristic, complemented, Hall
62	C₅²	C₂×C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
63	ΓL(1,𝔽₈)	—	25	1	no	Hall
64	C₂²×C₅	—	21	2	no	—
65	C₁₇	C₂×D₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₇	C₂×C₁₇×C₅²⋊C₃	1	1	yes	characteristic, complemented
67	D₇	C₂×C₁₇×C₅²⋊C₃	1	2	yes	complemented
68	C₂²×C₃	—	175	1	no	Hall
69	C₂×C₅	—	3	2	no	—
70	C₂×C₅	—	21	4	no	—
71	C₇	C₂²×C₁₇×C₅²⋊C₃	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₃	—	175	2	no	—
73	C₂×C₃	—	175	1	no	—
74	C₅	—	3	2	no	—
75	C₂²	—	7	1	no	Sylow
76	C₃	—	175	1	no	Sylow
77	C₂	—	7	2	no	—
78	C₂	C₁₇×D₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₁₇	D₇⋊(C₅²⋊C₃)
commutator, cosocle, hypocenter	C₅²×C₇	C₂²×C₃×C₁₇
Frattini	C₁	G
Fitting, socle	C₂×C₅²×C₇×C₁₇	C₂×C₃
LCS	G ▷ C₅²×C₇
UCS	C₁ ◁ C₂×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₇×C₇⋊(C₅²⋊C₃)	C₂	semidirect
2. C₁₇×D₇⋊(C₅²⋊C₃)	C₂	direct (N × H)
3. C₂×C₅²×D₇×C₁₇	C₃	semidirect
4. C₁₇×C₇⋊(C₅²⋊C₃)	C₂²	semidirect
5. C₅²×D₇×C₁₇	C₂×C₃	semidirect
6. C₂×C₅²×C₇×C₁₇	C₂×C₃	semidirect
8. C₅²×C₇×C₁₇	C₂²×C₃	semidirect
12. C₂×D₇⋊(C₅²⋊C₃)	C₁₇	direct (N × H)
18. C₂×C₇⋊(C₅²⋊C₃)	C₂×C₁₇	semidirect
19. D₇⋊(C₅²⋊C₃)	C₂×C₁₇	direct (N × H)
21. C₂×C₅²×C₁₇	AGL(1,𝔽₇)	semidirect
24. C₂×C₅²×D₇	C₃×C₁₇	semidirect
26. C₇⋊(C₅²⋊C₃)	C₂²×C₁₇	semidirect
27. C₂×D₇×C₁₇	C₅²⋊C₃	semidirect
28. C₅²×C₁₇	C₂×AGL(1,𝔽₇)	semidirect
30. C₂×C₅²×C₇	C₂×C₃×C₁₇	semidirect
31. C₅²×D₇	C₂×C₃×C₁₇	semidirect
34. C₂×C₇×C₁₇	C₂×C₅²⋊C₃	semidirect
35. D₇×C₁₇	C₂×C₅²⋊C₃	semidirect
37. C₅²×C₇	C₂²×C₃×C₁₇	semidirect
43. C₇×C₁₇	C₂²×C₅²⋊C₃	semidirect
55. C₂×C₅²	C₁₇×AGL(1,𝔽₇)	semidirect
60. C₂×C₁₇	D₇⋊(C₅²⋊C₃)	direct (N × H)
61. C₂×D₇	C₁₇×C₅²⋊C₃	semidirect
62. C₅²	C₂×C₁₇×AGL(1,𝔽₇)	semidirect
65. C₁₇	C₂×D₇⋊(C₅²⋊C₃)	direct (N × H)
66. C₂×C₇	C₂×C₁₇×C₅²⋊C₃	semidirect
67. D₇	C₂×C₁₇×C₅²⋊C₃	semidirect
71. C₇	C₂²×C₁₇×C₅²⋊C₃	semidirect
78. C₂	C₁₇×D₇⋊(C₅²⋊C₃)	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×C₁₇×C₇⋊(C₅²⋊C₃)	2
C₁₇×D₇⋊(C₅²⋊C₃)	2
C₂×C₅²×D₇×C₁₇	3
C₂²×C₁₇×C₅²⋊C₃	7
C₂×D₇⋊(C₅²⋊C₃)	17
C₂×C₁₇×AGL(1,𝔽₇)	25

Subgroups — C2×C17×D7⋊(C52⋊C3) #49087834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇×D₇⋊(C₅²⋊C₃) #49087834