Group #44472626 — subgroups

Subgroups — C₂×C₇×C₁₇×D₆₃ #44472626

Order 29988.

156 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₁₇×D₆₃	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₇²×C₉×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×C₁₇×D₂₁	—	3	1	no	maximal
4	C₇²×C₉×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇×C₁₇×D₂₁	—	3	2	no	—
6	C₂×C₃×C₇²×C₁₇	D₃	1	1	yes	characteristic
7	C₂×C₁₇×D₆₃	C₇	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₇×C₁₇×D₉	—	7	1	no	maximal
9	C₂×C₇×D₇×C₁₇	—	9	1	no	Hall
10	C₃×C₇²×C₁₇	C₂×D₃	1	1	yes	characteristic
11	C₁₇×D₆₃	C₂×C₇	1	2	yes	direct product factor
12	C₇×C₁₇×D₉	—	7	2	no	—
13	C₂×C₇×C₉×C₁₇	—	2	3	no	—
14	C₂×C₇×C₉×C₁₇	D₇	1	1	yes	characteristic, complemented
15	C₂×C₇×C₉×C₁₇	C₂×C₇	1	1	yes	characteristic, complemented
16	C₂×C₇×D₆₃	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₇²×C₁₇	D₉	1	1	yes	characteristic, complemented
18	C₇×D₇×C₁₇	—	9	2	no	—
19	C₂×C₁₇×D₂₁	—	3	1	no	—
20	C₂×D₃×C₇×C₁₇	—	21	1	no	—
21	C₇×C₉×C₁₇	—	2	3	no	—
22	C₇×C₉×C₁₇	C₂×D₇	1	1	yes	characteristic, complemented
23	C₇×C₉×C₁₇	C₂²×C₇	1	1	yes	characteristic, complemented
24	C₂×C₇²×C₉	C₂×C₁₇	1	1	yes	characteristic, complemented
25	C₇×D₆₃	C₂×C₁₇	1	2	yes	direct product factor
26	C₇²×C₁₇	C₂×D₉	1	1	yes	characteristic, complemented, Hall
27	C₂×C₃×C₇×C₁₇	—	2	3	no	—
28	C₂×C₃×C₇×C₁₇	D₃×C₇	1	1	yes	characteristic
29	C₂×C₃×C₇×C₁₇	D₂₁	1	1	yes	characteristic
30	C₁₇×D₂₁	—	3	2	no	—
31	D₃×C₇×C₁₇	—	21	2	no	—
32	C₂×C₁₇×D₉	—	7	1	no	Hall
33	C₂×C₇×D₂₁	—	3	1	no	—
34	C₂²×C₇×C₁₇	—	63	1	no	—
35	C₂×D₇×C₁₇	—	9	1	no	—
36	C₇²×C₉	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
37	C₃×C₇×C₁₇	—	2	3	no	—
38	C₃×C₇×C₁₇	C₂×D₃×C₇	1	1	yes	characteristic
39	C₃×C₇×C₁₇	C₂×D₂₁	1	1	yes	characteristic
40	C₂×C₉×C₁₇	C₇×D₇	1	1	yes	characteristic, complemented
41	C₁₇×D₉	—	7	2	no	—
42	C₂×C₃×C₇²	D₃×C₁₇	1	1	yes	characteristic
43	C₇×D₂₁	—	3	2	no	—
44	C₂×D₆₃	C₇×C₁₇	1	1	yes	characteristic, direct product factor
45	C₂×C₇×D₉	—	7	1	no	—
46	C₂×C₇×C₁₇	—	2	3	no	—
47	C₂×C₇×C₁₇	—	63	2	no	—
48	C₂×C₇×C₁₇	C₇×D₉	1	1	yes	characteristic, complemented
49	C₂×C₇×C₁₇	D₆₃	1	1	yes	characteristic, direct product factor
50	D₇×C₁₇	—	9	2	no	—
51	C₂×D₃×C₁₇	—	21	1	no	—
52	C₂×C₇×D₇	—	9	1	no	Hall
53	C₉×C₁₇	C₂×C₇×D₇	1	1	yes	characteristic, complemented, Hall
54	C₃×C₇²	C₂×D₃×C₁₇	1	1	yes	characteristic
55	C₂×C₇×C₉	—	2	3	no	—
56	C₂×C₇×C₉	D₇×C₁₇	1	1	yes	characteristic, complemented
57	C₂×C₇×C₉	C₂×C₇×C₁₇	1	1	yes	characteristic, complemented
58	D₆₃	C₂×C₇×C₁₇	1	2	yes	direct product factor
59	C₇×D₉	—	7	2	no	—
60	C₇×C₁₇	—	2	3	no	—
61	C₇×C₁₇	C₂×C₇×D₉	1	1	yes	characteristic, complemented
62	C₇×C₁₇	C₂×D₆₃	1	1	yes	characteristic, direct product factor
63	C₂×C₃×C₁₇	C₇×D₂₁	1	1	yes	characteristic
64	D₃×C₁₇	—	21	2	no	—
65	C₂×C₇²	C₁₇×D₉	1	1	yes	characteristic, complemented
66	C₇×D₇	—	9	2	no	—
67	C₂×D₂₁	—	3	1	no	—
68	C₂×D₃×C₇	—	21	1	no	—
69	C₂²×C₁₇	—	63	1	no	Hall
70	C₇×C₉	—	2	3	no	—
71	C₇×C₉	C₂×D₇×C₁₇	1	1	yes	characteristic, complemented
72	C₇×C₉	C₂²×C₇×C₁₇	1	1	yes	characteristic, complemented
73	C₃×C₁₇	C₂×C₇×D₂₁	1	1	yes	characteristic
74	C₇²	C₂×C₁₇×D₉	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₃×C₇	—	2	3	no	—
76	C₂×C₃×C₇	D₃×C₇×C₁₇	1	1	yes	characteristic
77	C₂×C₃×C₇	C₁₇×D₂₁	1	1	yes	characteristic
78	D₂₁	—	3	2	no	—
79	D₃×C₇	—	21	2	no	—
80	C₂×D₉	—	7	1	no	Hall
81	C₂×C₁₇	—	63	2	no	—
82	C₂×C₁₇	C₇×D₆₃	1	1	yes	characteristic, direct product factor
83	C₂²×C₇	—	63	1	no	—
84	C₂×D₇	—	9	1	no	—
85	C₃×C₇	—	2	3	no	—
86	C₃×C₇	C₂×D₃×C₇×C₁₇	1	1	yes	characteristic
87	C₃×C₇	C₂×C₁₇×D₂₁	1	1	yes	characteristic
88	C₂×C₉	C₇×D₇×C₁₇	1	1	yes	characteristic, complemented
89	D₉	—	7	2	no	—
90	C₁₇	C₂×C₇×D₆₃	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×C₇	—	2	3	no	—
92	C₂×C₇	—	63	2	no	—
93	C₂×C₇	C₇×C₁₇×D₉	1	1	yes	characteristic, complemented
94	C₂×C₇	C₁₇×D₆₃	1	1	yes	characteristic, direct product factor
95	D₇	—	9	2	no	—
96	C₂×D₃	—	21	1	no	—
97	C₉	C₂×C₇×D₇×C₁₇	1	1	yes	characteristic, complemented, Sylow
98	C₇	—	2	3	no	—
99	C₇	C₂×C₇×C₁₇×D₉	1	1	yes	characteristic, complemented
100	C₇	C₂×C₁₇×D₆₃	1	1	yes	characteristic, direct product factor
101	C₂×C₃	C₇×C₁₇×D₂₁	1	1	yes	characteristic
102	D₃	—	21	2	no	—
103	C₂²	—	63	1	no	Sylow
104	C₃	C₂×C₇×C₁₇×D₂₁	1	1	yes	characteristic
105	C₂	—	63	2	no	—
106	C₂	C₇×C₁₇×D₆₃	1	1	yes	characteristic, direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×C₁₇×D₆₃

C₂

yes

maximal, direct product factor

C₂×C₇²×C₉×C₁₇

C₂

yes

maximal, characteristic, complemented

C₂×C₇×C₁₇×D₂₁

—

maximal

C₇²×C₉×C₁₇

C₂²

yes

characteristic, complemented, Hall

C₇×C₁₇×D₂₁

—

C₂×C₃×C₇²×C₁₇

D₃

yes

characteristic

C₂×C₁₇×D₆₃

C₇

yes

maximal, characteristic, direct product factor

C₂×C₇×C₁₇×D₉

—

maximal

C₂×C₇×D₇×C₁₇

—

Hall

C₃×C₇²×C₁₇

C₂×D₃

yes

characteristic

C₁₇×D₆₃

C₂×C₇

yes

direct product factor

C₇×C₁₇×D₉

—

C₂×C₇×C₉×C₁₇

—

C₂×C₇×C₉×C₁₇

D₇

yes

characteristic, complemented

C₂×C₇×C₉×C₁₇

C₂×C₇

yes

characteristic, complemented

C₂×C₇×D₆₃

C₁₇

yes

maximal, characteristic, direct product factor, Hall

C₂×C₇²×C₁₇

D₉

yes

characteristic, complemented

C₇×D₇×C₁₇

—

C₂×C₁₇×D₂₁

—

C₂×D₃×C₇×C₁₇

—

C₇×C₉×C₁₇

—

C₇×C₉×C₁₇

C₂×D₇

yes

characteristic, complemented

C₇×C₉×C₁₇

C₂²×C₇

yes

characteristic, complemented

C₂×C₇²×C₉

C₂×C₁₇

yes

characteristic, complemented

C₇×D₆₃

C₂×C₁₇

yes

direct product factor

C₇²×C₁₇

C₂×D₉

yes

characteristic, complemented, Hall

C₂×C₃×C₇×C₁₇

—

C₂×C₃×C₇×C₁₇

D₃×C₇

yes

characteristic

C₂×C₃×C₇×C₁₇

D₂₁

yes

characteristic

C₁₇×D₂₁

—

D₃×C₇×C₁₇

—

C₂×C₁₇×D₉

—

Hall

C₂×C₇×D₂₁

—

C₂²×C₇×C₁₇

—

C₂×D₇×C₁₇

—

C₇²×C₉

C₂²×C₁₇

yes

characteristic, complemented, Hall

C₃×C₇×C₁₇

—

C₃×C₇×C₁₇

C₂×D₃×C₇

yes

characteristic

C₃×C₇×C₁₇

C₂×D₂₁

yes

characteristic

C₂×C₉×C₁₇

C₇×D₇

yes

characteristic, complemented

C₁₇×D₉

—

C₂×C₃×C₇²

D₃×C₁₇

yes

characteristic

C₇×D₂₁

—

C₂×D₆₃

C₇×C₁₇

yes

characteristic, direct product factor

C₂×C₇×D₉

—

C₂×C₇×C₁₇

—

C₂×C₇×C₁₇

—

C₂×C₇×C₁₇

C₇×D₉

yes

characteristic, complemented

C₂×C₇×C₁₇

D₆₃

yes

characteristic, direct product factor

D₇×C₁₇

—

C₂×D₃×C₁₇

—

C₂×C₇×D₇

—

Hall

C₉×C₁₇

C₂×C₇×D₇

yes

characteristic, complemented, Hall

C₃×C₇²

C₂×D₃×C₁₇

yes

characteristic

C₂×C₇×C₉

—

C₂×C₇×C₉

D₇×C₁₇

yes

characteristic, complemented

C₂×C₇×C₉

C₂×C₇×C₁₇

yes

characteristic, complemented

D₆₃

C₂×C₇×C₁₇

yes

direct product factor

C₇×D₉

—

C₇×C₁₇

—

C₇×C₁₇

C₂×C₇×D₉

yes

characteristic, complemented

C₇×C₁₇

C₂×D₆₃

yes

characteristic, direct product factor

C₂×C₃×C₁₇

C₇×D₂₁

yes

characteristic

D₃×C₁₇

—

C₂×C₇²

C₁₇×D₉

yes

characteristic, complemented

C₇×D₇

—

C₂×D₂₁

—

C₂×D₃×C₇

—

C₂²×C₁₇

—

Hall

C₇×C₉

—

C₇×C₉

C₂×D₇×C₁₇

yes

characteristic, complemented

C₇×C₉

C₂²×C₇×C₁₇

yes

characteristic, complemented

C₃×C₁₇

C₂×C₇×D₂₁

yes

characteristic

C₇²

C₂×C₁₇×D₉

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

C₂×C₃×C₇

D₃×C₇×C₁₇

yes

characteristic

C₂×C₃×C₇

C₁₇×D₂₁

yes

characteristic

D₂₁

—

D₃×C₇

—

C₂×D₉

—

Hall

C₂×C₁₇

—

C₂×C₁₇

C₇×D₆₃

yes

characteristic, direct product factor

C₂²×C₇

—

C₂×D₇

—

C₃×C₇

—

C₃×C₇

C₂×D₃×C₇×C₁₇

yes

characteristic

C₃×C₇

C₂×C₁₇×D₂₁

yes

characteristic

C₂×C₉

C₇×D₇×C₁₇

yes

characteristic, complemented

D₉

—

C₁₇

C₂×C₇×D₆₃

yes

characteristic, direct product factor, Sylow

C₂×C₇

—

C₂×C₇

—

C₂×C₇

C₇×C₁₇×D₉

yes

characteristic, complemented

C₂×C₇

C₁₇×D₆₃

yes

characteristic, direct product factor

D₇

—

C₂×D₃

—

C₉

C₂×C₇×D₇×C₁₇

yes

characteristic, complemented, Sylow

C₇

—

C₇

C₂×C₇×C₁₇×D₉

yes

characteristic, complemented

100

C₇

C₂×C₁₇×D₆₃

yes

characteristic, direct product factor

101

C₂×C₃

C₇×C₁₇×D₂₁

yes

characteristic

102

D₃

—

103

C₂²

—

Sylow

104

C₃

C₂×C₇×C₁₇×D₂₁

yes

characteristic

105

C₂

—

106

C₂

C₇×C₁₇×D₆₃

yes

characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₇×C₁₇	D₆₃
commutator, cosocle, hypocenter	C₇×C₉	C₂²×C₇×C₁₇
Frattini	C₃	C₂×C₇×C₁₇×D₂₁
Fitting	C₂×C₇²×C₉×C₁₇	C₂
socle	C₂×C₃×C₇²×C₁₇	D₃
LCS	G ▷ C₇×C₉
UCS	C₁ ◁ C₂×C₇×C₁₇

subgroup

G/H

center, hypercenter

C₂×C₇×C₁₇

D₆₃

commutator, cosocle, hypocenter

C₇×C₉

C₂²×C₇×C₁₇

Frattini

C₃

C₂×C₇×C₁₇×D₂₁

Fitting

C₂×C₇²×C₉×C₁₇

C₂

socle

C₂×C₃×C₇²×C₁₇

D₃

LCS

G ▷ C₇×C₉

UCS

C₁ ◁ C₂×C₇×C₁₇

N	H ≅ G/N	type
1. C₇×C₁₇×D₆₃	C₂	direct (N × H)
2. C₂×C₇²×C₉×C₁₇	C₂	semidirect
4. C₇²×C₉×C₁₇	C₂²	semidirect
7. C₂×C₁₇×D₆₃	C₇	direct (N × H)
11. C₁₇×D₆₃	C₂×C₇	direct (N × H)
14. C₂×C₇×C₉×C₁₇	D₇	semidirect
15. C₂×C₇×C₉×C₁₇	C₂×C₇	semidirect
16. C₂×C₇×D₆₃	C₁₇	direct (N × H)
17. C₂×C₇²×C₁₇	D₉	semidirect
22. C₇×C₉×C₁₇	C₂×D₇	semidirect
23. C₇×C₉×C₁₇	C₂²×C₇	semidirect
24. C₂×C₇²×C₉	C₂×C₁₇	semidirect
25. C₇×D₆₃	C₂×C₁₇	direct (N × H)
26. C₇²×C₁₇	C₂×D₉	semidirect
36. C₇²×C₉	C₂²×C₁₇	semidirect
40. C₂×C₉×C₁₇	C₇×D₇	semidirect
44. C₂×D₆₃	C₇×C₁₇	direct (N × H)
48. C₂×C₇×C₁₇	C₇×D₉	semidirect
49. C₂×C₇×C₁₇	D₆₃	direct (N × H)
53. C₉×C₁₇	C₂×C₇×D₇	semidirect
56. C₂×C₇×C₉	D₇×C₁₇	semidirect
57. C₂×C₇×C₉	C₂×C₇×C₁₇	semidirect
58. D₆₃	C₂×C₇×C₁₇	direct (N × H)
61. C₇×C₁₇	C₂×C₇×D₉	semidirect
62. C₇×C₁₇	C₂×D₆₃	direct (N × H)
65. C₂×C₇²	C₁₇×D₉	semidirect
71. C₇×C₉	C₂×D₇×C₁₇	semidirect
72. C₇×C₉	C₂²×C₇×C₁₇	semidirect
74. C₇²	C₂×C₁₇×D₉	semidirect
82. C₂×C₁₇	C₇×D₆₃	direct (N × H)
88. C₂×C₉	C₇×D₇×C₁₇	semidirect
90. C₁₇	C₂×C₇×D₆₃	direct (N × H)
93. C₂×C₇	C₇×C₁₇×D₉	semidirect
94. C₂×C₇	C₁₇×D₆₃	direct (N × H)
97. C₉	C₂×C₇×D₇×C₁₇	semidirect
99. C₇	C₂×C₇×C₁₇×D₉	semidirect
100. C₇	C₂×C₁₇×D₆₃	direct (N × H)
106. C₂	C₇×C₁₇×D₆₃	direct (N × H)

H ≅ G/N

type

1. C₇×C₁₇×D₆₃

C₂

direct (N × H)

2. C₂×C₇²×C₉×C₁₇

C₂

semidirect

4. C₇²×C₉×C₁₇

C₂²

semidirect

7. C₂×C₁₇×D₆₃

C₇

direct (N × H)

11. C₁₇×D₆₃

C₂×C₇

direct (N × H)

14. C₂×C₇×C₉×C₁₇

D₇

semidirect

15. C₂×C₇×C₉×C₁₇

C₂×C₇

semidirect

16. C₂×C₇×D₆₃

C₁₇

direct (N × H)

17. C₂×C₇²×C₁₇

D₉

semidirect

22. C₇×C₉×C₁₇

C₂×D₇

semidirect

23. C₇×C₉×C₁₇

C₂²×C₇

semidirect

24. C₂×C₇²×C₉

C₂×C₁₇

semidirect

25. C₇×D₆₃

C₂×C₁₇

direct (N × H)

26. C₇²×C₁₇

C₂×D₉

semidirect

36. C₇²×C₉

C₂²×C₁₇

semidirect

40. C₂×C₉×C₁₇

C₇×D₇

semidirect

44. C₂×D₆₃

C₇×C₁₇

direct (N × H)

48. C₂×C₇×C₁₇

C₇×D₉

semidirect

49. C₂×C₇×C₁₇

D₆₃

direct (N × H)

53. C₉×C₁₇

C₂×C₇×D₇

semidirect

56. C₂×C₇×C₉

D₇×C₁₇

semidirect

57. C₂×C₇×C₉

C₂×C₇×C₁₇

semidirect

58. D₆₃

C₂×C₇×C₁₇

direct (N × H)

61. C₇×C₁₇

C₂×C₇×D₉

semidirect

62. C₇×C₁₇

C₂×D₆₃

direct (N × H)

65. C₂×C₇²

C₁₇×D₉

semidirect

71. C₇×C₉

C₂×D₇×C₁₇

semidirect

72. C₇×C₉

C₂²×C₇×C₁₇

semidirect

74. C₇²

C₂×C₁₇×D₉

semidirect

82. C₂×C₁₇

C₇×D₆₃

direct (N × H)

88. C₂×C₉

C₇×D₇×C₁₇

semidirect

90. C₁₇

C₂×C₇×D₆₃

direct (N × H)

93. C₂×C₇

C₇×C₁₇×D₉

semidirect

94. C₂×C₇

C₁₇×D₆₃

direct (N × H)

97. C₉

C₂×C₇×D₇×C₁₇

semidirect

99. C₇

C₂×C₇×C₁₇×D₉

semidirect

100. C₇

C₂×C₁₇×D₆₃

direct (N × H)

106. C₂

C₇×C₁₇×D₆₃

direct (N × H)

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

subgroup	index
C₇×C₁₇×D₆₃	2
C₂×C₇²×C₉×C₁₇	2
C₂×C₇×C₁₇×D₂₁	3
C₂×C₁₇×D₆₃	7
C₂×C₇×C₁₇×D₉	7
C₂×C₇×D₆₃	17

subgroup

index

C₇×C₁₇×D₆₃

C₂×C₇²×C₉×C₁₇

C₂×C₇×C₁₇×D₂₁

C₂×C₁₇×D₆₃

C₂×C₇×C₁₇×D₉

C₂×C₇×D₆₃

Subgroups — C₂×C₇×C₁₇×D₆₃ #44472626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C7×C17×D63 #44472626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×C₁₇×D₆₃ #44472626