Group #44472595 — subgroups

Subgroups — C₇×D₁₇×D₆₃ #44472595

Order 29988.

156 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 44 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×D₁₀₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×C₁₇×D₆₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇²×C₉×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₁₇×D₂₁	—	3	1	no	maximal
5	C₇²×C₉×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇×D₃₅₇	—	3	1	no	—
7	C₇×C₁₇×D₂₁	—	3	1	no	—
8	C₃×C₇²×D₁₇	D₃	1	1	yes	characteristic
9	D₁₇×D₆₃	C₇	1	1	yes	maximal, characteristic, direct product factor
10	C₇×D₉×D₁₇	—	7	1	no	maximal
11	C₇×D₇×D₁₇	—	9	1	no	Hall
12	C₃×C₇²×C₁₇	C₂×D₃	1	1	yes	characteristic
13	D₁₀₇₁	C₂×C₇	1	1	yes	characteristic, complemented
14	C₁₇×D₆₃	C₂×C₇	1	1	yes	characteristic, complemented
15	C₇×D₁₅₃	—	7	1	no	—
16	C₇×C₁₇×D₉	—	7	1	no	—
17	C₇×C₉×D₁₇	—	2	3	no	—
18	C₇×C₉×D₁₇	D₇	1	1	yes	characteristic, complemented
19	C₇×C₉×D₁₇	C₂×C₇	1	1	yes	characteristic, complemented
20	C₂×C₇×D₆₃	—	17	1	no	maximal, Hall
21	C₇×D₁₁₉	—	9	1	no	—
22	C₇×D₇×C₁₇	—	9	1	no	—
23	C₇²×D₁₇	D₉	1	1	yes	characteristic, complemented
24	D₁₇×D₂₁	—	3	1	no	—
25	D₃×C₇×D₁₇	—	21	1	no	—
26	C₇×C₉×C₁₇	—	2	3	no	—
27	C₇×C₉×C₁₇	C₂×D₇	1	1	yes	characteristic, complemented
28	C₇×C₉×C₁₇	C₂²×C₇	1	1	yes	characteristic, complemented
29	C₂×C₇²×C₉	—	17	1	no	—
30	C₇×D₆₃	—	17	1	no	—
31	C₇×D₆₃	D₁₇	1	1	yes	characteristic, direct product factor
32	C₇²×C₁₇	C₂×D₉	1	1	yes	characteristic, complemented, Hall
33	D₃₅₇	—	3	1	no	—
34	C₁₇×D₂₁	—	3	1	no	—
35	C₇×D₅₁	—	21	1	no	—
36	D₃×C₇×C₁₇	—	21	1	no	—
37	C₃×C₇×D₁₇	—	2	3	no	—
38	C₃×C₇×D₁₇	D₃×C₇	1	1	yes	characteristic
39	C₃×C₇×D₁₇	D₂₁	1	1	yes	characteristic
40	D₉×D₁₇	—	7	1	no	Hall
41	C₂×C₇×D₂₁	—	51	1	no	—
42	C₂×C₇×D₁₇	—	63	1	no	—
43	D₇×D₁₇	—	9	1	no	—
44	C₇²×C₉	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
45	C₃×C₇×C₁₇	—	2	3	no	—
46	C₃×C₇×C₁₇	C₂×D₃×C₇	1	1	yes	characteristic
47	C₃×C₇×C₁₇	C₂×D₂₁	1	1	yes	characteristic
48	D₁₅₃	—	7	1	no	—
49	C₉×D₁₇	C₇×D₇	1	1	yes	characteristic, complemented
50	C₁₇×D₉	—	7	1	no	—
51	C₂×C₃×C₇²	—	17	1	no	—
52	C₇×D₂₁	—	3	1	no	—
53	C₇×D₂₁	—	51	1	no	—
54	C₂×D₆₃	—	17	1	no	—
55	C₂×C₇×D₉	—	119	1	no	—
56	C₂×C₇×C₁₇	—	63	1	no	—
57	D₁₁₉	—	9	1	no	—
58	C₇×D₁₇	—	2	3	no	—
59	C₇×D₁₇	—	63	1	no	—
60	C₇×D₁₇	C₇×D₉	1	1	yes	characteristic, complemented
61	C₇×D₁₇	D₆₃	1	1	yes	characteristic, direct product factor
62	D₇×C₁₇	—	9	1	no	—
63	D₃×D₁₇	—	21	1	no	—
64	C₂×C₇×D₇	—	153	1	no	Hall
65	C₉×C₁₇	C₂×C₇×D₇	1	1	yes	characteristic, complemented, Hall
66	C₃×C₇²	D₃×D₁₇	1	1	yes	characteristic
67	C₂×C₇×C₉	—	17	1	no	—
68	C₂×C₇×C₉	—	17	1	no	—
69	C₂×C₇×C₉	—	34	3	no	—
70	D₆₃	—	17	1	no	—
71	D₆₃	C₇×D₁₇	1	1	yes	characteristic, direct product factor
72	C₇×D₉	—	7	1	no	—
73	C₇×D₉	—	119	1	no	—
74	C₇×C₁₇	—	2	3	no	—
75	C₇×C₁₇	C₂×C₇×D₉	1	1	yes	characteristic, complemented
76	C₇×C₁₇	C₂×D₆₃	1	1	yes	characteristic, complemented
77	D₅₁	—	21	1	no	—
78	C₃×D₁₇	C₇×D₂₁	1	1	yes	characteristic
79	D₃×C₁₇	—	21	1	no	—
80	C₂×C₇²	—	17	1	no	—
81	C₇×D₇	—	9	1	no	—
82	C₇×D₇	—	153	1	no	—
83	C₂×D₂₁	—	51	1	no	—
84	C₂×D₃×C₇	—	357	1	no	—
85	C₂×D₁₇	—	63	1	no	Hall
86	C₇×C₉	—	2	3	no	—
87	C₇×C₉	D₇×D₁₇	1	1	yes	characteristic, complemented
88	C₇×C₉	C₂×C₇×D₁₇	1	1	yes	characteristic, complemented
89	C₃×C₁₇	C₂×C₇×D₂₁	1	1	yes	characteristic
90	C₇²	D₉×D₁₇	1	1	yes	characteristic, complemented, Sylow
91	C₂×C₃×C₇	—	17	1	no	—
92	C₂×C₃×C₇	—	17	1	no	—
93	C₂×C₃×C₇	—	34	3	no	—
94	D₂₁	—	3	1	no	—
95	D₂₁	—	51	1	no	—
96	D₃×C₇	—	21	1	no	—
97	D₃×C₇	—	357	1	no	—
98	C₂×D₉	—	119	1	no	Hall
99	C₂×C₁₇	—	63	1	no	—
100	D₁₇	—	63	1	no	—
101	D₁₇	C₇×D₆₃	1	1	yes	characteristic, direct product factor
102	C₂²×C₇	—	1071	1	no	—
103	C₂×D₇	—	153	1	no	—
104	C₃×C₇	—	2	3	no	—
105	C₃×C₇	D₃×C₇×D₁₇	1	1	yes	characteristic
106	C₃×C₇	D₁₇×D₂₁	1	1	yes	characteristic
107	C₂×C₉	—	17	1	no	—
108	D₉	—	7	1	no	—
109	D₉	—	119	1	no	—
110	C₁₇	C₂×C₇×D₆₃	1	1	yes	characteristic, complemented, Sylow
111	C₂×C₇	—	17	1	no	—
112	C₂×C₇	—	17	1	no	—
113	C₂×C₇	—	34	3	no	—
114	C₂×C₇	—	1071	1	no	—
115	C₂×C₇	—	63	1	no	—
116	D₇	—	9	1	no	—
117	D₇	—	153	1	no	—
118	C₂×D₃	—	357	1	no	—
119	C₉	C₇×D₇×D₁₇	1	1	yes	characteristic, complemented, Sylow
120	C₇	—	2	3	no	—
121	C₇	C₇×D₉×D₁₇	1	1	yes	characteristic, complemented
122	C₇	D₁₇×D₆₃	1	1	yes	characteristic, direct product factor
123	C₂×C₃	—	17	1	no	—
124	D₃	—	21	1	no	—
125	D₃	—	357	1	no	—
126	C₂²	—	1071	1	no	Sylow
127	C₃	C₇×D₁₇×D₂₁	1	1	yes	characteristic
128	C₂	—	17	1	no	—
129	C₂	—	1071	1	no	—
130	C₂	—	63	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×D₁₀₇₁

C₂

yes

maximal, characteristic, complemented

C₇×C₁₇×D₆₃

C₂

yes

maximal, characteristic, complemented

C₇²×C₉×D₁₇

C₂

yes

maximal, characteristic, complemented

C₇×D₁₇×D₂₁

—

maximal

C₇²×C₉×C₁₇

C₂²

yes

characteristic, complemented, Hall

C₇×D₃₅₇

—

C₇×C₁₇×D₂₁

—

C₃×C₇²×D₁₇

D₃

yes

characteristic

D₁₇×D₆₃

C₇

yes

maximal, characteristic, direct product factor

C₇×D₉×D₁₇

—

maximal

C₇×D₇×D₁₇

—

Hall

C₃×C₇²×C₁₇

C₂×D₃

yes

characteristic

D₁₀₇₁

C₂×C₇

yes

characteristic, complemented

C₁₇×D₆₃

C₂×C₇

yes

characteristic, complemented

C₇×D₁₅₃

—

C₇×C₁₇×D₉

—

C₇×C₉×D₁₇

—

C₇×C₉×D₁₇

D₇

yes

characteristic, complemented

C₇×C₉×D₁₇

C₂×C₇

yes

characteristic, complemented

C₂×C₇×D₆₃

—

maximal, Hall

C₇×D₁₁₉

—

C₇×D₇×C₁₇

—

C₇²×D₁₇

D₉

yes

characteristic, complemented

D₁₇×D₂₁

—

D₃×C₇×D₁₇

—

C₇×C₉×C₁₇

—

C₇×C₉×C₁₇

C₂×D₇

yes

characteristic, complemented

C₇×C₉×C₁₇

C₂²×C₇

yes

characteristic, complemented

C₂×C₇²×C₉

—

C₇×D₆₃

—

C₇×D₆₃

D₁₇

yes

characteristic, direct product factor

C₇²×C₁₇

C₂×D₉

yes

characteristic, complemented, Hall

D₃₅₇

—

C₁₇×D₂₁

—

C₇×D₅₁

—

D₃×C₇×C₁₇

—

C₃×C₇×D₁₇

—

C₃×C₇×D₁₇

D₃×C₇

yes

characteristic

C₃×C₇×D₁₇

D₂₁

yes

characteristic

D₉×D₁₇

—

Hall

C₂×C₇×D₂₁

—

C₂×C₇×D₁₇

—

D₇×D₁₇

—

C₇²×C₉

C₂×D₁₇

yes

characteristic, complemented, Hall

C₃×C₇×C₁₇

—

C₃×C₇×C₁₇

C₂×D₃×C₇

yes

characteristic

C₃×C₇×C₁₇

C₂×D₂₁

yes

characteristic

D₁₅₃

—

C₉×D₁₇

C₇×D₇

yes

characteristic, complemented

C₁₇×D₉

—

C₂×C₃×C₇²

—

C₇×D₂₁

—

C₇×D₂₁

—

C₂×D₆₃

—

C₂×C₇×D₉

—

119

—

C₂×C₇×C₁₇

—

D₁₁₉

—

C₇×D₁₇

—

C₇×D₁₇

—

C₇×D₁₇

C₇×D₉

yes

characteristic, complemented

C₇×D₁₇

D₆₃

yes

characteristic, direct product factor

D₇×C₁₇

—

D₃×D₁₇

—

C₂×C₇×D₇

—

153

Hall

C₉×C₁₇

C₂×C₇×D₇

yes

characteristic, complemented, Hall

C₃×C₇²

D₃×D₁₇

yes

characteristic

C₂×C₇×C₉

—

C₂×C₇×C₉

—

C₂×C₇×C₉

—

D₆₃

—

D₆₃

C₇×D₁₇

yes

characteristic, direct product factor

C₇×D₉

—

C₇×D₉

—

119

—

C₇×C₁₇

—

C₇×C₁₇

C₂×C₇×D₉

yes

characteristic, complemented

C₇×C₁₇

C₂×D₆₃

yes

characteristic, complemented

D₅₁

—

C₃×D₁₇

C₇×D₂₁

yes

characteristic

D₃×C₁₇

—

C₂×C₇²

—

C₇×D₇

—

C₇×D₇

—

153

—

C₂×D₂₁

—

C₂×D₃×C₇

—

357

—

C₂×D₁₇

—

Hall

C₇×C₉

—

C₇×C₉

D₇×D₁₇

yes

characteristic, complemented

C₇×C₉

C₂×C₇×D₁₇

yes

characteristic, complemented

C₃×C₁₇

C₂×C₇×D₂₁

yes

characteristic

C₇²

D₉×D₁₇

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

C₂×C₃×C₇

—

C₂×C₃×C₇

—

D₂₁

—

D₂₁

—

D₃×C₇

—

D₃×C₇

—

357

—

C₂×D₉

—

119

Hall

C₂×C₁₇

—

100

D₁₇

—

101

D₁₇

C₇×D₆₃

yes

characteristic, direct product factor

102

C₂²×C₇

—

1071

—

103

C₂×D₇

—

153

—

104

C₃×C₇

—

105

C₃×C₇

D₃×C₇×D₁₇

yes

characteristic

106

C₃×C₇

D₁₇×D₂₁

yes

characteristic

107

C₂×C₉

—

108

D₉

—

109

D₉

—

119

—

110

C₁₇

C₂×C₇×D₆₃

yes

characteristic, complemented, Sylow

111

C₂×C₇

—

112

C₂×C₇

—

113

C₂×C₇

—

114

C₂×C₇

—

1071

—

115

C₂×C₇

—

116

D₇

—

117

D₇

—

153

—

118

C₂×D₃

—

357

—

119

C₉

C₇×D₇×D₁₇

yes

characteristic, complemented, Sylow

120

C₇

—

121

C₇

C₇×D₉×D₁₇

yes

characteristic, complemented

122

C₇

D₁₇×D₆₃

yes

characteristic, direct product factor

123

C₂×C₃

—

124

D₃

—

125

D₃

—

357

—

126

C₂²

—

1071

Sylow

127

C₃

C₇×D₁₇×D₂₁

yes

characteristic

128

C₂

—

129

C₂

—

1071

—

130

C₂

—

	subgroup	G/H
center, hypercenter	C₇	D₁₇×D₆₃
commutator, cosocle, hypocenter	C₇×C₉×C₁₇	C₂²×C₇
Frattini	C₃	C₇×D₁₇×D₂₁
Fitting	C₇²×C₉×C₁₇	C₂²
socle	C₃×C₇²×C₁₇	C₂×D₃
LCS	G ▷ C₇×C₉×C₁₇
UCS	C₁ ◁ C₇

subgroup

G/H

center, hypercenter

C₇

D₁₇×D₆₃

commutator, cosocle, hypocenter

C₇×C₉×C₁₇

C₂²×C₇

Frattini

C₃

C₇×D₁₇×D₂₁

Fitting

C₇²×C₉×C₁₇

C₂²

socle

C₃×C₇²×C₁₇

C₂×D₃

LCS

G ▷ C₇×C₉×C₁₇

UCS

C₁ ◁ C₇

N	H ≅ G/N	type
1. C₇×D₁₀₇₁	C₂	semidirect
2. C₇×C₁₇×D₆₃	C₂	semidirect
3. C₇²×C₉×D₁₇	C₂	semidirect
5. C₇²×C₉×C₁₇	C₂²	semidirect
9. D₁₇×D₆₃	C₇	direct (N × H)
13. D₁₀₇₁	C₂×C₇	semidirect
14. C₁₇×D₆₃	C₂×C₇	semidirect
18. C₇×C₉×D₁₇	D₇	semidirect
19. C₇×C₉×D₁₇	C₂×C₇	semidirect
23. C₇²×D₁₇	D₉	semidirect
27. C₇×C₉×C₁₇	C₂×D₇	semidirect
28. C₇×C₉×C₁₇	C₂²×C₇	semidirect
31. C₇×D₆₃	D₁₇	direct (N × H)
32. C₇²×C₁₇	C₂×D₉	semidirect
44. C₇²×C₉	C₂×D₁₇	semidirect
49. C₉×D₁₇	C₇×D₇	semidirect
60. C₇×D₁₇	C₇×D₉	semidirect
61. C₇×D₁₇	D₆₃	direct (N × H)
65. C₉×C₁₇	C₂×C₇×D₇	semidirect
71. D₆₃	C₇×D₁₇	direct (N × H)
75. C₇×C₁₇	C₂×C₇×D₉	semidirect
76. C₇×C₁₇	C₂×D₆₃	semidirect
87. C₇×C₉	D₇×D₁₇	semidirect
88. C₇×C₉	C₂×C₇×D₁₇	semidirect
90. C₇²	D₉×D₁₇	semidirect
101. D₁₇	C₇×D₆₃	direct (N × H)
110. C₁₇	C₂×C₇×D₆₃	semidirect
119. C₉	C₇×D₇×D₁₇	semidirect
121. C₇	C₇×D₉×D₁₇	semidirect
122. C₇	D₁₇×D₆₃	direct (N × H)

H ≅ G/N

type

1. C₇×D₁₀₇₁

C₂

semidirect

2. C₇×C₁₇×D₆₃

C₂

semidirect

3. C₇²×C₉×D₁₇

C₂

semidirect

5. C₇²×C₉×C₁₇

C₂²

semidirect

9. D₁₇×D₆₃

C₇

direct (N × H)

13. D₁₀₇₁

C₂×C₇

semidirect

14. C₁₇×D₆₃

C₂×C₇

semidirect

18. C₇×C₉×D₁₇

D₇

semidirect

19. C₇×C₉×D₁₇

C₂×C₇

semidirect

23. C₇²×D₁₇

D₉

semidirect

27. C₇×C₉×C₁₇

C₂×D₇

semidirect

28. C₇×C₉×C₁₇

C₂²×C₇

semidirect

31. C₇×D₆₃

D₁₇

direct (N × H)

32. C₇²×C₁₇

C₂×D₉

semidirect

44. C₇²×C₉

C₂×D₁₇

semidirect

49. C₉×D₁₇

C₇×D₇

semidirect

60. C₇×D₁₇

C₇×D₉

semidirect

61. C₇×D₁₇

D₆₃

direct (N × H)

65. C₉×C₁₇

C₂×C₇×D₇

semidirect

71. D₆₃

C₇×D₁₇

direct (N × H)

75. C₇×C₁₇

C₂×C₇×D₉

semidirect

76. C₇×C₁₇

C₂×D₆₃

semidirect

87. C₇×C₉

D₇×D₁₇

semidirect

88. C₇×C₉

C₂×C₇×D₁₇

semidirect

90. C₇²

D₉×D₁₇

semidirect

101. D₁₇

C₇×D₆₃

direct (N × H)

110. C₁₇

C₂×C₇×D₆₃

semidirect

119. C₉

C₇×D₇×D₁₇

semidirect

121. C₇

C₇×D₉×D₁₇

semidirect

122. C₇

D₁₇×D₆₃

direct (N × H)

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

subgroup	index
C₇×D₁₀₇₁	2
C₇×C₁₇×D₆₃	2
C₇²×C₉×D₁₇	2
C₇×D₁₇×D₂₁	3
D₁₇×D₆₃	7
C₇×D₉×D₁₇	7
C₂×C₇×D₆₃	17

subgroup

index

C₇×D₁₀₇₁

C₇×C₁₇×D₆₃

C₇²×C₉×D₁₇

C₇×D₁₇×D₂₁

D₁₇×D₆₃

C₇×D₉×D₁₇

C₂×C₇×D₆₃

Subgroups — C₇×D₁₇×D₆₃ #44472595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×D17×D63 #44472595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×D₁₇×D₆₃ #44472595