Group #44472606 — subgroups

Subgroups — D₇×C₁₇×D₆₃ #44472606

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	C₁₇×C₇⋊D₆₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×C₁₇×D₆₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₉×D₇×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×C₁₇×D₂₁	—	3	1	no	maximal
5	C₇²×C₉×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₁₇×C₃⋊(C₇⋊D₇)	—	3	1	no	—
7	C₇×C₁₇×D₂₁	—	3	1	no	—
8	C₃×C₇×D₇×C₁₇	D₃	1	1	yes	characteristic
9	C₂×C₁₇×D₆₃	—	7	1	no	maximal
10	D₇×C₁₇×D₉	—	7	1	no	maximal
11	D₇²×C₁₇	—	9	1	no	Hall
12	C₃×C₇²×C₁₇	C₂×D₃	1	1	yes	characteristic
13	C₁₇×D₆₃	—	7	1	no	—
14	C₁₇×D₆₃	—	7	1	no	—
15	C₁₇×D₆₃	—	14	3	no	—
16	C₁₇×D₆₃	D₇	1	1	yes	characteristic, direct product factor
17	C₇×C₁₇×D₉	—	7	1	no	—
18	C₉×D₇×C₁₇	D₇	1	1	yes	characteristic, complemented
19	C₂×C₇×C₉×C₁₇	—	7	1	no	—
20	D₇×D₆₃	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₁₇×C₇⋊D₇	—	9	1	no	—
22	C₇×D₇×C₁₇	—	9	1	no	—
23	C₇×D₇×C₁₇	D₉	1	1	yes	characteristic, complemented
24	C₂×C₁₇×D₂₁	—	21	1	no	—
25	D₃×D₇×C₁₇	—	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	C₂×D₇	1	1	yes	characteristic, complemented
29	C₇⋊D₆₃	C₂×C₁₇	1	1	yes	characteristic, complemented
30	C₇×D₆₃	C₂×C₁₇	1	1	yes	characteristic, complemented
31	C₇×C₉×D₇	C₂×C₁₇	1	1	yes	characteristic, complemented
32	C₇²×C₁₇	C₂×D₉	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×C₇×C₁₇	—	7	1	no	—
34	C₁₇×D₂₁	—	3	1	no	—
35	C₁₇×D₂₁	—	21	1	no	—
36	C₁₇×D₂₁	—	21	1	no	—
37	C₁₇×D₂₁	—	42	3	no	—
38	D₃×C₇×C₁₇	—	21	1	no	—
39	C₃×D₇×C₁₇	D₂₁	1	1	yes	characteristic
40	C₂×C₁₇×D₉	—	49	1	no	Hall
41	D₇×D₂₁	—	3	1	no	—
42	C₂×D₇×C₁₇	—	63	1	no	—
43	C₂×D₇×C₁₇	—	63	1	no	—
44	C₇²×C₉	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
45	C₃×C₇×C₁₇	—	2	3	no	—
46	C₃×C₇×C₁₇	D₃×D₇	1	1	yes	characteristic
47	C₃×C₇×C₁₇	C₂×D₂₁	1	1	yes	characteristic
48	C₂×C₉×C₁₇	—	7	1	no	—
49	C₁₇×D₉	—	7	1	no	—
50	C₁₇×D₉	—	49	1	no	—
51	C₃⋊(C₇⋊D₇)	—	3	1	no	—
52	C₇×D₂₁	—	3	1	no	—
53	C₃×C₇×D₇	D₃×C₁₇	1	1	yes	characteristic
54	C₂×D₆₃	—	7	1	no	—
55	D₇×D₉	—	7	1	no	—
56	C₂×C₇×C₁₇	—	7	1	no	—
57	C₂×C₇×C₁₇	—	63	1	no	—
58	D₇×C₁₇	—	9	1	no	—
59	D₇×C₁₇	—	63	1	no	—
60	D₇×C₁₇	—	63	1	no	—
61	D₇×C₁₇	—	126	3	no	—
62	D₇×C₁₇	D₆₃	1	1	yes	characteristic, direct product factor
63	C₂×D₃×C₁₇	—	147	1	no	—
64	D₇²	—	9	1	no	Hall
65	C₉×C₁₇	D₇²	1	1	yes	characteristic, complemented, Hall
66	C₃×C₇²	C₂×D₃×C₁₇	1	1	yes	characteristic
67	C₂×C₇×C₉	—	7	1	no	—
68	D₆₃	—	7	1	no	—
69	D₆₃	—	7	1	no	—
70	D₆₃	—	14	3	no	—
71	D₆₃	D₇×C₁₇	1	1	yes	characteristic, direct product factor
72	C₉×D₇	D₇×C₁₇	1	1	yes	characteristic, complemented
73	C₇×D₉	—	7	1	no	—
74	C₇×C₁₇	—	2	3	no	—
75	C₇×C₁₇	D₇×D₉	1	1	yes	characteristic, complemented
76	C₇×C₁₇	C₂×D₆₃	1	1	yes	characteristic, complemented
77	C₂×C₃×C₁₇	—	7	1	no	—
78	D₃×C₁₇	—	21	1	no	—
79	D₃×C₁₇	—	147	1	no	—
80	C₇⋊D₇	—	9	1	no	—
81	C₇×D₇	—	9	1	no	—
82	C₇×D₇	C₁₇×D₉	1	1	yes	characteristic, complemented
83	C₂×D₂₁	—	21	1	no	—
84	D₃×D₇	—	21	1	no	—
85	C₂²×C₁₇	—	441	1	no	Hall
86	C₇×C₉	—	2	3	no	—
87	C₇×C₉	C₂×D₇×C₁₇	1	1	yes	characteristic, complemented
88	C₇×C₉	C₂×D₇×C₁₇	1	1	yes	characteristic, complemented
89	C₃×C₁₇	D₇×D₂₁	1	1	yes	characteristic
90	C₇²	C₂×C₁₇×D₉	1	1	yes	characteristic, complemented, Sylow
91	C₂×C₃×C₇	—	7	1	no	—
92	D₂₁	—	3	1	no	—
93	D₂₁	—	21	1	no	—
94	D₂₁	—	21	1	no	—
95	D₂₁	—	42	3	no	—
96	C₃×D₇	C₁₇×D₂₁	1	1	yes	characteristic
97	D₃×C₇	—	21	1	no	—
98	C₂×D₉	—	49	1	no	Hall
99	C₂×C₁₇	—	7	1	no	—
100	C₂×C₁₇	—	63	1	no	—
101	C₂×C₁₇	—	441	1	no	—
102	C₂×D₇	—	63	1	no	—
103	C₂×D₇	—	63	1	no	—
104	C₃×C₇	—	2	3	no	—
105	C₃×C₇	D₃×D₇×C₁₇	1	1	yes	characteristic
106	C₃×C₇	C₂×C₁₇×D₂₁	1	1	yes	characteristic
107	C₂×C₉	—	7	1	no	—
108	D₉	—	7	1	no	—
109	D₉	—	49	1	no	—
110	C₁₇	D₇×D₆₃	1	1	yes	characteristic, direct product factor, Sylow
111	C₂×C₇	—	7	1	no	—
112	C₂×C₇	—	63	1	no	—
113	D₇	—	9	1	no	—
114	D₇	—	63	1	no	—
115	D₇	—	63	1	no	—
116	D₇	—	126	3	no	—
117	D₇	C₁₇×D₆₃	1	1	yes	characteristic, direct product factor
118	C₂×D₃	—	147	1	no	—
119	C₉	D₇²×C₁₇	1	1	yes	characteristic, complemented, Sylow
120	C₇	—	2	3	no	—
121	C₇	D₇×C₁₇×D₉	1	1	yes	characteristic, complemented
122	C₇	C₂×C₁₇×D₆₃	1	1	yes	characteristic, complemented
123	C₂×C₃	—	7	1	no	—
124	D₃	—	21	1	no	—
125	D₃	—	147	1	no	—
126	C₂²	—	441	1	no	Sylow
127	C₃	D₇×C₁₇×D₂₁	1	1	yes	characteristic
128	C₂	—	7	1	no	—
129	C₂	—	63	1	no	—
130	C₂	—	441	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×C₇⋊D₆₃

C₂

yes

maximal, characteristic, complemented

C₇×C₁₇×D₆₃

C₂

yes

maximal, characteristic, complemented

C₇×C₉×D₇×C₁₇

C₂

yes

maximal, characteristic, complemented

D₇×C₁₇×D₂₁

—

maximal

C₇²×C₉×C₁₇

C₂²

yes

characteristic, complemented, Hall

C₁₇×C₃⋊(C₇⋊D₇)

—

C₇×C₁₇×D₂₁

—

C₃×C₇×D₇×C₁₇

D₃

yes

characteristic

C₂×C₁₇×D₆₃

—

maximal

D₇×C₁₇×D₉

—

maximal

D₇²×C₁₇

—

Hall

C₃×C₇²×C₁₇

C₂×D₃

yes

characteristic

C₁₇×D₆₃

—

C₁₇×D₆₃

—

C₁₇×D₆₃

—

C₁₇×D₆₃

D₇

yes

characteristic, direct product factor

C₇×C₁₇×D₉

—

C₉×D₇×C₁₇

D₇

yes

characteristic, complemented

C₂×C₇×C₉×C₁₇

—

D₇×D₆₃

C₁₇

yes

maximal, characteristic, direct product factor, Hall

C₁₇×C₇⋊D₇

—

C₇×D₇×C₁₇

—

C₇×D₇×C₁₇

D₉

yes

characteristic, complemented

C₂×C₁₇×D₂₁

—

D₃×D₇×C₁₇

—

C₇×C₉×C₁₇

—

C₇×C₉×C₁₇

C₂×D₇

yes

characteristic, complemented

C₇×C₉×C₁₇

C₂×D₇

yes

characteristic, complemented

C₇⋊D₆₃

C₂×C₁₇

yes

characteristic, complemented

C₇×D₆₃

C₂×C₁₇

yes

characteristic, complemented

C₇×C₉×D₇

C₂×C₁₇

yes

characteristic, complemented

C₇²×C₁₇

C₂×D₉

yes

characteristic, complemented, Hall

C₂×C₃×C₇×C₁₇

—

C₁₇×D₂₁

—

C₁₇×D₂₁

—

C₁₇×D₂₁

—

C₁₇×D₂₁

—

D₃×C₇×C₁₇

—

C₃×D₇×C₁₇

D₂₁

yes

characteristic

C₂×C₁₇×D₉

—

Hall

D₇×D₂₁

—

C₂×D₇×C₁₇

—

C₂×D₇×C₁₇

—

C₇²×C₉

C₂²×C₁₇

yes

characteristic, complemented, Hall

C₃×C₇×C₁₇

—

C₃×C₇×C₁₇

D₃×D₇

yes

characteristic

C₃×C₇×C₁₇

C₂×D₂₁

yes

characteristic

C₂×C₉×C₁₇

—

C₁₇×D₉

—

C₁₇×D₉

—

C₃⋊(C₇⋊D₇)

—

C₇×D₂₁

—

C₃×C₇×D₇

D₃×C₁₇

yes

characteristic

C₂×D₆₃

—

D₇×D₉

—

C₂×C₇×C₁₇

—

C₂×C₇×C₁₇

—

D₇×C₁₇

—

D₇×C₁₇

—

D₇×C₁₇

—

D₇×C₁₇

—

126

—

D₇×C₁₇

D₆₃

yes

characteristic, direct product factor

C₂×D₃×C₁₇

—

147

—

D₇²

—

Hall

C₉×C₁₇

D₇²

yes

characteristic, complemented, Hall

C₃×C₇²

C₂×D₃×C₁₇

yes

characteristic

C₂×C₇×C₉

—

D₆₃

—

D₆₃

—

D₆₃

—

D₆₃

D₇×C₁₇

yes

characteristic, direct product factor

C₉×D₇

D₇×C₁₇

yes

characteristic, complemented

C₇×D₉

—

C₇×C₁₇

—

C₇×C₁₇

D₇×D₉

yes

characteristic, complemented

C₇×C₁₇

C₂×D₆₃

yes

characteristic, complemented

C₂×C₃×C₁₇

—

D₃×C₁₇

—

D₃×C₁₇

—

147

—

C₇⋊D₇

—

C₇×D₇

—

C₇×D₇

C₁₇×D₉

yes

characteristic, complemented

C₂×D₂₁

—

D₃×D₇

—

C₂²×C₁₇

—

441

Hall

C₇×C₉

—

C₇×C₉

C₂×D₇×C₁₇

yes

characteristic, complemented

C₇×C₉

C₂×D₇×C₁₇

yes

characteristic, complemented

C₃×C₁₇

D₇×D₂₁

yes

characteristic

C₇²

C₂×C₁₇×D₉

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

D₂₁

—

D₂₁

—

D₂₁

—

D₂₁

—

C₃×D₇

C₁₇×D₂₁

yes

characteristic

D₃×C₇

—

C₂×D₉

—

Hall

C₂×C₁₇

—

100

C₂×C₁₇

—

101

C₂×C₁₇

—

441

—

102

C₂×D₇

—

103

C₂×D₇

—

104

C₃×C₇

—

105

C₃×C₇

D₃×D₇×C₁₇

yes

characteristic

106

C₃×C₇

C₂×C₁₇×D₂₁

yes

characteristic

107

C₂×C₉

—

108

D₉

—

109

D₉

—

110

C₁₇

D₇×D₆₃

yes

characteristic, direct product factor, Sylow

111

C₂×C₇

—

112

C₂×C₇

—

113

D₇

—

114

D₇

—

115

D₇

—

116

D₇

—

126

—

117

D₇

C₁₇×D₆₃

yes

characteristic, direct product factor

118

C₂×D₃

—

147

—

119

C₉

D₇²×C₁₇

yes

characteristic, complemented, Sylow

120

C₇

—

121

C₇

D₇×C₁₇×D₉

yes

characteristic, complemented

122

C₇

C₂×C₁₇×D₆₃

yes

characteristic, complemented

123

C₂×C₃

—

124

D₃

—

125

D₃

—

147

—

126

C₂²

—

441

Sylow

127

C₃

D₇×C₁₇×D₂₁

yes

characteristic

128

C₂

—

129

C₂

—

130

C₂

—

441

—

	subgroup	G/H
center, hypercenter	C₁₇	D₇×D₆₃
commutator, cosocle, hypocenter	C₇²×C₉	C₂²×C₁₇
Frattini	C₃	D₇×C₁₇×D₂₁
Fitting	C₇²×C₉×C₁₇	C₂²
socle	C₃×C₇²×C₁₇	C₂×D₃
LCS	G ▷ C₇²×C₉
UCS	C₁ ◁ C₁₇

subgroup

G/H

center, hypercenter

C₁₇

D₇×D₆₃

commutator, cosocle, hypocenter

C₇²×C₉

C₂²×C₁₇

Frattini

C₃

D₇×C₁₇×D₂₁

Fitting

C₇²×C₉×C₁₇

C₂²

socle

C₃×C₇²×C₁₇

C₂×D₃

LCS

G ▷ C₇²×C₉

UCS

C₁ ◁ C₁₇

N	H ≅ G/N	type
1. C₁₇×C₇⋊D₆₃	C₂	semidirect
2. C₇×C₁₇×D₆₃	C₂	semidirect
3. C₇×C₉×D₇×C₁₇	C₂	semidirect
5. C₇²×C₉×C₁₇	C₂²	semidirect
16. C₁₇×D₆₃	D₇	direct (N × H)
18. C₉×D₇×C₁₇	D₇	semidirect
20. D₇×D₆₃	C₁₇	direct (N × H)
23. C₇×D₇×C₁₇	D₉	semidirect
27. C₇×C₉×C₁₇	C₂×D₇	semidirect
28. C₇×C₉×C₁₇	C₂×D₇	semidirect
29. C₇⋊D₆₃	C₂×C₁₇	semidirect
30. C₇×D₆₃	C₂×C₁₇	semidirect
31. C₇×C₉×D₇	C₂×C₁₇	semidirect
32. C₇²×C₁₇	C₂×D₉	semidirect
44. C₇²×C₉	C₂²×C₁₇	semidirect
62. D₇×C₁₇	D₆₃	direct (N × H)
65. C₉×C₁₇	D₇²	semidirect
71. D₆₃	D₇×C₁₇	direct (N × H)
72. C₉×D₇	D₇×C₁₇	semidirect
75. C₇×C₁₇	D₇×D₉	semidirect
76. C₇×C₁₇	C₂×D₆₃	semidirect
82. C₇×D₇	C₁₇×D₉	semidirect
87. C₇×C₉	C₂×D₇×C₁₇	semidirect
88. C₇×C₉	C₂×D₇×C₁₇	semidirect
90. C₇²	C₂×C₁₇×D₉	semidirect
110. C₁₇	D₇×D₆₃	direct (N × H)
117. D₇	C₁₇×D₆₃	direct (N × H)
119. C₉	D₇²×C₁₇	semidirect
121. C₇	D₇×C₁₇×D₉	semidirect
122. C₇	C₂×C₁₇×D₆₃	semidirect

H ≅ G/N

type

1. C₁₇×C₇⋊D₆₃

C₂

semidirect

2. C₇×C₁₇×D₆₃

C₂

semidirect

3. C₇×C₉×D₇×C₁₇

C₂

semidirect

5. C₇²×C₉×C₁₇

C₂²

semidirect

16. C₁₇×D₆₃

D₇

direct (N × H)

18. C₉×D₇×C₁₇

D₇

semidirect

20. D₇×D₆₃

C₁₇

direct (N × H)

23. C₇×D₇×C₁₇

D₉

semidirect

27. C₇×C₉×C₁₇

C₂×D₇

semidirect

28. C₇×C₉×C₁₇

C₂×D₇

semidirect

29. C₇⋊D₆₃

C₂×C₁₇

semidirect

30. C₇×D₆₃

C₂×C₁₇

semidirect

31. C₇×C₉×D₇

C₂×C₁₇

semidirect

32. C₇²×C₁₇

C₂×D₉

semidirect

44. C₇²×C₉

C₂²×C₁₇

semidirect

62. D₇×C₁₇

D₆₃

direct (N × H)

65. C₉×C₁₇

D₇²

semidirect

71. D₆₃

D₇×C₁₇

direct (N × H)

72. C₉×D₇

D₇×C₁₇

semidirect

75. C₇×C₁₇

D₇×D₉

semidirect

76. C₇×C₁₇

C₂×D₆₃

semidirect

82. C₇×D₇

C₁₇×D₉

semidirect

87. C₇×C₉

C₂×D₇×C₁₇

semidirect

88. C₇×C₉

C₂×D₇×C₁₇

semidirect

90. C₇²

C₂×C₁₇×D₉

semidirect

110. C₁₇

D₇×D₆₃

direct (N × H)

117. D₇

C₁₇×D₆₃

direct (N × H)

119. C₉

D₇²×C₁₇

semidirect

121. C₇

D₇×C₁₇×D₉

semidirect

122. C₇

C₂×C₁₇×D₆₃

semidirect

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

subgroup	index
C₁₇×C₇⋊D₆₃	2
C₇×C₁₇×D₆₃	2
C₇×C₉×D₇×C₁₇	2
D₇×C₁₇×D₂₁	3
C₂×C₁₇×D₆₃	7
D₇×C₁₇×D₉	7
D₇×D₆₃	17

subgroup

index

C₁₇×C₇⋊D₆₃

C₇×C₁₇×D₆₃

C₇×C₉×D₇×C₁₇

D₇×C₁₇×D₂₁

C₂×C₁₇×D₆₃

D₇×C₁₇×D₉

D₇×D₆₃

Subgroups — D₇×C₁₇×D₆₃ #44472606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7×C17×D63 #44472606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×C₁₇×D₆₃ #44472606