Group #44472550 — subgroups

Subgroups — D₇×D₁₇×C₇⋊C₉ #44472550

Order 29988.

128 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 50 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⋊C₉×D₁₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	D₇×C₁₇×C₇⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×D₁₇×C₇⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×D₇×D₁₇	C₃	1	1	yes	maximal, characteristic
5	C₇×C₁₇×C₇⋊C₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₇×D₁₁₉	C₂×C₃	1	1	yes	characteristic
7	C₃×C₇×D₇×C₁₇	C₂×C₃	1	1	yes	characteristic
8	C₃×C₇²×D₁₇	C₂×C₃	1	1	yes	characteristic
9	C₉×D₇×D₁₇	—	7	1	no	maximal
10	C₂×D₁₇×C₇⋊C₉	—	7	1	no	maximal
11	C₇×D₇×D₁₇	C₉	1	1	yes	characteristic, complemented, Hall
12	C₃×C₇²×C₁₇	C₂²×C₃	1	1	yes	characteristic
13	C₉×D₁₁₉	—	7	1	no	—
14	C₉×D₇×C₁₇	—	7	1	no	—
15	C₇×C₉×D₁₇	—	7	1	no	—
16	C₂×C₁₇×C₇⋊C₉	—	7	1	no	—
17	D₁₇×C₇⋊C₉	—	7	1	no	—
18	D₁₇×C₇⋊C₉	D₇	1	1	yes	characteristic, direct product factor
19	C₂×D₇×C₇⋊C₉	—	17	1	no	maximal, Hall
20	C₇×D₁₁₉	C₂×C₉	1	1	yes	characteristic, complemented
21	C₇×D₇×C₁₇	C₂×C₉	1	1	yes	characteristic, complemented
22	C₇²×D₁₇	C₂×C₉	1	1	yes	characteristic, complemented
23	C₂×C₃×C₇×D₁₇	—	7	1	no	—
24	C₃×D₇×D₁₇	ΓL(1,𝔽₈)	1	1	yes	characteristic
25	C₇×C₉×C₁₇	—	7	1	no	—
26	C₁₇×C₇⋊C₉	C₂×D₇	1	1	yes	characteristic, complemented
27	C₂×C₇×C₇⋊C₉	—	17	1	no	—
28	D₇×C₇⋊C₉	—	17	1	no	—
29	D₇×C₇⋊C₉	D₁₇	1	1	yes	characteristic, direct product factor
30	C₇²×C₁₇	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
31	C₂×C₃×C₇×C₁₇	—	7	1	no	—
32	C₃×D₁₁₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
33	C₃×D₇×C₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
34	C₃×C₇×D₁₇	—	6	1	no	—
35	C₃×C₇×D₁₇	—	7	1	no	—
36	C₃×C₇×D₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
37	C₃×C₇×D₁₇	C₃×D₇	1	1	yes	characteristic
38	C₂×C₉×D₁₇	—	49	1	no	Hall
39	C₂×C₃×C₇×D₇	—	17	1	no	—
40	C₂×C₇×D₁₇	—	7	1	no	—
41	D₇×D₁₇	C₇⋊C₉	1	1	yes	characteristic, direct product factor
42	C₇×C₇⋊C₉	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
43	C₃×C₇×C₁₇	—	6	1	no	—
44	C₃×C₇×C₁₇	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
45	C₃×C₇×C₁₇	C₂×C₃×D₇	1	1	yes	characteristic
46	C₂×C₉×C₁₇	—	49	1	no	—
47	C₉×D₁₇	—	7	1	no	—
48	C₉×D₁₇	—	49	1	no	—
49	C₂×C₃×C₇²	—	17	1	no	—
50	C₃×C₇×D₇	—	17	1	no	—
51	C₃×C₇×D₇	C₃×D₁₇	1	1	yes	characteristic
52	C₂×C₉×D₇	—	119	1	no	—
53	C₂²×C₇⋊C₉	—	119	1	no	—
54	C₂×C₇×C₁₇	—	7	1	no	—
55	D₁₁₉	C₂×C₇⋊C₉	1	1	yes	characteristic, complemented
56	C₇×D₁₇	—	6	1	no	—
57	C₇×D₁₇	—	7	1	no	—
58	C₇×D₁₇	C₂×C₇⋊C₉	1	1	yes	characteristic, complemented
59	C₇×D₁₇	C₉×D₇	1	1	yes	characteristic, complemented
60	D₇×C₁₇	C₂×C₇⋊C₉	1	1	yes	characteristic, complemented
61	C₂×C₃×D₁₇	—	7	1	no	—
62	C₂×C₇×D₇	—	17	1	no	Hall
63	C₉×C₁₇	—	7	1	no	Hall
64	C₃×C₇²	C₂×C₃×D₁₇	1	1	yes	characteristic
65	C₂×C₇×C₉	—	119	1	no	—
66	C₉×D₇	—	7	1	no	—
67	C₉×D₇	—	119	1	no	—
68	C₂×C₇⋊C₉	—	7	1	no	—
69	C₂×C₇⋊C₉	—	17	1	no	—
70	C₂×C₇⋊C₉	—	119	1	no	—
71	C₇×C₁₇	—	6	1	no	—
72	C₇×C₁₇	C₂²×C₇⋊C₉	1	1	yes	characteristic, complemented
73	C₇×C₁₇	C₂×C₉×D₇	1	1	yes	characteristic, complemented
74	C₂×C₃×C₁₇	—	7	1	no	—
75	C₃×D₁₇	—	7	1	no	—
76	C₃×D₁₇	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
77	C₂×C₇²	—	17	1	no	—
78	C₇×D₇	—	17	1	no	—
79	C₇×D₇	C₉×D₁₇	1	1	yes	characteristic, complemented
80	C₂²×C₃×C₇	—	119	1	no	—
81	C₂×C₃×D₇	—	17	1	no	—
82	C₂×D₁₇	—	7	1	no	Hall
83	C₇×C₉	—	7	1	no	—
84	C₇⋊C₉	D₇×D₁₇	1	1	yes	characteristic, direct product factor
85	C₃×C₁₇	C₂×D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
86	C₇²	C₂×C₉×D₁₇	1	1	yes	characteristic, complemented, Sylow
87	C₂×C₃×C₇	—	7	1	no	—
88	C₂×C₃×C₇	—	17	1	no	—
89	C₂×C₃×C₇	—	17	1	no	—
90	C₂×C₃×C₇	—	102	1	no	—
91	C₂×C₃×C₇	—	119	1	no	—
92	C₃×D₇	—	17	1	no	—
93	C₃×D₇	ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic
94	C₂²×C₉	—	833	1	no	Hall
95	C₂×C₁₇	—	7	1	no	—
96	D₁₇	—	7	1	no	—
97	D₁₇	D₇×C₇⋊C₉	1	1	yes	characteristic, direct product factor
98	C₂²×C₇	—	119	1	no	—
99	C₂×D₇	—	17	1	no	—
100	C₃×C₇	—	6	1	no	—
101	C₃×C₇	C₂×ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic
102	C₃×C₇	C₃×D₇×D₁₇	1	1	yes	characteristic
103	C₂×C₉	—	49	1	no	—
104	C₂×C₉	—	833	1	no	—
105	C₂×C₉	—	119	1	no	—
106	C₁₇	C₂×D₇×C₇⋊C₉	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₇	—	7	1	no	—
108	C₂×C₇	—	17	1	no	—
109	C₂×C₇	—	17	1	no	—
110	C₂×C₇	—	102	1	no	—
111	C₂×C₇	—	119	1	no	—
112	D₇	—	17	1	no	—
113	D₇	D₁₇×C₇⋊C₉	1	1	yes	characteristic, direct product factor
114	C₂²×C₃	—	119	1	no	—
115	C₉	—	7	1	no	Sylow
116	C₇	—	6	1	no	—
117	C₇	C₂×D₁₇×C₇⋊C₉	1	1	yes	characteristic, complemented
118	C₇	C₉×D₇×D₁₇	1	1	yes	characteristic, complemented
119	C₂×C₃	—	7	1	no	—
120	C₂×C₃	—	17	1	no	—
121	C₂×C₃	—	119	1	no	—
122	C₂²	—	119	1	no	Sylow
123	C₃	D₇×ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic
124	C₂	—	7	1	no	—
125	C₂	—	17	1	no	—
126	C₂	—	119	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇⋊C₉×D₁₁₉

C₂

yes

maximal, characteristic, complemented

D₇×C₁₇×C₇⋊C₉

C₂

yes

maximal, characteristic, complemented

C₇×D₁₇×C₇⋊C₉

C₂

yes

maximal, characteristic, complemented

C₃×C₇×D₇×D₁₇

C₃

yes

maximal, characteristic

C₇×C₁₇×C₇⋊C₉

C₂²

yes

characteristic, complemented, Hall

C₃×C₇×D₁₁₉

C₂×C₃

yes

characteristic

C₃×C₇×D₇×C₁₇

C₂×C₃

yes

characteristic

C₃×C₇²×D₁₇

C₂×C₃

yes

characteristic

C₉×D₇×D₁₇

—

maximal

C₂×D₁₇×C₇⋊C₉

—

maximal

C₇×D₇×D₁₇

C₉

yes

characteristic, complemented, Hall

C₃×C₇²×C₁₇

C₂²×C₃

yes

characteristic

C₉×D₁₁₉

—

C₉×D₇×C₁₇

—

C₇×C₉×D₁₇

—

C₂×C₁₇×C₇⋊C₉

—

D₁₇×C₇⋊C₉

—

D₁₇×C₇⋊C₉

D₇

yes

characteristic, direct product factor

C₂×D₇×C₇⋊C₉

—

maximal, Hall

C₇×D₁₁₉

C₂×C₉

yes

characteristic, complemented

C₇×D₇×C₁₇

C₂×C₉

yes

characteristic, complemented

C₇²×D₁₇

C₂×C₉

yes

characteristic, complemented

C₂×C₃×C₇×D₁₇

—

C₃×D₇×D₁₇

ΓL(1,𝔽₈)

yes

characteristic

C₇×C₉×C₁₇

—

C₁₇×C₇⋊C₉

C₂×D₇

yes

characteristic, complemented

C₂×C₇×C₇⋊C₉

—

D₇×C₇⋊C₉

—

D₇×C₇⋊C₉

D₁₇

yes

characteristic, direct product factor

C₇²×C₁₇

C₂²×C₉

yes

characteristic, complemented, Hall

C₂×C₃×C₇×C₁₇

—

C₃×D₁₁₉

yes

C₃×D₇×C₁₇

yes

C₃×C₇×D₁₇

—

C₃×C₇×D₁₇

—

C₃×C₇×D₁₇

C₂×ΓL(1,𝔽₈)

yes

characteristic

C₃×C₇×D₁₇

C₃×D₇

yes

characteristic

C₂×C₉×D₁₇

—

Hall

C₂×C₃×C₇×D₇

—

C₂×C₇×D₁₇

—

D₇×D₁₇

C₇⋊C₉

yes

characteristic, direct product factor

C₇×C₇⋊C₉

C₂×D₁₇

yes

characteristic, complemented, Hall

C₃×C₇×C₁₇

—

C₃×C₇×C₁₇

C₂²×ΓL(1,𝔽₈)

yes

characteristic

C₃×C₇×C₁₇

C₂×C₃×D₇

yes

characteristic

C₂×C₉×C₁₇

—

C₉×D₁₇

—

C₉×D₁₇

—

C₂×C₃×C₇²

—

C₃×C₇×D₇

—

C₃×C₇×D₇

C₃×D₁₇

yes

characteristic

C₂×C₉×D₇

—

119

—

C₂²×C₇⋊C₉

—

119

—

C₂×C₇×C₁₇

—

D₁₁₉

C₂×C₇⋊C₉

yes

characteristic, complemented

C₇×D₁₇

—

C₇×D₁₇

—

C₇×D₁₇

C₂×C₇⋊C₉

yes

characteristic, complemented

C₇×D₁₇

C₉×D₇

yes

characteristic, complemented

D₇×C₁₇

C₂×C₇⋊C₉

yes

characteristic, complemented

C₂×C₃×D₁₇

—

C₂×C₇×D₇

—

Hall

C₉×C₁₇

—

Hall

C₃×C₇²

C₂×C₃×D₁₇

yes

characteristic

C₂×C₇×C₉

—

119

—

C₉×D₇

—

C₉×D₇

—

119

—

C₂×C₇⋊C₉

—

C₂×C₇⋊C₉

—

C₂×C₇⋊C₉

—

119

—

C₇×C₁₇

—

C₇×C₁₇

C₂²×C₇⋊C₉

yes

characteristic, complemented

C₇×C₁₇

C₂×C₉×D₇

yes

characteristic, complemented

C₂×C₃×C₁₇

—

C₃×D₁₇

—

C₃×D₁₇

D₇×ΓL(1,𝔽₈)

yes

characteristic

C₂×C₇²

—

C₇×D₇

—

C₇×D₇

C₉×D₁₇

yes

characteristic, complemented

C₂²×C₃×C₇

—

119

—

C₂×C₃×D₇

—

C₂×D₁₇

—

Hall

C₇×C₉

—

C₇⋊C₉

D₇×D₁₇

yes

characteristic, direct product factor

C₃×C₁₇

C₂×D₇×ΓL(1,𝔽₈)

yes

characteristic

C₇²

C₂×C₉×D₁₇

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

C₂×C₃×C₇

—

C₂×C₃×C₇

—

C₂×C₃×C₇

—

102

—

C₂×C₃×C₇

—

119

—

C₃×D₇

—

C₃×D₇

ΓL(1,𝔽₈)×D₁₇

yes

characteristic

C₂²×C₉

—

833

Hall

C₂×C₁₇

—

D₁₇

—

D₁₇

D₇×C₇⋊C₉

yes

characteristic, direct product factor

C₂²×C₇

—

119

—

C₂×D₇

—

100

C₃×C₇

—

101

C₃×C₇

C₂×ΓL(1,𝔽₈)×D₁₇

yes

characteristic

102

C₃×C₇

C₃×D₇×D₁₇

yes

characteristic

103

C₂×C₉

—

104

C₂×C₉

—

833

—

105

C₂×C₉

—

119

—

106

C₁₇

C₂×D₇×C₇⋊C₉

yes

characteristic, complemented, Sylow

107

C₂×C₇

—

108

C₂×C₇

—

109

C₂×C₇

—

110

C₂×C₇

—

102

—

111

C₂×C₇

—

119

—

112

D₇

—

113

D₇

D₁₇×C₇⋊C₉

yes

characteristic, direct product factor

114

C₂²×C₃

—

119

—

115

C₉

—

Sylow

116

C₇

—

117

C₇

C₂×D₁₇×C₇⋊C₉

yes

characteristic, complemented

118

C₇

C₉×D₇×D₁₇

yes

characteristic, complemented

119

C₂×C₃

—

120

C₂×C₃

—

121

C₂×C₃

—

119

—

122

C₂²

—

119

Sylow

123

C₃

D₇×ΓL(1,𝔽₈)×D₁₇

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

—

119

—

	subgroup	G/H
center, Frattini, hypercenter	C₃	D₇×ΓL(1,𝔽₈)×D₁₇
commutator, hypocenter	C₇²×C₁₇	C₂²×C₉
Fitting, socle, cosocle	C₃×C₇²×C₁₇	C₂²×C₃
prime radical	C₃×C₇×D₇×D₁₇	C₃
LCS	G ▷ C₇²×C₁₇
UCS	C₁ ◁ C₃

subgroup

G/H

center, Frattini, hypercenter

C₃

D₇×ΓL(1,𝔽₈)×D₁₇

commutator, hypocenter

C₇²×C₁₇

C₂²×C₉

Fitting, socle, cosocle

C₃×C₇²×C₁₇

C₂²×C₃

prime radical

C₃×C₇×D₇×D₁₇

C₃

LCS

G ▷ C₇²×C₁₇

UCS

C₁ ◁ C₃

N	H ≅ G/N	type
1. C₇⋊C₉×D₁₁₉	C₂	semidirect
2. D₇×C₁₇×C₇⋊C₉	C₂	semidirect
3. C₇×D₁₇×C₇⋊C₉	C₂	semidirect
5. C₇×C₁₇×C₇⋊C₉	C₂²	semidirect
11. C₇×D₇×D₁₇	C₉	semidirect
18. D₁₇×C₇⋊C₉	D₇	direct (N × H)
20. C₇×D₁₁₉	C₂×C₉	semidirect
21. C₇×D₇×C₁₇	C₂×C₉	semidirect
22. C₇²×D₁₇	C₂×C₉	semidirect
26. C₁₇×C₇⋊C₉	C₂×D₇	semidirect
29. D₇×C₇⋊C₉	D₁₇	direct (N × H)
30. C₇²×C₁₇	C₂²×C₉	semidirect
41. D₇×D₁₇	C₇⋊C₉	direct (N × H)
42. C₇×C₇⋊C₉	C₂×D₁₇	semidirect
55. D₁₁₉	C₂×C₇⋊C₉	semidirect
58. C₇×D₁₇	C₂×C₇⋊C₉	semidirect
59. C₇×D₁₇	C₉×D₇	semidirect
60. D₇×C₁₇	C₂×C₇⋊C₉	semidirect
72. C₇×C₁₇	C₂²×C₇⋊C₉	semidirect
73. C₇×C₁₇	C₂×C₉×D₇	semidirect
79. C₇×D₇	C₉×D₁₇	semidirect
84. C₇⋊C₉	D₇×D₁₇	direct (N × H)
86. C₇²	C₂×C₉×D₁₇	semidirect
97. D₁₇	D₇×C₇⋊C₉	direct (N × H)
106. C₁₇	C₂×D₇×C₇⋊C₉	semidirect
113. D₇	D₁₇×C₇⋊C₉	direct (N × H)
117. C₇	C₂×D₁₇×C₇⋊C₉	semidirect
118. C₇	C₉×D₇×D₁₇	semidirect

H ≅ G/N

type

1. C₇⋊C₉×D₁₁₉

C₂

semidirect

2. D₇×C₁₇×C₇⋊C₉

C₂

semidirect

3. C₇×D₁₇×C₇⋊C₉

C₂

semidirect

5. C₇×C₁₇×C₇⋊C₉

C₂²

semidirect

11. C₇×D₇×D₁₇

C₉

semidirect

18. D₁₇×C₇⋊C₉

D₇

direct (N × H)

20. C₇×D₁₁₉

C₂×C₉

semidirect

21. C₇×D₇×C₁₇

C₂×C₉

semidirect

22. C₇²×D₁₇

C₂×C₉

semidirect

26. C₁₇×C₇⋊C₉

C₂×D₇

semidirect

29. D₇×C₇⋊C₉

D₁₇

direct (N × H)

30. C₇²×C₁₇

C₂²×C₉

semidirect

41. D₇×D₁₇

C₇⋊C₉

direct (N × H)

42. C₇×C₇⋊C₉

C₂×D₁₇

semidirect

55. D₁₁₉

C₂×C₇⋊C₉

semidirect

58. C₇×D₁₇

C₂×C₇⋊C₉

semidirect

59. C₇×D₁₇

C₉×D₇

semidirect

60. D₇×C₁₇

C₂×C₇⋊C₉

semidirect

72. C₇×C₁₇

C₂²×C₇⋊C₉

semidirect

73. C₇×C₁₇

C₂×C₉×D₇

semidirect

79. C₇×D₇

C₉×D₁₇

semidirect

84. C₇⋊C₉

D₇×D₁₇

direct (N × H)

86. C₇²

C₂×C₉×D₁₇

semidirect

97. D₁₇

D₇×C₇⋊C₉

direct (N × H)

106. C₁₇

C₂×D₇×C₇⋊C₉

semidirect

113. D₇

D₁₇×C₇⋊C₉

direct (N × H)

117. C₇

C₂×D₁₇×C₇⋊C₉

semidirect

118. C₇

C₉×D₇×D₁₇

semidirect

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

subgroup	index
C₇⋊C₉×D₁₁₉	2
D₇×C₁₇×C₇⋊C₉	2
C₇×D₁₇×C₇⋊C₉	2
C₃×C₇×D₇×D₁₇	3
C₉×D₇×D₁₇	7
C₂×D₁₇×C₇⋊C₉	7
C₂×D₇×C₇⋊C₉	17

subgroup

index

C₇⋊C₉×D₁₁₉

D₇×C₁₇×C₇⋊C₉

C₇×D₁₇×C₇⋊C₉

C₃×C₇×D₇×D₁₇

C₉×D₇×D₁₇

C₂×D₁₇×C₇⋊C₉

C₂×D₇×C₇⋊C₉

Subgroups — D₇×D₁₇×C₇⋊C₉ #44472550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7×D17×C7⋊C9 #44472550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×D₁₇×C₇⋊C₉ #44472550