Group #44472541 — subgroups

Subgroups — D₇×C₁₇×D₇⋊C₉ #44472541

Order 29988.

128 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 54 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	D₇×C₁₇×C₇⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×(C₇⋊C₉)⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₁₇×D₇⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₇²×C₁₇	C₃	1	1	yes	maximal, characteristic
5	C₇×C₁₇×C₇⋊C₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₁₇×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic
7	C₃×C₇×D₇×C₁₇	C₂×C₃	1	1	yes	characteristic
8	C₃×C₇×D₇×C₁₇	C₂×C₃	1	1	yes	characteristic
9	C₂×C₉×D₇×C₁₇	—	7	1	no	maximal
10	C₂×C₁₇×D₇⋊C₉	—	7	1	no	maximal
11	D₇²×C₁₇	C₉	1	1	yes	characteristic, complemented, Hall
12	C₃×C₇²×C₁₇	C₂²×C₃	1	1	yes	characteristic
13	C₉×D₇×C₁₇	—	7	1	no	—
14	C₉×D₇×C₁₇	—	7	1	no	—
15	C₂×C₁₇×C₇⋊C₉	—	7	1	no	—
16	C₁₇×D₇⋊C₉	—	7	1	no	—
17	C₁₇×D₇⋊C₉	D₇	1	1	yes	characteristic, direct product factor
18	C₂×C₇×C₉×C₁₇	—	7	1	no	—
19	D₇×D₇⋊C₉	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₁₇×C₇⋊D₇	C₂×C₉	1	1	yes	characteristic, complemented
21	C₇×D₇×C₁₇	C₂×C₉	1	1	yes	characteristic, complemented
22	C₇×D₇×C₁₇	C₂×C₉	1	1	yes	characteristic, complemented
23	C₂×C₃×D₇×C₁₇	—	7	1	no	—
24	C₂×C₃×D₇×C₁₇	—	7	1	no	—
25	C₇×C₉×C₁₇	—	7	1	no	—
26	C₁₇×C₇⋊C₉	C₂×D₇	1	1	yes	characteristic, complemented
27	(C₇⋊C₉)⋊D₇	C₂×C₁₇	1	1	yes	characteristic, complemented
28	D₇×C₇⋊C₉	C₂×C₁₇	1	1	yes	characteristic, complemented
29	C₇×D₇⋊C₉	C₂×C₁₇	1	1	yes	characteristic, complemented
30	C₇²×C₁₇	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
31	C₂×C₃×C₇×C₁₇	—	7	1	no	—
32	C₂×C₃×C₇×C₁₇	—	7	1	no	—
33	C₃×D₇×C₁₇	—	7	1	no	—
34	C₃×D₇×C₁₇	—	7	1	no	—
35	C₃×D₇×C₁₇	—	42	1	no	—
36	C₃×D₇×C₁₇	AGL(1,𝔽₇)	1	1	yes	characteristic
37	C₃×D₇×C₁₇	C₃×D₇	1	1	yes	characteristic
38	C₂²×C₉×C₁₇	—	49	1	no	Hall
39	C₃×D₇²	C₃×C₁₇	1	1	yes	characteristic
40	C₂×D₇×C₁₇	—	7	1	no	—
41	C₂×D₇×C₁₇	—	7	1	no	—
42	C₇×C₇⋊C₉	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
43	C₃×C₇×C₁₇	—	6	1	no	—
44	C₃×C₇×C₁₇	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
45	C₃×C₇×C₁₇	C₂×C₃×D₇	1	1	yes	characteristic
46	C₂×C₉×C₁₇	—	7	1	no	—
47	C₂×C₉×C₁₇	—	49	1	no	—
48	C₂×C₉×C₁₇	—	49	1	no	—
49	C₃×C₇⋊D₇	C₂×C₃×C₁₇	1	1	yes	characteristic
50	C₃×C₇×D₇	C₂×C₃×C₁₇	1	1	yes	characteristic
51	C₃×C₇×D₇	C₂×C₃×C₁₇	1	1	yes	characteristic
52	C₂×C₉×D₇	—	7	1	no	—
53	C₂×D₇⋊C₉	—	7	1	no	—
54	C₂×C₇×C₁₇	—	7	1	no	—
55	C₂×C₇×C₁₇	—	7	1	no	—
56	D₇×C₁₇	—	7	1	no	—
57	D₇×C₁₇	—	7	1	no	—
58	D₇×C₁₇	—	42	1	no	—
59	D₇×C₁₇	D₇⋊C₉	1	1	yes	characteristic, direct product factor
60	D₇×C₁₇	C₉×D₇	1	1	yes	characteristic, complemented
61	C₂²×C₃×C₁₇	—	49	1	no	—
62	D₇²	C₉×C₁₇	1	1	yes	characteristic, complemented, Hall
63	C₉×C₁₇	—	7	1	no	Hall
64	C₃×C₇²	C₂²×C₃×C₁₇	1	1	yes	characteristic
65	C₂×C₇×C₉	—	7	1	no	—
66	C₉×D₇	—	7	1	no	—
67	C₉×D₇	—	7	1	no	—
68	C₂×C₇⋊C₉	—	7	1	no	—
69	D₇⋊C₉	—	7	1	no	—
70	D₇⋊C₉	D₇×C₁₇	1	1	yes	characteristic, direct product factor
71	C₇×C₁₇	—	6	1	no	—
72	C₇×C₁₇	C₂×D₇⋊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	C₂×C₃×C₁₇	—	7	1	no	—
76	C₂×C₃×C₁₇	—	49	1	no	—
77	C₇⋊D₇	C₂×C₉×C₁₇	1	1	yes	characteristic, complemented
78	C₇×D₇	C₂×C₉×C₁₇	1	1	yes	characteristic, complemented
79	C₇×D₇	C₂×C₉×C₁₇	1	1	yes	characteristic, complemented
80	C₂×C₃×D₇	—	7	1	no	—
81	C₂×C₃×D₇	—	7	1	no	—
82	C₂²×C₁₇	—	49	1	no	Hall
83	C₇×C₉	—	7	1	no	—
84	C₇⋊C₉	C₂×D₇×C₁₇	1	1	yes	characteristic, complemented
85	C₃×C₁₇	D₇×AGL(1,𝔽₇)	1	1	yes	characteristic
86	C₇²	C₂²×C₉×C₁₇	1	1	yes	characteristic, complemented, Sylow
87	C₂×C₃×C₇	—	7	1	no	—
88	C₂×C₃×C₇	—	7	1	no	—
89	C₃×D₇	—	7	1	no	—
90	C₃×D₇	—	7	1	no	—
91	C₃×D₇	—	42	1	no	—
92	C₃×D₇	C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic
93	C₃×D₇	C₃×D₇×C₁₇	1	1	yes	characteristic
94	C₂²×C₉	—	49	1	no	Hall
95	C₂×C₁₇	—	7	1	no	—
96	C₂×C₁₇	—	7	1	no	—
97	C₂×C₁₇	—	49	1	no	—
98	C₂×D₇	—	7	1	no	—
99	C₂×D₇	—	7	1	no	—
100	C₃×C₇	—	6	1	no	—
101	C₃×C₇	C₂×C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic
102	C₃×C₇	C₂×C₃×D₇×C₁₇	1	1	yes	characteristic
103	C₂×C₉	—	7	1	no	—
104	C₂×C₉	—	49	1	no	—
105	C₂×C₉	—	49	1	no	—
106	C₁₇	D₇×D₇⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
107	C₂×C₇	—	7	1	no	—
108	C₂×C₇	—	7	1	no	—
109	D₇	—	7	1	no	—
110	D₇	—	7	1	no	—
111	D₇	—	42	1	no	—
112	D₇	C₁₇×D₇⋊C₉	1	1	yes	characteristic, direct product factor
113	D₇	C₉×D₇×C₁₇	1	1	yes	characteristic, complemented
114	C₂²×C₃	—	49	1	no	—
115	C₉	—	7	1	no	Sylow
116	C₇	—	6	1	no	—
117	C₇	C₂×C₁₇×D₇⋊C₉	1	1	yes	characteristic, complemented
118	C₇	C₂×C₉×D₇×C₁₇	1	1	yes	characteristic, complemented
119	C₂×C₃	—	7	1	no	—
120	C₂×C₃	—	7	1	no	—
121	C₂×C₃	—	49	1	no	—
122	C₂²	—	49	1	no	Sylow
123	C₃	D₇×C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic
124	C₂	—	7	1	no	—
125	C₂	—	7	1	no	—
126	C₂	—	49	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

D₇×C₁₇×C₇⋊C₉

C₂

yes

maximal, characteristic, complemented

C₁₇×(C₇⋊C₉)⋊D₇

C₂

yes

maximal, characteristic, complemented

C₇×C₁₇×D₇⋊C₉

C₂

yes

maximal, characteristic, complemented

C₃×D₇²×C₁₇

C₃

yes

maximal, characteristic

C₇×C₁₇×C₇⋊C₉

C₂²

yes

characteristic, complemented, Hall

C₃×C₁₇×C₇⋊D₇

C₂×C₃

yes

characteristic

C₃×C₇×D₇×C₁₇

C₂×C₃

yes

characteristic

C₃×C₇×D₇×C₁₇

C₂×C₃

yes

characteristic

C₂×C₉×D₇×C₁₇

—

maximal

C₂×C₁₇×D₇⋊C₉

—

maximal

D₇²×C₁₇

C₉

yes

characteristic, complemented, Hall

C₃×C₇²×C₁₇

C₂²×C₃

yes

characteristic

C₉×D₇×C₁₇

—

C₉×D₇×C₁₇

—

C₂×C₁₇×C₇⋊C₉

—

C₁₇×D₇⋊C₉

—

C₁₇×D₇⋊C₉

D₇

yes

characteristic, direct product factor

C₂×C₇×C₉×C₁₇

—

D₇×D₇⋊C₉

C₁₇

yes

maximal, characteristic, direct product factor, Hall

C₁₇×C₇⋊D₇

C₂×C₉

yes

characteristic, complemented

C₇×D₇×C₁₇

C₂×C₉

yes

characteristic, complemented

C₇×D₇×C₁₇

C₂×C₉

yes

characteristic, complemented

C₂×C₃×D₇×C₁₇

—

C₂×C₃×D₇×C₁₇

—

C₇×C₉×C₁₇

—

C₁₇×C₇⋊C₉

C₂×D₇

yes

characteristic, complemented

(C₇⋊C₉)⋊D₇

C₂×C₁₇

yes

characteristic, complemented

D₇×C₇⋊C₉

C₂×C₁₇

yes

characteristic, complemented

C₇×D₇⋊C₉

C₂×C₁₇

yes

characteristic, complemented

C₇²×C₁₇

C₂²×C₉

yes

characteristic, complemented, Hall

C₂×C₃×C₇×C₁₇

—

C₂×C₃×C₇×C₁₇

—

C₃×D₇×C₁₇

—

C₃×D₇×C₁₇

—

C₃×D₇×C₁₇

—

C₃×D₇×C₁₇

AGL(1,𝔽₇)

yes

characteristic

C₃×D₇×C₁₇

C₃×D₇

yes

characteristic

C₂²×C₉×C₁₇

—

Hall

C₃×D₇²

C₃×C₁₇

yes

characteristic

C₂×D₇×C₁₇

—

C₂×D₇×C₁₇

—

C₇×C₇⋊C₉

C₂²×C₁₇

yes

characteristic, complemented, Hall

C₃×C₇×C₁₇

—

C₃×C₇×C₁₇

C₂×AGL(1,𝔽₇)

yes

characteristic

C₃×C₇×C₁₇

C₂×C₃×D₇

yes

characteristic

C₂×C₉×C₁₇

—

C₂×C₉×C₁₇

—

C₂×C₉×C₁₇

—

C₃×C₇⋊D₇

C₂×C₃×C₁₇

yes

characteristic

C₃×C₇×D₇

C₂×C₃×C₁₇

yes

characteristic

C₃×C₇×D₇

C₂×C₃×C₁₇

yes

characteristic

C₂×C₉×D₇

—

C₂×D₇⋊C₉

—

C₂×C₇×C₁₇

—

C₂×C₇×C₁₇

—

D₇×C₁₇

—

D₇×C₁₇

—

D₇×C₁₇

—

D₇×C₁₇

D₇⋊C₉

yes

characteristic, direct product factor

D₇×C₁₇

C₉×D₇

yes

characteristic, complemented

C₂²×C₃×C₁₇

—

D₇²

C₉×C₁₇

yes

characteristic, complemented, Hall

C₉×C₁₇

—

Hall

C₃×C₇²

C₂²×C₃×C₁₇

yes

characteristic

C₂×C₇×C₉

—

C₉×D₇

—

C₉×D₇

—

C₂×C₇⋊C₉

—

D₇⋊C₉

—

D₇⋊C₉

D₇×C₁₇

yes

characteristic, direct product factor

C₇×C₁₇

—

C₇×C₁₇

C₂×D₇⋊C₉

yes

characteristic, complemented

C₇×C₁₇

C₂×C₉×D₇

yes

characteristic, complemented

C₂×C₃×C₁₇

—

C₂×C₃×C₁₇

—

C₂×C₃×C₁₇

—

C₇⋊D₇

C₂×C₉×C₁₇

yes

characteristic, complemented

C₇×D₇

C₂×C₉×C₁₇

yes

characteristic, complemented

C₇×D₇

C₂×C₉×C₁₇

yes

characteristic, complemented

C₂×C₃×D₇

—

C₂×C₃×D₇

—

C₂²×C₁₇

—

Hall

C₇×C₉

—

C₇⋊C₉

C₂×D₇×C₁₇

yes

characteristic, complemented

C₃×C₁₇

D₇×AGL(1,𝔽₇)

yes

characteristic

C₇²

C₂²×C₉×C₁₇

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

C₂×C₃×C₇

—

C₃×D₇

—

C₃×D₇

—

C₃×D₇

—

C₃×D₇

C₁₇×AGL(1,𝔽₇)

yes

characteristic

C₃×D₇

C₃×D₇×C₁₇

yes

characteristic

C₂²×C₉

—

Hall

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×D₇

—

C₂×D₇

—

100

C₃×C₇

—

101

C₃×C₇

C₂×C₁₇×AGL(1,𝔽₇)

yes

characteristic

102

C₃×C₇

C₂×C₃×D₇×C₁₇

yes

characteristic

103

C₂×C₉

—

104

C₂×C₉

—

105

C₂×C₉

—

106

C₁₇

D₇×D₇⋊C₉

yes

characteristic, direct product factor, Sylow

107

C₂×C₇

—

108

C₂×C₇

—

109

D₇

—

110

D₇

—

111

D₇

—

112

D₇

C₁₇×D₇⋊C₉

yes

characteristic, direct product factor

113

D₇

C₉×D₇×C₁₇

yes

characteristic, complemented

114

C₂²×C₃

—

115

C₉

—

Sylow

116

C₇

—

117

C₇

C₂×C₁₇×D₇⋊C₉

yes

characteristic, complemented

118

C₇

C₂×C₉×D₇×C₁₇

yes

characteristic, complemented

119

C₂×C₃

—

120

C₂×C₃

—

121

C₂×C₃

—

122

C₂²

—

Sylow

123

C₃

D₇×C₁₇×AGL(1,𝔽₇)

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

—

	subgroup	G/H
center, hypercenter	C₃×C₁₇	D₇×AGL(1,𝔽₇)
commutator, hypocenter	C₇²	C₂²×C₉×C₁₇
Frattini	C₃	D₇×C₁₇×AGL(1,𝔽₇)
Fitting, socle	C₃×C₇²×C₁₇	C₂²×C₃
prime radical	C₃×D₇²×C₁₇	C₃
cosocle	C₃×C₇²	C₂²×C₃×C₁₇
LCS	G ▷ C₇²
UCS	C₁ ◁ C₃×C₁₇

subgroup

G/H

center, hypercenter

C₃×C₁₇

D₇×AGL(1,𝔽₇)

commutator, hypocenter

C₇²

C₂²×C₉×C₁₇

Frattini

C₃

D₇×C₁₇×AGL(1,𝔽₇)

Fitting, socle

C₃×C₇²×C₁₇

C₂²×C₃

prime radical

C₃×D₇²×C₁₇

C₃

cosocle

C₃×C₇²

C₂²×C₃×C₁₇

LCS

G ▷ C₇²

UCS

C₁ ◁ C₃×C₁₇

N	H ≅ G/N	type
1. D₇×C₁₇×C₇⋊C₉	C₂	semidirect
2. C₁₇×(C₇⋊C₉)⋊D₇	C₂	semidirect
3. C₇×C₁₇×D₇⋊C₉	C₂	semidirect
5. C₇×C₁₇×C₇⋊C₉	C₂²	semidirect
11. D₇²×C₁₇	C₉	semidirect
17. C₁₇×D₇⋊C₉	D₇	direct (N × H)
19. D₇×D₇⋊C₉	C₁₇	direct (N × H)
20. C₁₇×C₇⋊D₇	C₂×C₉	semidirect
21. C₇×D₇×C₁₇	C₂×C₉	semidirect
22. C₇×D₇×C₁₇	C₂×C₉	semidirect
26. C₁₇×C₇⋊C₉	C₂×D₇	semidirect
27. (C₇⋊C₉)⋊D₇	C₂×C₁₇	semidirect
28. D₇×C₇⋊C₉	C₂×C₁₇	semidirect
29. C₇×D₇⋊C₉	C₂×C₁₇	semidirect
30. C₇²×C₁₇	C₂²×C₉	semidirect
42. C₇×C₇⋊C₉	C₂²×C₁₇	semidirect
59. D₇×C₁₇	D₇⋊C₉	direct (N × H)
60. D₇×C₁₇	C₉×D₇	semidirect
62. D₇²	C₉×C₁₇	semidirect
70. D₇⋊C₉	D₇×C₁₇	direct (N × H)
72. C₇×C₁₇	C₂×D₇⋊C₉	semidirect
73. C₇×C₁₇	C₂×C₉×D₇	semidirect
77. C₇⋊D₇	C₂×C₉×C₁₇	semidirect
78. C₇×D₇	C₂×C₉×C₁₇	semidirect
79. C₇×D₇	C₂×C₉×C₁₇	semidirect
84. C₇⋊C₉	C₂×D₇×C₁₇	semidirect
86. C₇²	C₂²×C₉×C₁₇	semidirect
106. C₁₇	D₇×D₇⋊C₉	direct (N × H)
112. D₇	C₁₇×D₇⋊C₉	direct (N × H)
113. D₇	C₉×D₇×C₁₇	semidirect
117. C₇	C₂×C₁₇×D₇⋊C₉	semidirect
118. C₇	C₂×C₉×D₇×C₁₇	semidirect

H ≅ G/N

type

1. D₇×C₁₇×C₇⋊C₉

C₂

semidirect

2. C₁₇×(C₇⋊C₉)⋊D₇

C₂

semidirect

3. C₇×C₁₇×D₇⋊C₉

C₂

semidirect

5. C₇×C₁₇×C₇⋊C₉

C₂²

semidirect

11. D₇²×C₁₇

C₉

semidirect

17. C₁₇×D₇⋊C₉

D₇

direct (N × H)

19. D₇×D₇⋊C₉

C₁₇

direct (N × H)

20. C₁₇×C₇⋊D₇

C₂×C₉

semidirect

21. C₇×D₇×C₁₇

C₂×C₉

semidirect

22. C₇×D₇×C₁₇

C₂×C₉

semidirect

26. C₁₇×C₇⋊C₉

C₂×D₇

semidirect

27. (C₇⋊C₉)⋊D₇

C₂×C₁₇

semidirect

28. D₇×C₇⋊C₉

C₂×C₁₇

semidirect

29. C₇×D₇⋊C₉

C₂×C₁₇

semidirect

30. C₇²×C₁₇

C₂²×C₉

semidirect

42. C₇×C₇⋊C₉

C₂²×C₁₇

semidirect

59. D₇×C₁₇

D₇⋊C₉

direct (N × H)

60. D₇×C₁₇

C₉×D₇

semidirect

62. D₇²

C₉×C₁₇

semidirect

70. D₇⋊C₉

D₇×C₁₇

direct (N × H)

72. C₇×C₁₇

C₂×D₇⋊C₉

semidirect

73. C₇×C₁₇

C₂×C₉×D₇

semidirect

77. C₇⋊D₇

C₂×C₉×C₁₇

semidirect

78. C₇×D₇

C₂×C₉×C₁₇

semidirect

79. C₇×D₇

C₂×C₉×C₁₇

semidirect

84. C₇⋊C₉

C₂×D₇×C₁₇

semidirect

86. C₇²

C₂²×C₉×C₁₇

semidirect

106. C₁₇

D₇×D₇⋊C₉

direct (N × H)

112. D₇

C₁₇×D₇⋊C₉

direct (N × H)

113. D₇

C₉×D₇×C₁₇

semidirect

117. C₇

C₂×C₁₇×D₇⋊C₉

semidirect

118. C₇

C₂×C₉×D₇×C₁₇

semidirect

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

subgroup	index
D₇×C₁₇×C₇⋊C₉	2
C₁₇×(C₇⋊C₉)⋊D₇	2
C₇×C₁₇×D₇⋊C₉	2
C₃×D₇²×C₁₇	3
C₂×C₉×D₇×C₁₇	7
C₂×C₁₇×D₇⋊C₉	7
D₇×D₇⋊C₉	17

subgroup

index

D₇×C₁₇×C₇⋊C₉

C₁₇×(C₇⋊C₉)⋊D₇

C₇×C₁₇×D₇⋊C₉

C₃×D₇²×C₁₇

C₂×C₉×D₇×C₁₇

C₂×C₁₇×D₇⋊C₉

D₇×D₇⋊C₉

Subgroups — D₇×C₁₇×D₇⋊C₉ #44472541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7×C17×D7⋊C9 #44472541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×C₁₇×D₇⋊C₉ #44472541