Group #45691680 — subgroups

Subgroups — D₇×D₁₁₉₇ #45691680

Order 33516.

156 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 40 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⋊D₁₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×D₁₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₉×D₇×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×D₃₉₉	—	3	1	no	maximal
5	C₇²×C₉×C₁₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃⋊(C₇⋊D₁₃₃)	—	3	1	no	—
7	C₇×D₃₉₉	—	3	1	no	—
8	C₃×C₇×D₇×C₁₉	D₃	1	1	yes	characteristic
9	C₂×D₁₁₉₇	—	7	1	no	maximal
10	D₇×D₁₇₁	—	7	1	no	maximal
11	D₇×D₁₃₃	—	9	1	no	Hall
12	C₃×C₇²×C₁₉	C₂×D₃	1	1	yes	characteristic
13	D₁₁₉₇	—	7	1	no	—
14	D₁₁₉₇	—	7	1	no	—
15	D₁₁₉₇	—	14	3	no	—
16	D₁₁₉₇	D₇	1	1	yes	characteristic, direct product factor
17	C₇×D₁₇₁	—	7	1	no	—
18	C₉×D₇×C₁₉	D₇	1	1	yes	characteristic, complemented
19	C₂×C₇×C₉×C₁₉	—	7	1	no	—
20	C₇⋊D₁₃₃	—	9	1	no	—
21	C₇×D₁₃₃	—	9	1	no	—
22	C₇×D₇×C₁₉	D₉	1	1	yes	characteristic, complemented
23	D₇×D₆₃	—	19	1	no	maximal, Hall
24	C₂×D₃₉₉	—	21	1	no	—
25	D₇×D₅₇	—	21	1	no	—
26	C₇×C₉×C₁₉	—	2	3	no	—
27	C₇×C₉×C₁₉	C₂×D₇	1	1	yes	characteristic, complemented
28	C₇×C₉×C₁₉	C₂×D₇	1	1	yes	characteristic, complemented
29	C₇²×C₁₉	C₂×D₉	1	1	yes	characteristic, complemented, Hall
30	C₇⋊D₆₃	—	19	1	no	—
31	C₇×D₆₃	—	19	1	no	—
32	C₇×C₉×D₇	D₁₉	1	1	yes	characteristic, complemented
33	C₂×C₃×C₇×C₁₉	—	7	1	no	—
34	D₃₉₉	—	3	1	no	—
35	D₃₉₉	—	21	1	no	—
36	D₃₉₉	—	21	1	no	—
37	D₃₉₉	—	42	3	no	—
38	C₇×D₅₇	—	21	1	no	—
39	C₃×D₇×C₁₉	D₂₁	1	1	yes	characteristic
40	C₂×D₁₇₁	—	49	1	no	Hall
41	D₇×D₂₁	—	57	1	no	—
42	C₂×D₁₃₃	—	63	1	no	—
43	D₇×D₁₉	—	63	1	no	—
44	C₇²×C₉	C₂×D₁₉	1	1	yes	characteristic, complemented, Hall
45	C₃×C₇×C₁₉	—	2	3	no	—
46	C₃×C₇×C₁₉	D₃×D₇	1	1	yes	characteristic
47	C₃×C₇×C₁₉	C₂×D₂₁	1	1	yes	characteristic
48	C₂×C₉×C₁₉	—	7	1	no	—
49	D₁₇₁	—	7	1	no	—
50	D₁₇₁	—	49	1	no	—
51	C₃⋊(C₇⋊D₇)	—	57	1	no	—
52	C₇×D₂₁	—	57	1	no	—
53	C₃×C₇×D₇	D₅₇	1	1	yes	characteristic
54	C₂×C₇×C₁₉	—	7	1	no	—
55	D₁₃₃	—	9	1	no	—
56	D₁₃₃	—	63	1	no	—
57	D₁₃₃	—	63	1	no	—
58	D₁₃₃	—	126	3	no	—
59	C₇×D₁₉	—	63	1	no	—
60	D₇×C₁₉	D₆₃	1	1	yes	characteristic, complemented
61	C₂×D₆₃	—	133	1	no	—
62	D₇×D₉	—	133	1	no	—
63	C₂×D₅₇	—	147	1	no	—
64	D₇²	—	171	1	no	Hall
65	C₉×C₁₉	D₇²	1	1	yes	characteristic, complemented, Hall
66	C₃×C₇²	C₂×D₅₇	1	1	yes	characteristic
67	C₇×C₁₉	—	2	3	no	—
68	C₇×C₁₉	D₇×D₉	1	1	yes	characteristic, complemented
69	C₇×C₁₉	C₂×D₆₃	1	1	yes	characteristic, complemented
70	C₂×C₇×C₉	—	7	1	no	—
71	D₆₃	—	266	3	no	—
72	D₆₃	—	19	1	no	—
73	D₆₃	—	133	1	no	—
74	D₆₃	—	133	1	no	—
75	C₉×D₇	D₁₃₃	1	1	yes	characteristic, complemented
76	C₇×D₉	—	133	1	no	—
77	C₂×C₃×C₁₉	—	7	1	no	—
78	D₅₇	—	21	1	no	—
79	D₅₇	—	147	1	no	—
80	C₇⋊D₇	—	171	1	no	—
81	C₇×D₇	—	171	1	no	—
82	C₇×D₇	D₁₇₁	1	1	yes	characteristic, complemented
83	C₂×D₂₁	—	399	1	no	—
84	D₃×D₇	—	399	1	no	—
85	C₂×D₁₉	—	441	1	no	Hall
86	C₇×C₉	—	2	3	no	—
87	C₇×C₉	D₇×D₁₉	1	1	yes	characteristic, complemented
88	C₇×C₉	C₂×D₁₃₃	1	1	yes	characteristic, complemented
89	C₃×C₁₉	D₇×D₂₁	1	1	yes	characteristic
90	C₇²	C₂×D₁₇₁	1	1	yes	characteristic, complemented, Sylow
91	C₂×C₃×C₇	—	7	1	no	—
92	D₂₁	—	798	3	no	—
93	D₂₁	—	57	1	no	—
94	D₂₁	—	399	1	no	—
95	D₂₁	—	399	1	no	—
96	C₃×D₇	D₃₉₉	1	1	yes	characteristic
97	D₃×C₇	—	399	1	no	—
98	C₂×C₁₉	—	7	1	no	—
99	D₁₉	—	63	1	no	—
100	D₁₉	—	441	1	no	—
101	C₂×D₉	—	931	1	no	Hall
102	C₂×D₇	—	1197	1	no	—
103	C₂×D₇	—	1197	1	no	—
104	C₃×C₇	—	2	3	no	—
105	C₃×C₇	D₇×D₅₇	1	1	yes	characteristic
106	C₃×C₇	C₂×D₃₉₉	1	1	yes	characteristic
107	C₁₉	D₇×D₆₃	1	1	yes	characteristic, complemented, Sylow
108	C₂×C₉	—	7	1	no	—
109	D₉	—	133	1	no	—
110	D₉	—	931	1	no	—
111	C₂×C₇	—	7	1	no	—
112	C₂×C₇	—	1197	1	no	—
113	D₇	—	2394	3	no	—
114	D₇	—	171	1	no	—
115	D₇	—	1197	1	no	—
116	D₇	—	1197	1	no	—
117	D₇	D₁₁₉₇	1	1	yes	characteristic, direct product factor
118	C₂×D₃	—	2793	1	no	—
119	C₉	D₇×D₁₃₃	1	1	yes	characteristic, complemented, Sylow
120	C₇	—	2	3	no	—
121	C₇	D₇×D₁₇₁	1	1	yes	characteristic, complemented
122	C₇	C₂×D₁₁₉₇	1	1	yes	characteristic, complemented
123	C₂×C₃	—	7	1	no	—
124	D₃	—	399	1	no	—
125	D₃	—	2793	1	no	—
126	C₂²	—	8379	1	no	Sylow
127	C₃	D₇×D₃₉₉	1	1	yes	characteristic
128	C₂	—	7	1	no	—
129	C₂	—	1197	1	no	—
130	C₂	—	8379	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇⋊D₁₁₉₇

C₂

yes

maximal, characteristic, complemented

C₇×D₁₁₉₇

C₂

yes

maximal, characteristic, complemented

C₇×C₉×D₇×C₁₉

C₂

yes

maximal, characteristic, complemented

D₇×D₃₉₉

—

maximal

C₇²×C₉×C₁₉

C₂²

yes

characteristic, complemented, Hall

C₃⋊(C₇⋊D₁₃₃)

—

C₇×D₃₉₉

—

C₃×C₇×D₇×C₁₉

D₃

yes

characteristic

C₂×D₁₁₉₇

—

maximal

D₇×D₁₇₁

—

maximal

D₇×D₁₃₃

—

Hall

C₃×C₇²×C₁₉

C₂×D₃

yes

characteristic

D₁₁₉₇

—

D₁₁₉₇

—

D₁₁₉₇

—

D₁₁₉₇

D₇

yes

characteristic, direct product factor

C₇×D₁₇₁

—

C₉×D₇×C₁₉

D₇

yes

characteristic, complemented

C₂×C₇×C₉×C₁₉

—

C₇⋊D₁₃₃

—

C₇×D₁₃₃

—

C₇×D₇×C₁₉

D₉

yes

characteristic, complemented

D₇×D₆₃

—

maximal, Hall

C₂×D₃₉₉

—

D₇×D₅₇

—

C₇×C₉×C₁₉

—

C₇×C₉×C₁₉

C₂×D₇

yes

characteristic, complemented

C₇×C₉×C₁₉

C₂×D₇

yes

characteristic, complemented

C₇²×C₁₉

C₂×D₉

yes

characteristic, complemented, Hall

C₇⋊D₆₃

—

C₇×D₆₃

—

C₇×C₉×D₇

D₁₉

yes

characteristic, complemented

C₂×C₃×C₇×C₁₉

—

D₃₉₉

—

D₃₉₉

—

D₃₉₉

—

D₃₉₉

—

C₇×D₅₇

—

C₃×D₇×C₁₉

D₂₁

yes

characteristic

C₂×D₁₇₁

—

Hall

D₇×D₂₁

—

C₂×D₁₃₃

—

D₇×D₁₉

—

C₇²×C₉

C₂×D₁₉

yes

characteristic, complemented, Hall

C₃×C₇×C₁₉

—

C₃×C₇×C₁₉

D₃×D₇

yes

characteristic

C₃×C₇×C₁₉

C₂×D₂₁

yes

characteristic

C₂×C₉×C₁₉

—

D₁₇₁

—

D₁₇₁

—

C₃⋊(C₇⋊D₇)

—

C₇×D₂₁

—

C₃×C₇×D₇

D₅₇

yes

characteristic

C₂×C₇×C₁₉

—

D₁₃₃

—

D₁₃₃

—

D₁₃₃

—

D₁₃₃

—

126

—

C₇×D₁₉

—

D₇×C₁₉

D₆₃

yes

characteristic, complemented

C₂×D₆₃

—

133

—

D₇×D₉

—

133

—

C₂×D₅₇

—

147

—

D₇²

—

171

Hall

C₉×C₁₉

D₇²

yes

characteristic, complemented, Hall

C₃×C₇²

C₂×D₅₇

yes

characteristic

C₇×C₁₉

—

C₇×C₁₉

D₇×D₉

yes

characteristic, complemented

C₇×C₁₉

C₂×D₆₃

yes

characteristic, complemented

C₂×C₇×C₉

—

D₆₃

—

266

—

D₆₃

—

D₆₃

—

133

—

D₆₃

—

133

—

C₉×D₇

D₁₃₃

yes

characteristic, complemented

C₇×D₉

—

133

—

C₂×C₃×C₁₉

—

D₅₇

—

D₅₇

—

147

—

C₇⋊D₇

—

171

—

C₇×D₇

—

171

—

C₇×D₇

D₁₇₁

yes

characteristic, complemented

C₂×D₂₁

—

399

—

D₃×D₇

—

399

—

C₂×D₁₉

—

441

Hall

C₇×C₉

—

C₇×C₉

D₇×D₁₉

yes

characteristic, complemented

C₇×C₉

C₂×D₁₃₃

yes

characteristic, complemented

C₃×C₁₉

D₇×D₂₁

yes

characteristic

C₇²

C₂×D₁₇₁

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

D₂₁

—

798

—

D₂₁

—

D₂₁

—

399

—

D₂₁

—

399

—

C₃×D₇

D₃₉₉

yes

characteristic

D₃×C₇

—

399

—

C₂×C₁₉

—

D₁₉

—

100

D₁₉

—

441

—

101

C₂×D₉

—

931

Hall

102

C₂×D₇

—

1197

—

103

C₂×D₇

—

1197

—

104

C₃×C₇

—

105

C₃×C₇

D₇×D₅₇

yes

characteristic

106

C₃×C₇

C₂×D₃₉₉

yes

characteristic

107

C₁₉

D₇×D₆₃

yes

characteristic, complemented, Sylow

108

C₂×C₉

—

109

D₉

—

133

—

110

D₉

—

931

—

111

C₂×C₇

—

112

C₂×C₇

—

1197

—

113

D₇

—

2394

—

114

D₇

—

171

—

115

D₇

—

1197

—

116

D₇

—

1197

—

117

D₇

D₁₁₉₇

yes

characteristic, direct product factor

118

C₂×D₃

—

2793

—

119

C₉

D₇×D₁₃₃

yes

characteristic, complemented, Sylow

120

C₇

—

121

C₇

D₇×D₁₇₁

yes

characteristic, complemented

122

C₇

C₂×D₁₁₉₇

yes

characteristic, complemented

123

C₂×C₃

—

124

D₃

—

399

—

125

D₃

—

2793

—

126

C₂²

—

8379

Sylow

127

C₃

D₇×D₃₉₉

yes

characteristic

128

C₂

—

129

C₂

—

1197

—

130

C₂

—

8379

—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇²×C₉×C₁₉	C₂²
Frattini	C₃	D₇×D₃₉₉
socle	C₃×C₇²×C₁₉	C₂×D₃
LCS	G ▷ C₇²×C₉×C₁₉
UCS	C₁

subgroup

G/H

center, hypercenter

C₁

commutator, Fitting, cosocle, hypocenter

C₇²×C₉×C₁₉

C₂²

Frattini

C₃

D₇×D₃₉₉

socle

C₃×C₇²×C₁₉

C₂×D₃

LCS

G ▷ C₇²×C₉×C₁₉

UCS

C₁

N	H ≅ G/N	type
1. C₇⋊D₁₁₉₇	C₂	semidirect
2. C₇×D₁₁₉₇	C₂	semidirect
3. C₇×C₉×D₇×C₁₉	C₂	semidirect
5. C₇²×C₉×C₁₉	C₂²	semidirect
16. D₁₁₉₇	D₇	direct (N × H)
18. C₉×D₇×C₁₉	D₇	semidirect
22. C₇×D₇×C₁₉	D₉	semidirect
27. C₇×C₉×C₁₉	C₂×D₇	semidirect
28. C₇×C₉×C₁₉	C₂×D₇	semidirect
29. C₇²×C₁₉	C₂×D₉	semidirect
32. C₇×C₉×D₇	D₁₉	semidirect
44. C₇²×C₉	C₂×D₁₉	semidirect
60. D₇×C₁₉	D₆₃	semidirect
65. C₉×C₁₉	D₇²	semidirect
68. C₇×C₁₉	D₇×D₉	semidirect
69. C₇×C₁₉	C₂×D₆₃	semidirect
75. C₉×D₇	D₁₃₃	semidirect
82. C₇×D₇	D₁₇₁	semidirect
87. C₇×C₉	D₇×D₁₉	semidirect
88. C₇×C₉	C₂×D₁₃₃	semidirect
90. C₇²	C₂×D₁₇₁	semidirect
107. C₁₉	D₇×D₆₃	semidirect
117. D₇	D₁₁₉₇	direct (N × H)
119. C₉	D₇×D₁₃₃	semidirect
121. C₇	D₇×D₁₇₁	semidirect
122. C₇	C₂×D₁₁₉₇	semidirect

H ≅ G/N

type

1. C₇⋊D₁₁₉₇

C₂

semidirect

2. C₇×D₁₁₉₇

C₂

semidirect

3. C₇×C₉×D₇×C₁₉

C₂

semidirect

5. C₇²×C₉×C₁₉

C₂²

semidirect

16. D₁₁₉₇

D₇

direct (N × H)

18. C₉×D₇×C₁₉

D₇

semidirect

22. C₇×D₇×C₁₉

D₉

semidirect

27. C₇×C₉×C₁₉

C₂×D₇

semidirect

28. C₇×C₉×C₁₉

C₂×D₇

semidirect

29. C₇²×C₁₉

C₂×D₉

semidirect

32. C₇×C₉×D₇

D₁₉

semidirect

44. C₇²×C₉

C₂×D₁₉

semidirect

60. D₇×C₁₉

D₆₃

semidirect

65. C₉×C₁₉

D₇²

semidirect

68. C₇×C₁₉

D₇×D₉

semidirect

69. C₇×C₁₉

C₂×D₆₃

semidirect

75. C₉×D₇

D₁₃₃

semidirect

82. C₇×D₇

D₁₇₁

semidirect

87. C₇×C₉

D₇×D₁₉

semidirect

88. C₇×C₉

C₂×D₁₃₃

semidirect

90. C₇²

C₂×D₁₇₁

semidirect

107. C₁₉

D₇×D₆₃

semidirect

117. D₇

D₁₁₉₇

direct (N × H)

119. C₉

D₇×D₁₃₃

semidirect

121. C₇

D₇×D₁₇₁

semidirect

122. C₇

C₂×D₁₁₉₇

semidirect

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

subgroup	index
C₇⋊D₁₁₉₇	2
C₇×D₁₁₉₇	2
C₇×C₉×D₇×C₁₉	2
D₇×D₃₉₉	3
C₂×D₁₁₉₇	7
D₇×D₁₇₁	7
D₇×D₆₃	19

subgroup

index

C₇⋊D₁₁₉₇

C₇×D₁₁₉₇

C₇×C₉×D₇×C₁₉

D₇×D₃₉₉

C₂×D₁₁₉₇

D₇×D₁₇₁

D₇×D₆₃

Subgroups — D₇×D₁₁₉₇ #45691680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7×D1197 #45691680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×D₁₁₉₇ #45691680