Group #19100338 — subgroups

Subgroups — C₇×#10624929 #19100338

Order 5103.

960 conjugacy classes of subgroups; only the 470 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 470 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	C₃×C₇×#737	C₃	1	3	yes	maximal, complemented
2	C₃×C₇×#737	C₃	1	1	yes	maximal, characteristic, complemented
3	C₇×#6913	C₃	1	18	yes	maximal, complemented
4	C₇×#6912	C₃	1	2	yes	maximal, complemented
5	C₇×#6911	C₃	1	1	yes	maximal, characteristic, complemented
6	C₇×#6910	C₃	1	3	yes	maximal, complemented
7	C₇×#6909	C₃	1	3	yes	maximal, complemented
8	C₇×#6908	C₃	1	9	yes	maximal, complemented
9	#10624929	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₇×#737	—	3	12	no	—
11	C₇×#737	—	3	6	no	—
12	C₇×#737	—	3	6	no	—
13	C₇×#737	—	3	3	no	—
14	C₇×#737	C₃²	1	9	yes	complemented
15	C₇×#737	C₃²	1	18	yes	complemented
16	C₃×C₇×3₋¹⁺²	—	3	6	no	—
17	C₃×C₇×3₋¹⁺²	C₃²	1	6	yes	—
18	C₃×C₇×3₋¹⁺²	C₃²	1	1	yes	characteristic
19	C₃×C₇×3₋¹⁺²	C₃²	1	1	yes	characteristic
20	C₃×C₇×3₋¹⁺²	C₃²	1	6	yes	complemented
21	C₃×C₇×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
22	C₃×C₇×H(3,𝔽₃)	C₃²	1	3	yes	—
23	C₃²×C₇×C₉	—	3	3	no	—
24	C₃²×C₇×C₉	C₃²	1	1	yes	characteristic
25	C₃²×C₇×C₉	C₃²	1	3	yes	complemented
26	C₇×#733	C₃²	1	18	yes	complemented
27	C₇×#732	C₃²	1	9	yes	—
28	C₇×#731	C₃²	1	27	yes	complemented
29	C₇×AΣL(1,𝔽₂₇)	C₃²	1	27	yes	complemented
30	C₃×#737	C₃×C₇	1	3	yes	complemented
31	C₃×#737	C₃×C₇	1	1	yes	characteristic, complemented
32	#6913	C₃×C₇	1	18	yes	complemented
33	#6912	C₃×C₇	1	2	yes	complemented
34	#6911	C₃×C₇	1	1	yes	characteristic, complemented
35	#6910	C₃×C₇	1	3	yes	complemented
36	#6909	C₃×C₇	1	3	yes	complemented
37	#6908	C₃×C₇	1	9	yes	complemented
38	C₃³×C₇	—	3	3	no	—
39	C₃³×C₇	C₃³	1	3	yes	—
40	C₃³×C₇	C₃³	1	1	yes	characteristic
41	C₇×3₋¹⁺²	—	3	12	no	—
42	C₇×3₋¹⁺²	—	3	3	no	—
43	C₇×3₋¹⁺²	—	3	3	no	—
44	C₇×3₋¹⁺²	—	3	18	no	—
45	C₇×3₋¹⁺²	—	3	36	no	—
46	C₇×3₋¹⁺²	—	3	18	no	—
47	C₇×3₋¹⁺²	C₃³	1	18	yes	—
48	C₇×H(3,𝔽₃)	—	3	3	no	—
49	C₇×H(3,𝔽₃)	—	3	6	no	—
50	C₇×H(3,𝔽₃)	C₃³	1	9	yes	—
51	C₃×C₇×C₉	—	3	6	no	—
52	C₃×C₇×C₉	—	3	18	no	—
53	C₃×C₇×C₉	—	3	6	no	—
54	C₃×C₇×C₉	—	3	9	no	—
55	C₃×C₇×C₉	—	3	3	no	—
56	C₃×C₇×C₉	H(3,𝔽₃)	1	6	yes	—
57	C₃×C₇×C₉	H(3,𝔽₃)	1	3	yes	—
58	C₃×C₇×C₉	C₃³	1	3	yes	—
59	C₃×C₇×C₉	C₃³	1	3	yes	—
60	C₃×C₇×C₉	C₃³	1	2	yes	—
61	C₃×C₇×C₉	C₃³	1	1	yes	characteristic
62	#737	—	3	6	no	—
63	#737	—	3	12	no	—
64	#737	—	3	3	no	—
65	#737	—	3	6	no	—
66	#737	C₃²×C₇	1	18	yes	complemented
67	#737	C₃²×C₇	1	9	yes	complemented
68	C₃×3₋¹⁺²	—	3	6	no	—
69	C₃×3₋¹⁺²	C₃²×C₇	1	6	yes	—
70	C₃×3₋¹⁺²	C₃²×C₇	1	1	yes	characteristic
71	C₃×3₋¹⁺²	C₃²×C₇	1	1	yes	characteristic
72	C₃×3₋¹⁺²	C₃²×C₇	1	6	yes	complemented
73	C₃×H(3,𝔽₃)	C₃²×C₇	1	1	yes	characteristic
74	C₃×H(3,𝔽₃)	C₃²×C₇	1	3	yes	—
75	C₃²×C₉	—	3	3	no	—
76	C₃²×C₉	C₃²×C₇	1	1	yes	characteristic
77	C₃²×C₉	C₃²×C₇	1	3	yes	complemented
78	#733	C₃²×C₇	1	18	yes	complemented
79	#732	C₃²×C₇	1	9	yes	—
80	#731	C₃²×C₇	1	27	yes	complemented
81	AΣL(1,𝔽₂₇)	C₃²×C₇	1	27	yes	complemented
82	C₃²×C₇	—	3	3	no	—
83	C₃²×C₇	—	3	9	no	—
84	C₃²×C₇	—	9	1	no	—
85	C₃²×C₇	—	9	3	no	—
86	C₃²×C₇	—	9	9	no	—
87	C₃²×C₇	C₃×H(3,𝔽₃)	1	3	yes	—
88	C₃²×C₇	C₃⁴	1	1	yes	characteristic
89	C₇×C₉	—	3	18	no	—
90	C₇×C₉	—	3	3	no	—
91	C₇×C₉	—	3	3	no	—
92	C₇×C₉	C₃×H(3,𝔽₃)	1	6	yes	—
93	C₇×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
94	C₇×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
95	C₃³	—	3	3	no	—
96	C₃³	C₃³×C₇	1	3	yes	—
97	C₃³	C₃³×C₇	1	1	yes	characteristic
98	3₋¹⁺²	—	3	12	no	—
99	3₋¹⁺²	—	3	3	no	—
100	3₋¹⁺²	—	3	3	no	—
101	3₋¹⁺²	—	3	36	no	—
102	3₋¹⁺²	—	3	18	no	—
103	3₋¹⁺²	—	3	18	no	—
104	3₋¹⁺²	C₃³×C₇	1	18	yes	—
105	H(3,𝔽₃)	—	3	3	no	—
106	H(3,𝔽₃)	—	3	6	no	—
107	H(3,𝔽₃)	C₃³×C₇	1	9	yes	—
108	C₃×C₉	—	3	6	no	—
109	C₃×C₉	—	3	18	no	—
110	C₃×C₉	—	3	9	no	—
111	C₃×C₉	—	3	6	no	—
112	C₃×C₉	—	3	3	no	—
113	C₃×C₉	C₇×H(3,𝔽₃)	1	3	yes	—
114	C₃×C₉	C₇×H(3,𝔽₃)	1	6	yes	—
115	C₃×C₉	C₃³×C₇	1	3	yes	—
116	C₃×C₉	C₃³×C₇	1	3	yes	—
117	C₃×C₉	C₃³×C₇	1	1	yes	characteristic
118	C₃×C₉	C₃³×C₇	1	2	yes	—
119	C₃×C₇	—	3	3	no	—
120	C₃×C₇	—	3	1	no	—
121	C₃×C₇	—	9	3	no	—
122	C₃×C₇	—	9	9	no	—
123	C₃×C₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
124	C₃²	—	3	3	no	—
125	C₃²	—	3	9	no	—
126	C₃²	—	9	1	no	—
127	C₃²	—	9	3	no	—
128	C₃²	—	9	9	no	—
129	C₃²	C₃×C₇×H(3,𝔽₃)	1	3	yes	—
130	C₃²	C₃⁴×C₇	1	1	yes	characteristic
131	C₉	—	3	18	no	—
132	C₉	—	3	3	no	—
133	C₉	—	3	3	no	—
134	C₉	C₃×C₇×H(3,𝔽₃)	1	6	yes	—
135	C₉	C₃×C₇×H(3,𝔽₃)	1	2	yes	—
136	C₉	C₃×C₇×H(3,𝔽₃)	1	1	yes	characteristic
137	C₇	#10624929	1	1	yes	characteristic, direct product factor, Sylow
138	C₃	—	3	3	no	—
139	C₃	—	3	1	no	—
140	C₃	—	9	9	no	—
141	C₃	—	9	3	no	—
142	C₃	C₃²×C₇×H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₇×#737

C₃

yes

maximal, complemented

C₃×C₇×#737

C₃

yes

maximal, characteristic, complemented

C₇×#6913

C₃

yes

maximal, complemented

C₇×#6912

C₃

yes

maximal, complemented

C₇×#6911

C₃

yes

maximal, characteristic, complemented

C₇×#6910

C₃

yes

maximal, complemented

C₇×#6909

C₃

yes

maximal, complemented

C₇×#6908

C₃

yes

maximal, complemented

#10624929

C₇

yes

maximal, characteristic, direct product factor, Sylow

—

—

—

—

C₃²

yes

complemented

C₇×#737

C₃²

yes

complemented

C₃×C₇×3₋¹⁺²

—

C₃×C₇×3₋¹⁺²

C₃²

yes

—

C₃×C₇×3₋¹⁺²

C₃²

yes

characteristic

C₃×C₇×3₋¹⁺²

C₃²

yes

characteristic

C₃×C₇×3₋¹⁺²

C₃²

yes

complemented

C₃×C₇×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×C₇×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₇×C₉

—

C₃²×C₇×C₉

C₃²

yes

characteristic

C₃²×C₇×C₉

C₃²

yes

complemented

C₇×#733

C₃²

yes

complemented

C₇×#732

C₃²

yes

—

C₇×#731

C₃²

yes

complemented

C₇×AΣL(1,𝔽₂₇)

C₃²

yes

complemented

C₃×#737

C₃×C₇

yes

complemented

C₃×#737

C₃×C₇

yes

characteristic, complemented

#6913

C₃×C₇

yes

complemented

#6912

C₃×C₇

yes

complemented

#6911

C₃×C₇

yes

characteristic, complemented

#6910

C₃×C₇

yes

complemented

#6909

C₃×C₇

yes

complemented

#6908

C₃×C₇

yes

complemented

C₃³×C₇

—

C₃³×C₇

C₃³

yes

—

C₃³×C₇

C₃³

yes

characteristic

C₇×3₋¹⁺²

—

C₇×3₋¹⁺²

—

C₇×3₋¹⁺²

—

C₇×3₋¹⁺²

—

C₇×3₋¹⁺²

—

C₇×3₋¹⁺²

—

C₇×3₋¹⁺²

C₃³

yes

—

C₇×H(3,𝔽₃)

—

C₇×H(3,𝔽₃)

—

C₇×H(3,𝔽₃)

C₃³

yes

—

C₃×C₇×C₉

—

C₃×C₇×C₉

—

C₃×C₇×C₉

—

C₃×C₇×C₉

—

C₃×C₇×C₉

—

C₃×C₇×C₉

H(3,𝔽₃)

yes

—

C₃×C₇×C₉

H(3,𝔽₃)

yes

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

—

—

—

—

C₃²×C₇

yes

complemented

#737

C₃²×C₇

yes

complemented

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₇

yes

—

C₃×3₋¹⁺²

C₃²×C₇

yes

characteristic

C₃×3₋¹⁺²

C₃²×C₇

yes

characteristic

C₃×3₋¹⁺²

C₃²×C₇

yes

complemented

C₃×H(3,𝔽₃)

C₃²×C₇

yes

characteristic

C₃×H(3,𝔽₃)

C₃²×C₇

yes

—

C₃²×C₉

—

C₃²×C₉

C₃²×C₇

yes

characteristic

C₃²×C₉

C₃²×C₇

yes

complemented

#733

C₃²×C₇

yes

complemented

#732

C₃²×C₇

yes

—

#731

C₃²×C₇

yes

complemented

AΣL(1,𝔽₂₇)

C₃²×C₇

yes

complemented

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₇

C₃⁴

yes

characteristic

C₇×C₉

—

C₇×C₉

—

C₇×C₉

—

C₇×C₉

C₃×H(3,𝔽₃)

yes

—

C₇×C₉

C₃×H(3,𝔽₃)

yes

—

C₇×C₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

—

C₃³

C₃³×C₇

yes

—

C₃³

C₃³×C₇

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

100

3₋¹⁺²

—

101

3₋¹⁺²

—

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

C₃³×C₇

yes

—

105

H(3,𝔽₃)

—

106

H(3,𝔽₃)

—

107

H(3,𝔽₃)

C₃³×C₇

yes

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

C₇×H(3,𝔽₃)

yes

—

114

C₃×C₉

C₇×H(3,𝔽₃)

yes

—

115

C₃×C₉

C₃³×C₇

yes

—

116

C₃×C₉

C₃³×C₇

yes

—

117

C₃×C₉

C₃³×C₇

yes

characteristic

118

C₃×C₉

C₃³×C₇

yes

—

119

C₃×C₇

—

120

C₃×C₇

—

121

C₃×C₇

—

122

C₃×C₇

—

123

C₃×C₇

C₃²×H(3,𝔽₃)

yes

characteristic

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

C₃×C₇×H(3,𝔽₃)

yes

—

130

C₃²

C₃⁴×C₇

yes

characteristic

131

C₉

—

132

C₉

—

133

C₉

—

134

C₉

C₃×C₇×H(3,𝔽₃)

yes

—

135

C₉

C₃×C₇×H(3,𝔽₃)

yes

—

136

C₉

C₃×C₇×H(3,𝔽₃)

yes

characteristic

137

C₇

#10624929

yes

characteristic, direct product factor, Sylow

138

C₃

—

139

C₃

—

140

C₃

—

141

C₃

—

142

C₃

C₃²×C₇×H(3,𝔽₃)

yes

characteristic

	subgroup	G/H
center	C₇×C₉	C₃×H(3,𝔽₃)
commutator, Frattini	C₃²	C₃⁴×C₇
socle	C₃×C₇	C₃²×H(3,𝔽₃)
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₇×C₉ ◁ C₃²×C₇×C₉ ◁ G

subgroup

G/H

center

C₇×C₉

C₃×H(3,𝔽₃)

commutator, Frattini

C₃²

C₃⁴×C₇

socle

C₃×C₇

C₃²×H(3,𝔽₃)

LCS

G ▷ C₃² ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₇×C₉ ◁ C₃²×C₇×C₉ ◁ G

N	H ≅ G/N	type
1. C₃×C₇×#737	C₃	semidirect
2. C₃×C₇×#737	C₃	semidirect
3. C₇×#6913	C₃	semidirect
4. C₇×#6912	C₃	semidirect
5. C₇×#6911	C₃	semidirect
6. C₇×#6910	C₃	semidirect
7. C₇×#6909	C₃	semidirect
8. C₇×#6908	C₃	semidirect
9. #10624929	C₇	direct (N × H)
14. C₇×#737	C₃²	semidirect
15. C₇×#737	C₃²	semidirect
20. C₃×C₇×3₋¹⁺²	C₃²	semidirect
25. C₃²×C₇×C₉	C₃²	semidirect
26. C₇×#733	C₃²	semidirect
28. C₇×#731	C₃²	semidirect
29. C₇×AΣL(1,𝔽₂₇)	C₃²	semidirect
30. C₃×#737	C₃×C₇	semidirect
31. C₃×#737	C₃×C₇	semidirect
32. #6913	C₃×C₇	semidirect
33. #6912	C₃×C₇	semidirect
34. #6911	C₃×C₇	semidirect
35. #6910	C₃×C₇	semidirect
36. #6909	C₃×C₇	semidirect
37. #6908	C₃×C₇	semidirect
66. #737	C₃²×C₇	semidirect
67. #737	C₃²×C₇	semidirect
72. C₃×3₋¹⁺²	C₃²×C₇	semidirect
77. C₃²×C₉	C₃²×C₇	semidirect
78. #733	C₃²×C₇	semidirect
80. #731	C₃²×C₇	semidirect
81. AΣL(1,𝔽₂₇)	C₃²×C₇	semidirect
137. C₇	#10624929	direct (N × H)

H ≅ G/N

type

1. C₃×C₇×#737

C₃

semidirect

2. C₃×C₇×#737

C₃

semidirect

3. C₇×#6913

C₃

semidirect

4. C₇×#6912

C₃

semidirect

5. C₇×#6911

C₃

semidirect

6. C₇×#6910

C₃

semidirect

7. C₇×#6909

C₃

semidirect

8. C₇×#6908

C₃

semidirect

9. #10624929

C₇

direct (N × H)

14. C₇×#737

C₃²

semidirect

15. C₇×#737

C₃²

semidirect

20. C₃×C₇×3₋¹⁺²

C₃²

semidirect

25. C₃²×C₇×C₉

C₃²

semidirect

26. C₇×#733

C₃²

semidirect

28. C₇×#731

C₃²

semidirect

29. C₇×AΣL(1,𝔽₂₇)

C₃²

semidirect

30. C₃×#737

C₃×C₇

semidirect

31. C₃×#737

C₃×C₇

semidirect

32. #6913

C₃×C₇

semidirect

33. #6912

C₃×C₇

semidirect

34. #6911

C₃×C₇

semidirect

35. #6910

C₃×C₇

semidirect

36. #6909

C₃×C₇

semidirect

37. #6908

C₃×C₇

semidirect

66. #737

C₃²×C₇

semidirect

67. #737

C₃²×C₇

semidirect

72. C₃×3₋¹⁺²

C₃²×C₇

semidirect

77. C₃²×C₉

C₃²×C₇

semidirect

78. #733

C₃²×C₇

semidirect

80. #731

C₃²×C₇

semidirect

81. AΣL(1,𝔽₂₇)

C₃²×C₇

semidirect

137. C₇

#10624929

direct (N × H)

Subgroups — C₇×#10624929 #19100338

Conjugacy classes of subgroups

Complements

Subgroups — C7×#10624929 #19100338

Conjugacy classes of subgroups

Complements

Subgroups — C₇×#10624929 #19100338