Group #19100343 — subgroups

Subgroups — C₇×#10624934 #19100343

Order 5103.

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

normal subgroups only

This group has 980 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	C₇×3₋¹⁺⁴	C₃	1	3	yes	maximal, complemented
2	C₃²×C₇×3₋¹⁺²	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×#6908	C₃	1	9	yes	maximal, complemented
8	C₃×C₇×AΣL(1,𝔽₂₇)	C₃	1	3	yes	maximal, complemented
9	#10624934	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₃⁴×C₇	C₃²	1	1	yes	characteristic
11	C₇×#737	—	3	18	no	—
12	C₇×#737	C₃²	1	27	yes	complemented
13	C₃×C₇×3₋¹⁺²	—	3	9	no	—
14	C₃×C₇×3₋¹⁺²	—	3	9	no	—
15	C₃×C₇×3₋¹⁺²	C₃²	1	6	yes	—
16	C₃×C₇×3₋¹⁺²	C₃²	1	1	yes	characteristic
17	C₃×C₇×3₋¹⁺²	C₃²	1	1	yes	characteristic
18	C₃×C₇×3₋¹⁺²	C₃²	1	1	yes	characteristic
19	C₃×C₇×3₋¹⁺²	C₃²	1	6	yes	complemented
20	C₃×C₇×H(3,𝔽₃)	C₃²	1	3	yes	—
21	C₃²×C₇×C₉	C₃²	1	3	yes	complemented
22	C₇×#733	C₃²	1	18	yes	complemented
23	C₇×#732	C₃²	1	9	yes	complemented
24	C₇×#731	C₃²	1	27	yes	complemented
25	C₇×AΣL(1,𝔽₂₇)	C₃²	1	9	yes	complemented
26	C₇×AΣL(1,𝔽₂₇)	C₃²	1	18	yes	complemented
27	3₋¹⁺⁴	C₃×C₇	1	3	yes	complemented
28	C₃²×3₋¹⁺²	C₃×C₇	1	1	yes	characteristic, complemented
29	#6913	C₃×C₇	1	18	yes	complemented
30	#6912	C₃×C₇	1	2	yes	complemented
31	#6911	C₃×C₇	1	1	yes	characteristic, complemented
32	#6910	C₃×C₇	1	3	yes	complemented
33	#6908	C₃×C₇	1	9	yes	complemented
34	C₃×AΣL(1,𝔽₂₇)	C₃×C₇	1	3	yes	complemented
35	C₃³×C₇	—	3	3	no	—
36	C₃³×C₇	—	3	3	no	—
37	C₃³×C₇	—	9	3	no	—
38	C₃³×C₇	C₃³	1	1	yes	characteristic
39	C₃³×C₇	C₃³	1	2	yes	—
40	C₃³×C₇	C₃³	1	1	yes	characteristic
41	C₇×3₋¹⁺²	—	3	6	no	—
42	C₇×3₋¹⁺²	—	3	3	no	—
43	C₇×3₋¹⁺²	—	3	3	no	—
44	C₇×3₋¹⁺²	—	3	6	no	—
45	C₇×3₋¹⁺²	—	3	18	no	—
46	C₇×3₋¹⁺²	—	3	54	no	—
47	C₇×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	—
48	C₇×3₋¹⁺²	C₃³	1	18	yes	—
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	9	no	—
53	C₃×C₇×C₉	—	3	27	no	—
54	C₃×C₇×C₉	C₃³	1	3	yes	—
55	C₃×C₇×C₉	C₃³	1	3	yes	—
56	C₃×C₇×C₉	C₃³	1	3	yes	—
57	C₃⁴	C₃²×C₇	1	1	yes	characteristic
58	#737	—	3	18	no	—
59	#737	C₃²×C₇	1	27	yes	complemented
60	C₃×3₋¹⁺²	—	3	9	no	—
61	C₃×3₋¹⁺²	—	3	9	no	—
62	C₃×3₋¹⁺²	C₃²×C₇	1	6	yes	—
63	C₃×3₋¹⁺²	C₃²×C₇	1	1	yes	characteristic
64	C₃×3₋¹⁺²	C₃²×C₇	1	1	yes	characteristic
65	C₃×3₋¹⁺²	C₃²×C₇	1	1	yes	characteristic
66	C₃×3₋¹⁺²	C₃²×C₇	1	6	yes	complemented
67	C₃×H(3,𝔽₃)	C₃²×C₇	1	3	yes	—
68	C₃²×C₉	C₃²×C₇	1	3	yes	complemented
69	#733	C₃²×C₇	1	18	yes	complemented
70	#732	C₃²×C₇	1	9	yes	complemented
71	#731	C₃²×C₇	1	27	yes	complemented
72	AΣL(1,𝔽₂₇)	C₃²×C₇	1	18	yes	complemented
73	AΣL(1,𝔽₂₇)	C₃²×C₇	1	9	yes	complemented
74	C₃²×C₇	—	3	3	no	—
75	C₃²×C₇	—	3	1	no	—
76	C₃²×C₇	—	3	9	no	—
77	C₃²×C₇	—	3	2	no	—
78	C₃²×C₇	—	9	1	no	—
79	C₃²×C₇	—	9	9	no	—
80	C₃²×C₇	—	9	2	no	—
81	C₃²×C₇	—	27	3	no	—
82	C₃²×C₇	C₃×H(3,𝔽₃)	1	3	yes	—
83	C₃²×C₇	C₃⁴	1	1	yes	characteristic
84	C₇×C₉	—	3	18	no	—
85	C₇×C₉	—	3	3	no	—
86	C₇×C₉	—	3	3	no	—
87	C₇×C₉	C₃×H(3,𝔽₃)	1	9	yes	—
88	C₃³	—	3	3	no	—
89	C₃³	—	3	3	no	—
90	C₃³	—	9	3	no	—
91	C₃³	C₃³×C₇	1	1	yes	characteristic
92	C₃³	C₃³×C₇	1	2	yes	—
93	C₃³	C₃³×C₇	1	1	yes	characteristic
94	3₋¹⁺²	—	3	6	no	—
95	3₋¹⁺²	—	3	3	no	—
96	3₋¹⁺²	—	3	3	no	—
97	3₋¹⁺²	—	3	6	no	—
98	3₋¹⁺²	—	3	18	no	—
99	3₋¹⁺²	—	3	54	no	—
100	3₋¹⁺²	C₇×H(3,𝔽₃)	1	9	yes	—
101	3₋¹⁺²	C₃³×C₇	1	18	yes	—
102	H(3,𝔽₃)	—	3	6	no	—
103	H(3,𝔽₃)	C₃³×C₇	1	9	yes	—
104	C₃×C₉	—	3	6	no	—
105	C₃×C₉	—	3	9	no	—
106	C₃×C₉	—	3	27	no	—
107	C₃×C₉	C₃³×C₇	1	3	yes	—
108	C₃×C₉	C₃³×C₇	1	3	yes	—
109	C₃×C₉	C₃³×C₇	1	3	yes	—
110	C₃×C₇	—	3	3	no	—
111	C₃×C₇	—	3	1	no	—
112	C₃×C₇	—	3	3	no	—
113	C₃×C₇	—	9	2	no	—
114	C₃×C₇	—	9	9	no	—
115	C₃×C₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
116	C₃²	—	3	3	no	—
117	C₃²	—	3	1	no	—
118	C₃²	—	3	9	no	—
119	C₃²	—	3	2	no	—
120	C₃²	—	9	1	no	—
121	C₃²	—	9	9	no	—
122	C₃²	—	9	2	no	—
123	C₃²	—	27	3	no	—
124	C₃²	C₃×C₇×H(3,𝔽₃)	1	3	yes	—
125	C₃²	C₃⁴×C₇	1	1	yes	characteristic
126	C₉	—	3	18	no	—
127	C₉	—	3	3	no	—
128	C₉	—	3	3	no	—
129	C₉	C₃×C₇×H(3,𝔽₃)	1	9	yes	—
130	C₇	#10624934	1	1	yes	characteristic, direct product factor, Sylow
131	C₃	—	3	1	no	—
132	C₃	—	3	3	no	—
133	C₃	—	3	3	no	—
134	C₃	—	9	2	no	—
135	C₃	—	9	9	no	—
136	C₃	C₃²×C₇×H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×3₋¹⁺⁴

C₃

yes

maximal, complemented

C₃²×C₇×3₋¹⁺²

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×#6908

C₃

yes

maximal, complemented

C₃×C₇×AΣL(1,𝔽₂₇)

C₃

yes

maximal, complemented

#10624934

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₃⁴×C₇

C₃²

yes

characteristic

C₇×#737

—

C₇×#737

C₃²

yes

complemented

C₃×C₇×3₋¹⁺²

—

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

characteristic

C₃×C₇×3₋¹⁺²

C₃²

yes

complemented

C₃×C₇×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₇×C₉

C₃²

yes

complemented

C₇×#733

C₃²

yes

complemented

C₇×#732

C₃²

yes

complemented

C₇×#731

C₃²

yes

complemented

C₇×AΣL(1,𝔽₂₇)

C₃²

yes

complemented

C₇×AΣL(1,𝔽₂₇)

C₃²

yes

complemented

3₋¹⁺⁴

C₃×C₇

yes

complemented

C₃²×3₋¹⁺²

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

#6908

C₃×C₇

yes

complemented

C₃×AΣL(1,𝔽₂₇)

C₃×C₇

yes

complemented

C₃³×C₇

—

C₃³×C₇

—

C₃³×C₇

—

C₃³×C₇

C₃³

yes

characteristic

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

H(3,𝔽₃)

yes

—

C₇×3₋¹⁺²

C₃³

yes

—

C₇×H(3,𝔽₃)

—

C₇×H(3,𝔽₃)

C₃³

yes

—

C₃×C₇×C₉

—

C₃×C₇×C₉

—

C₃×C₇×C₉

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

—

C₃⁴

C₃²×C₇

yes

characteristic

#737

—

#737

C₃²×C₇

yes

complemented

C₃×3₋¹⁺²

—

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

characteristic

C₃×3₋¹⁺²

C₃²×C₇

yes

complemented

C₃×H(3,𝔽₃)

C₃²×C₇

yes

—

C₃²×C₉

C₃²×C₇

yes

complemented

#733

C₃²×C₇

yes

complemented

#732

C₃²×C₇

yes

complemented

#731

C₃²×C₇

yes

complemented

AΣL(1,𝔽₂₇)

C₃²×C₇

yes

complemented

AΣL(1,𝔽₂₇)

C₃²×C₇

yes

complemented

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

—

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

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃³×C₇

yes

characteristic

C₃³

C₃³×C₇

yes

—

C₃³

C₃³×C₇

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

100

3₋¹⁺²

C₇×H(3,𝔽₃)

yes

—

101

3₋¹⁺²

C₃³×C₇

yes

—

102

H(3,𝔽₃)

—

103

H(3,𝔽₃)

C₃³×C₇

yes

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

C₃³×C₇

yes

—

108

C₃×C₉

C₃³×C₇

yes

—

109

C₃×C₉

C₃³×C₇

yes

—

110

C₃×C₇

—

111

C₃×C₇

—

112

C₃×C₇

—

113

C₃×C₇

—

114

C₃×C₇

—

115

C₃×C₇

C₃²×H(3,𝔽₃)

yes

characteristic

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

C₃×C₇×H(3,𝔽₃)

yes

—

125

C₃²

C₃⁴×C₇

yes

characteristic

126

C₉

—

127

C₉

—

128

C₉

—

129

C₉

C₃×C₇×H(3,𝔽₃)

yes

—

130

C₇

#10624934

yes

characteristic, direct product factor, Sylow

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

—

135

C₃

—

136

C₃

C₃²×C₇×H(3,𝔽₃)

yes

characteristic

	subgroup	G/H
center, socle	C₃×C₇	C₃²×H(3,𝔽₃)
commutator, Frattini	C₃²	C₃⁴×C₇
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₇ ◁ C₃×C₇×3₋¹⁺² ◁ G

subgroup

G/H

center, socle

C₃×C₇

C₃²×H(3,𝔽₃)

commutator, Frattini

C₃²

C₃⁴×C₇

LCS

G ▷ C₃² ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃×C₇ ◁ C₃×C₇×3₋¹⁺² ◁ G

N	H ≅ G/N	type
1. C₇×3₋¹⁺⁴	C₃	semidirect
2. C₃²×C₇×3₋¹⁺²	C₃	semidirect
3. C₇×#6913	C₃	semidirect
4. C₇×#6912	C₃	semidirect
5. C₇×#6911	C₃	semidirect
6. C₇×#6910	C₃	semidirect
7. C₇×#6908	C₃	semidirect
8. C₃×C₇×AΣL(1,𝔽₂₇)	C₃	semidirect
9. #10624934	C₇	direct (N × H)
12. C₇×#737	C₃²	semidirect
19. C₃×C₇×3₋¹⁺²	C₃²	semidirect
21. C₃²×C₇×C₉	C₃²	semidirect
22. C₇×#733	C₃²	semidirect
23. C₇×#732	C₃²	semidirect
24. C₇×#731	C₃²	semidirect
25. C₇×AΣL(1,𝔽₂₇)	C₃²	semidirect
26. C₇×AΣL(1,𝔽₂₇)	C₃²	semidirect
27. 3₋¹⁺⁴	C₃×C₇	semidirect
28. C₃²×3₋¹⁺²	C₃×C₇	semidirect
29. #6913	C₃×C₇	semidirect
30. #6912	C₃×C₇	semidirect
31. #6911	C₃×C₇	semidirect
32. #6910	C₃×C₇	semidirect
33. #6908	C₃×C₇	semidirect
34. C₃×AΣL(1,𝔽₂₇)	C₃×C₇	semidirect
59. #737	C₃²×C₇	semidirect
66. C₃×3₋¹⁺²	C₃²×C₇	semidirect
68. C₃²×C₉	C₃²×C₇	semidirect
69. #733	C₃²×C₇	semidirect
70. #732	C₃²×C₇	semidirect
71. #731	C₃²×C₇	semidirect
72. AΣL(1,𝔽₂₇)	C₃²×C₇	semidirect
73. AΣL(1,𝔽₂₇)	C₃²×C₇	semidirect
130. C₇	#10624934	direct (N × H)

H ≅ G/N

type

1. C₇×3₋¹⁺⁴

C₃

semidirect

2. C₃²×C₇×3₋¹⁺²

C₃

semidirect

3. C₇×#6913

C₃

semidirect

4. C₇×#6912

C₃

semidirect

5. C₇×#6911

C₃

semidirect

6. C₇×#6910

C₃

semidirect

7. C₇×#6908

C₃

semidirect

8. C₃×C₇×AΣL(1,𝔽₂₇)

C₃

semidirect

9. #10624934

C₇

direct (N × H)

12. C₇×#737

C₃²

semidirect

19. C₃×C₇×3₋¹⁺²

C₃²

semidirect

21. C₃²×C₇×C₉

C₃²

semidirect

22. C₇×#733

C₃²

semidirect

23. C₇×#732

C₃²

semidirect

24. C₇×#731

C₃²

semidirect

25. C₇×AΣL(1,𝔽₂₇)

C₃²

semidirect

26. C₇×AΣL(1,𝔽₂₇)

C₃²

semidirect

27. 3₋¹⁺⁴

C₃×C₇

semidirect

28. C₃²×3₋¹⁺²

C₃×C₇

semidirect

29. #6913

C₃×C₇

semidirect

30. #6912

C₃×C₇

semidirect

31. #6911

C₃×C₇

semidirect

32. #6910

C₃×C₇

semidirect

33. #6908

C₃×C₇

semidirect

34. C₃×AΣL(1,𝔽₂₇)

C₃×C₇

semidirect

59. #737

C₃²×C₇

semidirect

66. C₃×3₋¹⁺²

C₃²×C₇

semidirect

68. C₃²×C₉

C₃²×C₇

semidirect

69. #733

C₃²×C₇

semidirect

70. #732

C₃²×C₇

semidirect

71. #731

C₃²×C₇

semidirect

72. AΣL(1,𝔽₂₇)

C₃²×C₇

semidirect

73. AΣL(1,𝔽₂₇)

C₃²×C₇

semidirect

130. C₇

#10624934

direct (N × H)

Subgroups — C₇×#10624934 #19100343

Conjugacy classes of subgroups

Complements

Subgroups — C7×#10624934 #19100343

Conjugacy classes of subgroups

Complements

Subgroups — C₇×#10624934 #19100343