Group #19100334 — subgroups

Subgroups — C₃×C₇×#6910 #19100334

Order 5103.

1088 conjugacy classes of subgroups; only the 476 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 476 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	C₃²×C₇×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
2	C₃²×C₇×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₇×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₇×#6910	C₃	1	27	yes	maximal, direct product factor
5	C₃×C₇×#731	C₃	1	9	yes	maximal, complemented
6	C₃×#6910	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₃⁴×C₇	—	3	1	no	—
8	C₃⁴×C₇	C₃²	1	1	yes	characteristic
9	C₃×C₇×3₋¹⁺²	—	3	12	no	—
10	C₃×C₇×3₋¹⁺²	—	3	6	no	—
11	C₃×C₇×3₋¹⁺²	—	3	2	no	—
12	C₃×C₇×3₋¹⁺²	—	3	2	no	—
13	C₃×C₇×3₋¹⁺²	—	3	2	no	—
14	C₃×C₇×3₋¹⁺²	—	3	1	no	—
15	C₃×C₇×3₋¹⁺²	C₃²	1	6	yes	complemented
16	C₃×C₇×3₋¹⁺²	C₃²	1	18	yes	complemented
17	C₃×C₇×3₋¹⁺²	C₃²	1	9	yes	complemented
18	C₃×C₇×H(3,𝔽₃)	—	3	6	no	—
19	C₃×C₇×H(3,𝔽₃)	—	3	2	no	—
20	C₃×C₇×H(3,𝔽₃)	C₃²	1	3	yes	—
21	C₃×C₇×H(3,𝔽₃)	C₃²	1	9	yes	—
22	C₃²×C₇×C₉	—	3	2	no	—
23	C₃²×C₇×C₉	C₃²	1	3	yes	complemented
24	C₇×#731	C₃²	1	81	yes	complemented
25	C₃²×3₋¹⁺²	C₃×C₇	1	2	yes	complemented
26	C₃²×3₋¹⁺²	C₃×C₇	1	1	yes	characteristic, complemented
27	C₃²×H(3,𝔽₃)	C₃×C₇	1	1	yes	characteristic
28	#6910	C₃×C₇	1	27	yes	direct product factor
29	C₃×#731	C₃×C₇	1	9	yes	complemented
30	C₃³×C₇	—	3	9	no	—
31	C₃³×C₇	—	3	3	no	—
32	C₃³×C₇	—	9	3	no	—
33	C₃³×C₇	—	9	9	no	—
34	C₃³×C₇	H(3,𝔽₃)	1	2	yes	—
35	C₃³×C₇	H(3,𝔽₃)	1	1	yes	characteristic
36	C₃³×C₇	H(3,𝔽₃)	1	6	yes	—
37	C₃³×C₇	C₃³	1	1	yes	characteristic
38	C₃³×C₇	C₃³	1	3	yes	—
39	C₇×3₋¹⁺²	—	3	12	no	—
40	C₇×3₋¹⁺²	—	3	18	no	—
41	C₇×3₋¹⁺²	—	3	18	no	—
42	C₇×3₋¹⁺²	—	3	18	no	—
43	C₇×3₋¹⁺²	—	3	9	no	—
44	C₇×3₋¹⁺²	C₃³	1	18	yes	complemented
45	C₇×H(3,𝔽₃)	—	3	6	no	—
46	C₇×H(3,𝔽₃)	—	3	18	no	—
47	C₇×H(3,𝔽₃)	C₃³	1	9	yes	—
48	C₃×C₇×C₉	—	3	18	no	—
49	C₃×C₇×C₉	—	3	6	no	—
50	C₃×C₇×C₉	—	3	6	no	—
51	C₃×C₇×C₉	—	3	3	no	—
52	C₃×C₇×C₉	C₃³	1	9	yes	complemented
53	C₃⁴	—	3	1	no	—
54	C₃⁴	C₃²×C₇	1	1	yes	characteristic
55	C₃×3₋¹⁺²	—	3	2	no	—
56	C₃×3₋¹⁺²	—	3	2	no	—
57	C₃×3₋¹⁺²	—	3	12	no	—
58	C₃×3₋¹⁺²	—	3	2	no	—
59	C₃×3₋¹⁺²	—	3	1	no	—
60	C₃×3₋¹⁺²	—	3	6	no	—
61	C₃×3₋¹⁺²	C₃²×C₇	1	6	yes	complemented
62	C₃×3₋¹⁺²	C₃²×C₇	1	18	yes	complemented
63	C₃×3₋¹⁺²	C₃²×C₇	1	9	yes	complemented
64	C₃×H(3,𝔽₃)	—	3	2	no	—
65	C₃×H(3,𝔽₃)	—	3	6	no	—
66	C₃×H(3,𝔽₃)	C₃²×C₇	1	3	yes	—
67	C₃×H(3,𝔽₃)	C₃²×C₇	1	9	yes	—
68	C₃²×C₉	—	3	2	no	—
69	C₃²×C₉	C₃²×C₇	1	3	yes	complemented
70	#731	C₃²×C₇	1	81	yes	complemented
71	C₃²×C₇	—	3	3	no	—
72	C₃²×C₇	—	3	6	no	—
73	C₃²×C₇	—	3	3	no	—
74	C₃²×C₇	—	3	9	no	—
75	C₃²×C₇	—	9	6	no	—
76	C₃²×C₇	—	9	3	no	—
77	C₃²×C₇	—	9	27	no	—
78	C₃²×C₇	—	9	9	no	—
79	C₃²×C₇	C₃×H(3,𝔽₃)	1	6	yes	—
80	C₃²×C₇	C₃×H(3,𝔽₃)	1	2	yes	—
81	C₃²×C₇	C₃×H(3,𝔽₃)	1	3	yes	—
82	C₃²×C₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
83	C₃²×C₇	C₃⁴	1	1	yes	characteristic
84	C₇×C₉	—	3	18	no	—
85	C₇×C₉	—	3	9	no	—
86	C₃³	—	3	9	no	—
87	C₃³	—	3	3	no	—
88	C₃³	—	9	3	no	—
89	C₃³	—	9	9	no	—
90	C₃³	C₇×H(3,𝔽₃)	1	2	yes	—
91	C₃³	C₇×H(3,𝔽₃)	1	1	yes	characteristic
92	C₃³	C₇×H(3,𝔽₃)	1	6	yes	—
93	C₃³	C₃³×C₇	1	1	yes	characteristic
94	C₃³	C₃³×C₇	1	3	yes	—
95	3₋¹⁺²	—	3	12	no	—
96	3₋¹⁺²	—	3	18	no	—
97	3₋¹⁺²	—	3	18	no	—
98	3₋¹⁺²	—	3	18	no	—
99	3₋¹⁺²	—	3	9	no	—
100	3₋¹⁺²	C₃³×C₇	1	18	yes	complemented
101	H(3,𝔽₃)	—	3	6	no	—
102	H(3,𝔽₃)	—	3	18	no	—
103	H(3,𝔽₃)	C₃³×C₇	1	9	yes	—
104	C₃×C₉	—	3	18	no	—
105	C₃×C₉	—	3	6	no	—
106	C₃×C₉	—	3	6	no	—
107	C₃×C₉	—	3	3	no	—
108	C₃×C₉	C₃³×C₇	1	9	yes	complemented
109	C₃×C₇	—	3	3	no	—
110	C₃×C₇	—	3	6	no	—
111	C₃×C₇	—	3	2	no	—
112	C₃×C₇	—	3	1	no	—
113	C₃×C₇	—	9	9	no	—
114	C₃×C₇	#6910	1	3	yes	direct product factor
115	C₃×C₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
116	C₃²	—	3	3	no	—
117	C₃²	—	3	6	no	—
118	C₃²	—	3	3	no	—
119	C₃²	—	3	9	no	—
120	C₃²	—	9	6	no	—
121	C₃²	—	9	3	no	—
122	C₃²	—	9	27	no	—
123	C₃²	—	9	9	no	—
124	C₃²	C₃×C₇×H(3,𝔽₃)	1	6	yes	—
125	C₃²	C₃×C₇×H(3,𝔽₃)	1	2	yes	—
126	C₃²	C₃×C₇×H(3,𝔽₃)	1	3	yes	—
127	C₃²	C₃×C₇×H(3,𝔽₃)	1	1	yes	characteristic
128	C₃²	C₃⁴×C₇	1	1	yes	characteristic
129	C₉	—	3	18	no	—
130	C₉	—	3	9	no	—
131	C₇	C₃×#6910	1	1	yes	characteristic, direct product factor, Sylow
132	C₃	—	3	3	no	—
133	C₃	—	3	6	no	—
134	C₃	—	3	2	no	—
135	C₃	—	3	1	no	—
136	C₃	—	9	9	no	—
137	C₃	C₇×#6910	1	3	yes	direct product factor
138	C₃	C₃²×C₇×H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₇×3₋¹⁺²

C₃

yes

maximal, complemented

C₃²×C₇×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃²×C₇×H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₇×#6910

C₃

yes

maximal, direct product factor

C₃×C₇×#731

C₃

yes

maximal, complemented

C₃×#6910

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₃⁴×C₇

—

C₃⁴×C₇

C₃²

yes

characteristic

C₃×C₇×3₋¹⁺²

—

C₃×C₇×3₋¹⁺²

—

C₃×C₇×3₋¹⁺²

—

C₃×C₇×3₋¹⁺²

—

C₃×C₇×3₋¹⁺²

—

C₃×C₇×3₋¹⁺²

—

C₃×C₇×3₋¹⁺²

C₃²

yes

complemented

C₃×C₇×3₋¹⁺²

C₃²

yes

complemented

C₃×C₇×3₋¹⁺²

C₃²

yes

complemented

C₃×C₇×H(3,𝔽₃)

—

C₃×C₇×H(3,𝔽₃)

—

C₃×C₇×H(3,𝔽₃)

C₃²

yes

—

C₃×C₇×H(3,𝔽₃)

C₃²

yes

—

C₃²×C₇×C₉

—

C₃²×C₇×C₉

C₃²

yes

complemented

C₇×#731

C₃²

yes

complemented

C₃²×3₋¹⁺²

C₃×C₇

yes

complemented

C₃²×3₋¹⁺²

C₃×C₇

yes

characteristic, complemented

C₃²×H(3,𝔽₃)

C₃×C₇

yes

characteristic

#6910

C₃×C₇

yes

direct product factor

C₃×#731

C₃×C₇

yes

complemented

C₃³×C₇

—

C₃³×C₇

—

C₃³×C₇

—

C₃³×C₇

—

C₃³×C₇

H(3,𝔽₃)

yes

—

C₃³×C₇

H(3,𝔽₃)

yes

characteristic

C₃³×C₇

H(3,𝔽₃)

yes

—

C₃³×C₇

C₃³

yes

characteristic

C₃³×C₇

C₃³

yes

—

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

complemented

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

C₃³

yes

complemented

C₃⁴

—

C₃⁴

C₃²×C₇

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₇

yes

complemented

C₃×3₋¹⁺²

C₃²×C₇

yes

complemented

C₃×3₋¹⁺²

C₃²×C₇

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃²×C₇

yes

—

C₃×H(3,𝔽₃)

C₃²×C₇

yes

—

C₃²×C₉

—

C₃²×C₉

C₃²×C₇

yes

complemented

#731

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

C₃×H(3,𝔽₃)

yes

—

C₃²×C₇

C₃×H(3,𝔽₃)

yes

—

C₃²×C₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₇

C₃⁴

yes

characteristic

C₇×C₉

—

C₇×C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₇×H(3,𝔽₃)

yes

—

C₃³

C₇×H(3,𝔽₃)

yes

characteristic

C₃³

C₇×H(3,𝔽₃)

yes

—

C₃³

C₃³×C₇

yes

characteristic

C₃³

C₃³×C₇

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

100

3₋¹⁺²

C₃³×C₇

yes

complemented

101

H(3,𝔽₃)

—

102

H(3,𝔽₃)

—

103

H(3,𝔽₃)

C₃³×C₇

yes

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

C₃³×C₇

yes

complemented

109

C₃×C₇

—

110

C₃×C₇

—

111

C₃×C₇

—

112

C₃×C₇

—

113

C₃×C₇

—

114

C₃×C₇

#6910

yes

direct product factor

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

C₃×C₇×H(3,𝔽₃)

yes

—

126

C₃²

C₃×C₇×H(3,𝔽₃)

yes

—

127

C₃²

C₃×C₇×H(3,𝔽₃)

yes

characteristic

128

C₃²

C₃⁴×C₇

yes

characteristic

129

C₉

—

130

C₉

—

131

C₇

C₃×#6910

yes

characteristic, direct product factor, Sylow

132

C₃

—

133

C₃

—

134

C₃

—

135

C₃

—

136

C₃

—

137

C₃

C₇×#6910

yes

direct product factor

138

C₃

C₃²×C₇×H(3,𝔽₃)

yes

characteristic

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

subgroup

G/H

center, socle

C₃²×C₇

C₃×H(3,𝔽₃)

commutator, Frattini

C₃²

C₃⁴×C₇

LCS

G ▷ C₃² ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₇ ◁ C₃⁴×C₇ ◁ G

N	H ≅ G/N	type
1. C₃²×C₇×3₋¹⁺²	C₃	semidirect
2. C₃²×C₇×3₋¹⁺²	C₃	semidirect
4. C₇×#6910	C₃	direct (N × H)
5. C₃×C₇×#731	C₃	semidirect
6. C₃×#6910	C₇	direct (N × H)
15. C₃×C₇×3₋¹⁺²	C₃²	semidirect
16. C₃×C₇×3₋¹⁺²	C₃²	semidirect
17. C₃×C₇×3₋¹⁺²	C₃²	semidirect
23. C₃²×C₇×C₉	C₃²	semidirect
24. C₇×#731	C₃²	semidirect
25. C₃²×3₋¹⁺²	C₃×C₇	semidirect
26. C₃²×3₋¹⁺²	C₃×C₇	semidirect
28. #6910	C₃×C₇	direct (N × H)
29. C₃×#731	C₃×C₇	semidirect
44. C₇×3₋¹⁺²	C₃³	semidirect
52. C₃×C₇×C₉	C₃³	semidirect
61. C₃×3₋¹⁺²	C₃²×C₇	semidirect
62. C₃×3₋¹⁺²	C₃²×C₇	semidirect
63. C₃×3₋¹⁺²	C₃²×C₇	semidirect
69. C₃²×C₉	C₃²×C₇	semidirect
70. #731	C₃²×C₇	semidirect
100. 3₋¹⁺²	C₃³×C₇	semidirect
108. C₃×C₉	C₃³×C₇	semidirect
114. C₃×C₇	#6910	direct (N × H)
131. C₇	C₃×#6910	direct (N × H)
137. C₃	C₇×#6910	direct (N × H)

H ≅ G/N

type

1. C₃²×C₇×3₋¹⁺²

C₃

semidirect

2. C₃²×C₇×3₋¹⁺²

C₃

semidirect

4. C₇×#6910

C₃

direct (N × H)

5. C₃×C₇×#731

C₃

semidirect

6. C₃×#6910

C₇

direct (N × H)

15. C₃×C₇×3₋¹⁺²

C₃²

semidirect

16. C₃×C₇×3₋¹⁺²

C₃²

semidirect

17. C₃×C₇×3₋¹⁺²

C₃²

semidirect

23. C₃²×C₇×C₉

C₃²

semidirect

24. C₇×#731

C₃²

semidirect

25. C₃²×3₋¹⁺²

C₃×C₇

semidirect

26. C₃²×3₋¹⁺²

C₃×C₇

semidirect

28. #6910

C₃×C₇

direct (N × H)

29. C₃×#731

C₃×C₇

semidirect

44. C₇×3₋¹⁺²

C₃³

semidirect

52. C₃×C₇×C₉

C₃³

semidirect

61. C₃×3₋¹⁺²

C₃²×C₇

semidirect

62. C₃×3₋¹⁺²

C₃²×C₇

semidirect

63. C₃×3₋¹⁺²

C₃²×C₇

semidirect

69. C₃²×C₉

C₃²×C₇

semidirect

70. #731

C₃²×C₇

semidirect

100. 3₋¹⁺²

C₃³×C₇

semidirect

108. C₃×C₉

C₃³×C₇

semidirect

114. C₃×C₇

#6910

direct (N × H)

131. C₇

C₃×#6910

direct (N × H)

137. C₃

C₇×#6910

direct (N × H)

Subgroups — C₃×C₇×#6910 #19100334

Conjugacy classes of subgroups

Complements

Subgroups — C3×C7×#6910 #19100334

Conjugacy classes of subgroups

Complements

Subgroups — C₃×C₇×#6910 #19100334