Group #19100288 — subgroups

Subgroups — C₇×#10624879 #19100288

Order 5103.

758 conjugacy classes of subgroups; only the 494 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 494 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	1	yes	maximal, characteristic
2	C₇×#6900	C₃	1	3	yes	maximal, complemented
3	C₇×#6899	C₃	1	6	yes	maximal, complemented
4	C₇×#6896	C₃	1	9	yes	maximal, complemented
5	C₇×#6894	C₃	1	6	yes	maximal, complemented
6	C₇×C₉⋊H(3,𝔽₃)	C₃	1	3	yes	maximal
7	C₇×C₉×3₋¹⁺²	C₃	1	9	yes	maximal, complemented
8	C₇×#6887	C₃	1	2	yes	maximal, complemented
9	C₃×C₇×C₉²	C₃	1	1	yes	maximal, characteristic, complemented
10	#10624879	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₇×#737	—	3	9	no	—
12	C₃×C₇×3₋¹⁺²	C₃²	1	9	yes	—
13	C₃×C₇×3₋¹⁺²	C₃²	1	2	yes	—
14	C₃×C₇×3₋¹⁺²	C₃²	1	3	yes	—
15	C₃×C₇×3₋¹⁺²	C₃²	1	3	yes	—
16	C₃×C₇×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
17	C₃²×C₇×C₉	C₃²	1	3	yes	—
18	C₃²×C₇×C₉	C₃²	1	3	yes	—
19	C₃²×C₇×C₉	C₃²	1	1	yes	characteristic
20	C₇×C₉⋊C₉	C₃²	1	6	yes	—
21	C₇×C₉⋊C₉	C₃²	1	6	yes	—
22	C₇×C₉⋊C₉	C₃²	1	36	yes	—
23	C₇×C₉⋊C₉	C₃²	1	18	yes	—
24	C₇×C₃²⋊C₉	C₃²	1	6	yes	—
25	C₇×C₃²⋊C₉	C₃²	1	18	yes	—
26	C₇×C₉²	C₃²	1	3	yes	—
27	C₇×C₉²	C₃²	1	2	yes	—
28	C₇×C₉²	C₃²	1	1	yes	characteristic
29	C₇×C₉²	C₃²	1	3	yes	—
30	C₇×C₉²	C₃²	1	6	yes	—
31	C₃×#737	C₃×C₇	1	1	yes	characteristic
32	#6900	C₃×C₇	1	3	yes	complemented
33	#6899	C₃×C₇	1	6	yes	complemented
34	#6896	C₃×C₇	1	9	yes	complemented
35	#6894	C₃×C₇	1	6	yes	complemented
36	C₉⋊H(3,𝔽₃)	C₃×C₇	1	3	yes	—
37	C₉×3₋¹⁺²	C₃×C₇	1	9	yes	complemented
38	#6887	C₃×C₇	1	2	yes	complemented
39	C₃×C₉²	C₃×C₇	1	1	yes	characteristic, complemented
40	C₃³×C₇	C₃³	1	3	yes	—
41	C₃³×C₇	C₃³	1	1	yes	characteristic
42	C₇×3₋¹⁺²	—	3	6	no	—
43	C₇×3₋¹⁺²	—	3	9	no	—
44	C₇×3₋¹⁺²	—	3	9	no	—
45	C₇×3₋¹⁺²	—	3	27	no	—
46	C₇×H(3,𝔽₃)	—	3	3	no	—
47	C₃×C₇×C₉	—	3	9	no	—
48	C₃×C₇×C₉	—	3	9	no	—
49	C₃×C₇×C₉	3₋¹⁺²	1	9	yes	complemented
50	C₃×C₇×C₉	C₃³	1	6	yes	—
51	C₃×C₇×C₉	C₃³	1	18	yes	—
52	C₃×C₇×C₉	C₃³	1	6	yes	—
53	C₃×C₇×C₉	C₃³	1	1	yes	characteristic
54	C₃×C₇×C₉	C₃³	1	1	yes	characteristic
55	C₃×C₇×C₉	C₃³	1	3	yes	—
56	C₃×C₇×C₉	C₃³	1	1	yes	characteristic
57	#737	—	3	9	no	—
58	C₃×3₋¹⁺²	C₃²×C₇	1	2	yes	—
59	C₃×3₋¹⁺²	C₃²×C₇	1	3	yes	—
60	C₃×3₋¹⁺²	C₃²×C₇	1	3	yes	—
61	C₃×3₋¹⁺²	C₃²×C₇	1	9	yes	—
62	C₃×H(3,𝔽₃)	C₃²×C₇	1	1	yes	characteristic
63	C₃²×C₉	C₃²×C₇	1	3	yes	—
64	C₃²×C₉	C₃²×C₇	1	3	yes	—
65	C₃²×C₉	C₃²×C₇	1	1	yes	characteristic
66	C₉⋊C₉	C₃²×C₇	1	6	yes	—
67	C₉⋊C₉	C₃²×C₇	1	6	yes	—
68	C₉⋊C₉	C₃²×C₇	1	36	yes	—
69	C₉⋊C₉	C₃²×C₇	1	18	yes	—
70	C₃²⋊C₉	C₃²×C₇	1	6	yes	—
71	C₃²⋊C₉	C₃²×C₇	1	18	yes	—
72	C₉²	C₃²×C₇	1	3	yes	—
73	C₉²	C₃²×C₇	1	3	yes	—
74	C₉²	C₃²×C₇	1	2	yes	—
75	C₉²	C₃²×C₇	1	1	yes	characteristic
76	C₉²	C₃²×C₇	1	6	yes	—
77	C₃²×C₇	—	3	3	no	—
78	C₃²×C₇	—	3	3	no	—
79	C₃²×C₇	—	3	9	no	—
80	C₃²×C₇	C₃×3₋¹⁺²	1	3	yes	—
81	C₃²×C₇	C₃⁴	1	1	yes	characteristic
82	C₇×C₉	—	3	6	no	—
83	C₇×C₉	—	3	6	no	—
84	C₇×C₉	—	3	18	no	—
85	C₇×C₉	C₃×3₋¹⁺²	1	3	yes	—
86	C₇×C₉	C₃×3₋¹⁺²	1	3	yes	—
87	C₇×C₉	C₃×3₋¹⁺²	1	3	yes	—
88	C₇×C₉	#737	1	9	yes	complemented
89	C₃³	C₃³×C₇	1	3	yes	—
90	C₃³	C₃³×C₇	1	1	yes	characteristic
91	3₋¹⁺²	—	3	6	no	—
92	3₋¹⁺²	—	3	9	no	—
93	3₋¹⁺²	—	3	9	no	—
94	3₋¹⁺²	—	3	27	no	—
95	H(3,𝔽₃)	—	3	3	no	—
96	C₃×C₉	—	3	9	no	—
97	C₃×C₉	—	3	9	no	—
98	C₃×C₉	C₇×3₋¹⁺²	1	9	yes	complemented
99	C₃×C₉	C₃³×C₇	1	6	yes	—
100	C₃×C₉	C₃³×C₇	1	18	yes	—
101	C₃×C₉	C₃³×C₇	1	6	yes	—
102	C₃×C₉	C₃³×C₇	1	1	yes	characteristic
103	C₃×C₉	C₃³×C₇	1	1	yes	characteristic
104	C₃×C₉	C₃³×C₇	1	1	yes	characteristic
105	C₃×C₉	C₃³×C₇	1	3	yes	—
106	C₃×C₇	—	3	3	no	—
107	C₃×C₇	—	9	3	no	—
108	C₃×C₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
109	C₃×C₇	C₃×#737	1	3	yes	—
110	C₃²	—	3	3	no	—
111	C₃²	—	3	3	no	—
112	C₃²	—	3	9	no	—
113	C₃²	C₃×C₇×3₋¹⁺²	1	3	yes	—
114	C₃²	C₃⁴×C₇	1	1	yes	characteristic
115	C₉	—	3	6	no	—
116	C₉	—	3	6	no	—
117	C₉	—	3	18	no	—
118	C₉	C₃×C₇×3₋¹⁺²	1	3	yes	—
119	C₉	C₃×C₇×3₋¹⁺²	1	3	yes	—
120	C₉	C₃×C₇×3₋¹⁺²	1	3	yes	—
121	C₉	C₇×#737	1	9	yes	complemented
122	C₇	#10624879	1	1	yes	characteristic, direct product factor, Sylow
123	C₃	—	3	3	no	—
124	C₃	—	9	3	no	—
125	C₃	C₃²×C₇×3₋¹⁺²	1	1	yes	characteristic
126	C₃	C₃×C₇×#737	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₇×#737

C₃

yes

maximal, characteristic

C₇×#6900

C₃

yes

maximal, complemented

C₇×#6899

C₃

yes

maximal, complemented

C₇×#6896

C₃

yes

maximal, complemented

C₇×#6894

C₃

yes

maximal, complemented

C₇×C₉⋊H(3,𝔽₃)

C₃

yes

maximal

C₇×C₉×3₋¹⁺²

C₃

yes

maximal, complemented

C₇×#6887

C₃

yes

maximal, complemented

C₃×C₇×C₉²

C₃

yes

maximal, characteristic, complemented

#10624879

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₇×#737

—

C₃×C₇×3₋¹⁺²

C₃²

yes

—

C₃×C₇×3₋¹⁺²

C₃²

yes

—

C₃×C₇×3₋¹⁺²

C₃²

yes

—

C₃×C₇×3₋¹⁺²

C₃²

yes

—

C₃×C₇×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×C₇×C₉

C₃²

yes

—

C₃²×C₇×C₉

C₃²

yes

—

C₃²×C₇×C₉

C₃²

yes

characteristic

C₇×C₉⋊C₉

C₃²

yes

—

C₇×C₉⋊C₉

C₃²

yes

—

C₇×C₉⋊C₉

C₃²

yes

—

C₇×C₉⋊C₉

C₃²

yes

—

C₇×C₃²⋊C₉

C₃²

yes

—

C₇×C₃²⋊C₉

C₃²

yes

—

C₇×C₉²

C₃²

yes

—

C₇×C₉²

C₃²

yes

—

C₇×C₉²

C₃²

yes

characteristic

C₇×C₉²

C₃²

yes

—

C₇×C₉²

C₃²

yes

—

C₃×#737

C₃×C₇

yes

characteristic

#6900

C₃×C₇

yes

complemented

#6899

C₃×C₇

yes

complemented

#6896

C₃×C₇

yes

complemented

#6894

C₃×C₇

yes

complemented

C₉⋊H(3,𝔽₃)

C₃×C₇

yes

—

C₉×3₋¹⁺²

C₃×C₇

yes

complemented

#6887

C₃×C₇

yes

complemented

C₃×C₉²

C₃×C₇

yes

characteristic, complemented

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×H(3,𝔽₃)

—

C₃×C₇×C₉

—

C₃×C₇×C₉

—

C₃×C₇×C₉

3₋¹⁺²

yes

complemented

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

characteristic

C₃×C₇×C₉

C₃³

yes

characteristic

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

characteristic

#737

—

C₃×3₋¹⁺²

C₃²×C₇

yes

—

C₃×3₋¹⁺²

C₃²×C₇

yes

—

C₃×3₋¹⁺²

C₃²×C₇

yes

—

C₃×3₋¹⁺²

C₃²×C₇

yes

—

C₃×H(3,𝔽₃)

C₃²×C₇

yes

characteristic

C₃²×C₉

C₃²×C₇

yes

—

C₃²×C₉

C₃²×C₇

yes

—

C₃²×C₉

C₃²×C₇

yes

characteristic

C₉⋊C₉

C₃²×C₇

yes

—

C₉⋊C₉

C₃²×C₇

yes

—

C₉⋊C₉

C₃²×C₇

yes

—

C₉⋊C₉

C₃²×C₇

yes

—

C₃²⋊C₉

C₃²×C₇

yes

—

C₃²⋊C₉

C₃²×C₇

yes

—

C₉²

C₃²×C₇

yes

—

C₉²

C₃²×C₇

yes

—

C₉²

C₃²×C₇

yes

—

C₉²

C₃²×C₇

yes

characteristic

C₉²

C₃²×C₇

yes

—

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

C₃×3₋¹⁺²

yes

—

C₃²×C₇

C₃⁴

yes

characteristic

C₇×C₉

—

C₇×C₉

—

C₇×C₉

—

C₇×C₉

C₃×3₋¹⁺²

yes

—

C₇×C₉

C₃×3₋¹⁺²

yes

—

C₇×C₉

C₃×3₋¹⁺²

yes

—

C₇×C₉

#737

yes

complemented

C₃³

C₃³×C₇

yes

—

C₃³

C₃³×C₇

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₇×3₋¹⁺²

yes

complemented

C₃×C₉

C₃³×C₇

yes

—

100

C₃×C₉

C₃³×C₇

yes

—

101

C₃×C₉

C₃³×C₇

yes

—

102

C₃×C₉

C₃³×C₇

yes

characteristic

103

C₃×C₉

C₃³×C₇

yes

characteristic

104

C₃×C₉

C₃³×C₇

yes

characteristic

105

C₃×C₉

C₃³×C₇

yes

—

106

C₃×C₇

—

107

C₃×C₇

—

108

C₃×C₇

C₃²×3₋¹⁺²

yes

characteristic

109

C₃×C₇

C₃×#737

yes

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

C₃×C₇×3₋¹⁺²

yes

—

114

C₃²

C₃⁴×C₇

yes

characteristic

115

C₉

—

116

C₉

—

117

C₉

—

118

C₉

C₃×C₇×3₋¹⁺²

yes

—

119

C₉

C₃×C₇×3₋¹⁺²

yes

—

120

C₉

C₃×C₇×3₋¹⁺²

yes

—

121

C₉

C₇×#737

yes

complemented

122

C₇

#10624879

yes

characteristic, direct product factor, Sylow

123

C₃

—

124

C₃

—

125

C₃

C₃²×C₇×3₋¹⁺²

yes

characteristic

126

C₃

C₃×C₇×#737

yes

—

	subgroup	G/H
center	C₃×C₇×C₉	C₃³
commutator, Frattini	C₃²	C₃⁴×C₇
socle	C₃²×C₇	C₃⁴
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃×C₇×C₉ ◁ G

subgroup

G/H

center

C₃×C₇×C₉

C₃³

commutator, Frattini

C₃²

C₃⁴×C₇

socle

C₃²×C₇

C₃⁴

LCS

G ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃×C₇×C₉ ◁ G

N	H ≅ G/N	type
2. C₇×#6900	C₃	semidirect
3. C₇×#6899	C₃	semidirect
4. C₇×#6896	C₃	semidirect
5. C₇×#6894	C₃	semidirect
7. C₇×C₉×3₋¹⁺²	C₃	semidirect
8. C₇×#6887	C₃	semidirect
9. C₃×C₇×C₉²	C₃	semidirect
10. #10624879	C₇	direct (N × H)
32. #6900	C₃×C₇	semidirect
33. #6899	C₃×C₇	semidirect
34. #6896	C₃×C₇	semidirect
35. #6894	C₃×C₇	semidirect
37. C₉×3₋¹⁺²	C₃×C₇	semidirect
38. #6887	C₃×C₇	semidirect
39. C₃×C₉²	C₃×C₇	semidirect
49. C₃×C₇×C₉	3₋¹⁺²	semidirect
88. C₇×C₉	#737	semidirect
98. C₃×C₉	C₇×3₋¹⁺²	semidirect
121. C₉	C₇×#737	semidirect
122. C₇	#10624879	direct (N × H)

H ≅ G/N

type

2. C₇×#6900

C₃

semidirect

3. C₇×#6899

C₃

semidirect

4. C₇×#6896

C₃

semidirect

5. C₇×#6894

C₃

semidirect

7. C₇×C₉×3₋¹⁺²

C₃

semidirect

8. C₇×#6887

C₃

semidirect

9. C₃×C₇×C₉²

C₃

semidirect

10. #10624879

C₇

direct (N × H)

32. #6900

C₃×C₇

semidirect

33. #6899

C₃×C₇

semidirect

34. #6896

C₃×C₇

semidirect

35. #6894

C₃×C₇

semidirect

37. C₉×3₋¹⁺²

C₃×C₇

semidirect

38. #6887

C₃×C₇

semidirect

39. C₃×C₉²

C₃×C₇

semidirect

49. C₃×C₇×C₉

3₋¹⁺²

semidirect

88. C₇×C₉

#737

semidirect

98. C₃×C₉

C₇×3₋¹⁺²

semidirect

121. C₉

C₇×#737

semidirect

122. C₇

#10624879

direct (N × H)

Subgroups — C₇×#10624879 #19100288

Conjugacy classes of subgroups

Complements

Subgroups — C7×#10624879 #19100288

Conjugacy classes of subgroups

Complements

Subgroups — C₇×#10624879 #19100288