Group #13090092 — subgroups

Subgroups — C₂×#10624606 #13090092

Order 1458.

204 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 80 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	#10624606	C₂	1	1	yes	maximal, characteristic, direct product factor, Sylow
2	C₂×#6894	C₃	1	2	yes	maximal, complemented
3	C₂×C₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₂×#6887	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×#6862	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×#6861	C₃	1	1	yes	maximal, characteristic, complemented
7	C₂×#6859	C₃	1	2	yes	maximal
8	C₂×#6858	C₃	1	2	yes	maximal, complemented
9	C₂×#6858	C₃	1	2	yes	maximal, complemented
10	C₂×#6857	C₃	1	1	yes	maximal, characteristic
11	#6894	C₂×C₃	1	2	yes	complemented
12	C₉×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
13	#6887	C₂×C₃	1	1	yes	characteristic, complemented
14	#6862	C₂×C₃	1	1	yes	characteristic, complemented
15	#6861	C₂×C₃	1	1	yes	characteristic, complemented
16	#6859	C₂×C₃	1	2	yes	—
17	#6858	C₂×C₃	1	2	yes	complemented
18	#6858	C₂×C₃	1	2	yes	complemented
19	#6857	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×3₋¹⁺²	C₃²	1	2	yes	—
21	C₂×C₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
22	C₂×C₃²×C₉	—	3	1	no	—
23	C₂×C₃²×C₉	C₃²	1	1	yes	characteristic
24	C₂×C₉⋊C₉	—	3	1	no	—
25	C₂×C₉⋊C₉	—	3	1	no	—
26	C₂×C₉⋊C₉	—	3	2	no	—
27	C₂×C₉⋊C₉	—	3	2	no	—
28	C₂×C₉⋊C₉	—	3	2	no	—
29	C₂×C₃²⋊C₉	—	3	1	no	—
30	C₂×C₃²⋊C₉	—	3	1	no	—
31	C₂×C₃²⋊C₉	C₃²	1	1	yes	characteristic
32	C₂×C₃²⋊C₉	C₃²	1	2	yes	—
33	C₂×C₃²⋊C₉	C₃²	1	1	yes	characteristic
34	C₂×C₃²⋊C₉	C₃²	1	1	yes	characteristic
35	C₂×C₃²⋊C₉	C₃²	1	2	yes	—
36	C₂×C₃²⋊C₉	C₃²	1	2	yes	—
37	C₂×C₉²	—	3	1	no	—
38	C₃×3₋¹⁺²	C₂×C₃²	1	2	yes	—
39	C₃×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
40	C₃²×C₉	—	3	1	no	—
41	C₃²×C₉	C₂×C₃²	1	1	yes	characteristic
42	C₉⋊C₉	—	3	1	no	—
43	C₉⋊C₉	—	3	1	no	—
44	C₉⋊C₉	—	3	2	no	—
45	C₉⋊C₉	—	3	2	no	—
46	C₉⋊C₉	—	3	2	no	—
47	C₃²⋊C₉	—	3	1	no	—
48	C₃²⋊C₉	—	3	1	no	—
49	C₃²⋊C₉	C₂×C₃²	1	1	yes	characteristic
50	C₃²⋊C₉	C₂×C₃²	1	2	yes	—
51	C₃²⋊C₉	C₂×C₃²	1	1	yes	characteristic
52	C₃²⋊C₉	C₂×C₃²	1	1	yes	characteristic
53	C₃²⋊C₉	C₂×C₃²	1	2	yes	—
54	C₃²⋊C₉	C₂×C₃²	1	2	yes	—
55	C₉²	—	3	1	no	—
56	C₂×C₃³	—	3	1	no	—
57	C₂×C₃³	C₃³	1	1	yes	characteristic
58	C₂×3₋¹⁺²	—	9	2	no	—
59	C₂×H(3,𝔽₃)	—	3	3	no	—
60	C₂×C₃×C₉	—	3	3	no	—
61	C₂×C₃×C₉	—	3	2	no	—
62	C₂×C₃×C₉	—	3	1	no	—
63	C₂×C₃×C₉	—	3	1	no	—
64	C₂×C₃×C₉	—	3	1	no	—
65	C₂×C₃×C₉	—	3	2	no	—
66	C₂×C₃×C₉	—	3	2	no	—
67	C₂×C₃×C₉	—	3	2	no	—
68	C₂×C₃×C₉	—	9	3	no	—
69	C₂×C₃×C₉	H(3,𝔽₃)	1	1	yes	characteristic
70	C₂×C₃×C₉	H(3,𝔽₃)	1	1	yes	characteristic
71	C₂×C₃×C₉	H(3,𝔽₃)	1	1	yes	characteristic
72	C₃³	—	3	1	no	—
73	C₃³	C₂×C₃³	1	1	yes	characteristic
74	3₋¹⁺²	—	9	2	no	—
75	H(3,𝔽₃)	—	3	3	no	—
76	C₃×C₉	—	3	3	no	—
77	C₃×C₉	—	3	1	no	—
78	C₃×C₉	—	3	2	no	—
79	C₃×C₉	—	3	1	no	—
80	C₃×C₉	—	3	1	no	—
81	C₃×C₉	—	3	2	no	—
82	C₃×C₉	—	3	2	no	—
83	C₃×C₉	—	3	2	no	—
84	C₃×C₉	—	9	3	no	—
85	C₃×C₉	C₂×H(3,𝔽₃)	1	1	yes	characteristic
86	C₃×C₉	C₂×H(3,𝔽₃)	1	1	yes	characteristic
87	C₃×C₉	C₂×H(3,𝔽₃)	1	1	yes	characteristic
88	C₂×C₃²	—	3	2	no	—
89	C₂×C₃²	—	3	1	no	—
90	C₂×C₃²	—	3	1	no	—
91	C₂×C₃²	—	3	3	no	—
92	C₂×C₃²	—	9	2	no	—
93	C₂×C₃²	—	9	1	no	—
94	C₂×C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
95	C₂×C₉	—	3	1	no	—
96	C₂×C₉	—	3	1	no	—
97	C₂×C₉	—	9	2	no	—
98	C₂×C₉	—	9	2	no	—
99	C₂×C₉	—	9	2	no	—
100	C₂×C₉	—	9	1	no	—
101	C₂×C₉	—	9	1	no	—
102	C₂×C₉	—	9	1	no	—
103	C₂×C₉	—	9	2	no	—
104	C₂×C₉	#731	1	3	yes	—
105	C₃²	—	3	1	no	—
106	C₃²	—	3	1	no	—
107	C₃²	—	3	3	no	—
108	C₃²	—	3	2	no	—
109	C₃²	—	9	1	no	—
110	C₃²	—	9	2	no	—
111	C₃²	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic
112	C₉	—	3	1	no	—
113	C₉	—	3	1	no	—
114	C₉	—	9	1	no	—
115	C₉	—	9	1	no	—
116	C₉	—	9	1	no	—
117	C₉	—	9	2	no	—
118	C₉	—	9	2	no	—
119	C₉	—	9	2	no	—
120	C₉	—	9	2	no	—
121	C₉	C₂×#731	1	3	yes	—
122	C₂×C₃	—	9	1	no	—
123	C₂×C₃	—	9	3	no	—
124	C₂×C₃	C₃×#731	1	1	yes	characteristic
125	C₂×C₃	#6908	1	1	yes	characteristic
126	C₂×C₃	#6913	1	2	yes	—
127	C₃	—	9	1	no	—
128	C₃	—	9	3	no	—
129	C₃	C₂×C₃×#731	1	1	yes	characteristic
130	C₃	C₂×#6908	1	1	yes	characteristic
131	C₃	C₂×#6913	1	2	yes	—
132	C₂	#10624606	1	1	yes	characteristic, direct product factor, Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624606

C₂

yes

maximal, characteristic, direct product factor, Sylow

C₂×#6894

C₃

yes

maximal, complemented

C₂×C₉×H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₂×#6887

C₃

yes

maximal, characteristic, complemented

C₂×#6862

C₃

yes

maximal, characteristic, complemented

C₂×#6861

C₃

yes

maximal, characteristic, complemented

C₂×#6859

C₃

yes

maximal

C₂×#6858

C₃

yes

maximal, complemented

C₂×#6858

C₃

yes

maximal, complemented

C₂×#6857

C₃

yes

maximal, characteristic

#6894

C₂×C₃

yes

complemented

C₉×H(3,𝔽₃)

C₂×C₃

yes

characteristic

#6887

C₂×C₃

yes

characteristic, complemented

#6862

C₂×C₃

yes

characteristic, complemented

#6861

C₂×C₃

yes

characteristic, complemented

#6859

C₂×C₃

yes

—

#6858

C₂×C₃

yes

complemented

#6858

C₂×C₃

yes

complemented

#6857

C₂×C₃

yes

characteristic

C₂×C₃×3₋¹⁺²

C₃²

yes

—

C₂×C₃×H(3,𝔽₃)

C₃²

yes

characteristic

C₂×C₃²×C₉

—

C₂×C₃²×C₉

C₃²

yes

characteristic

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₃²⋊C₉

—

C₂×C₃²⋊C₉

—

C₂×C₃²⋊C₉

C₃²

yes

characteristic

C₂×C₃²⋊C₉

C₃²

yes

—

C₂×C₃²⋊C₉

C₃²

yes

characteristic

C₂×C₃²⋊C₉

C₃²

yes

characteristic

C₂×C₃²⋊C₉

C₃²

yes

—

C₂×C₃²⋊C₉

C₃²

yes

—

C₂×C₉²

—

C₃×3₋¹⁺²

C₂×C₃²

yes

—

C₃×H(3,𝔽₃)

C₂×C₃²

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₂×C₃²

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₂×C₃²

yes

characteristic

C₃²⋊C₉

C₂×C₃²

yes

—

C₃²⋊C₉

C₂×C₃²

yes

characteristic

C₃²⋊C₉

C₂×C₃²

yes

characteristic

C₃²⋊C₉

C₂×C₃²

yes

—

C₃²⋊C₉

C₂×C₃²

yes

—

C₉²

—

C₂×C₃³

—

C₂×C₃³

C₃³

yes

characteristic

C₂×3₋¹⁺²

—

C₂×H(3,𝔽₃)

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

yes

C₂×C₃×C₉

yes

C₂×C₃×C₉

yes

C₃³

—

C₃³

C₂×C₃³

yes

characteristic

3₋¹⁺²

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₂×H(3,𝔽₃)

yes

characteristic

C₃×C₉

C₂×H(3,𝔽₃)

yes

characteristic

C₃×C₉

C₂×H(3,𝔽₃)

yes

characteristic

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₃×H(3,𝔽₃)

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

100

C₂×C₉

—

101

C₂×C₉

—

102

C₂×C₉

—

103

C₂×C₉

—

104

C₂×C₉

#731

yes

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

C₂×C₃×H(3,𝔽₃)

yes

characteristic

112

C₉

—

113

C₉

—

114

C₉

—

115

C₉

—

116

C₉

—

117

C₉

—

118

C₉

—

119

C₉

—

120

C₉

—

121

C₉

C₂×#731

yes

—

122

C₂×C₃

—

123

C₂×C₃

—

124

C₂×C₃

C₃×#731

yes

characteristic

125

C₂×C₃

#6908

yes

characteristic

126

C₂×C₃

#6913

yes

—

127

C₃

—

128

C₃

—

129

C₃

C₂×C₃×#731

yes

characteristic

130

C₃

C₂×#6908

yes

characteristic

131

C₃

C₂×#6913

yes

—

132

C₂

#10624606

yes

characteristic, direct product factor, Sylow

	subgroup	G/H
center, socle	C₂×C₃²	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₂×C₃³
prime radical	C₂×C₃×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₉ ◁ G

subgroup

G/H

center, socle

C₂×C₃²

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₂×C₃³

prime radical

C₂×C₃×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃³ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₉ ◁ G

N	H ≅ G/N	type
1. #10624606	C₂	direct (N × H)
2. C₂×#6894	C₃	semidirect
4. C₂×#6887	C₃	semidirect
5. C₂×#6862	C₃	semidirect
6. C₂×#6861	C₃	semidirect
8. C₂×#6858	C₃	semidirect
9. C₂×#6858	C₃	semidirect
11. #6894	C₂×C₃	semidirect
13. #6887	C₂×C₃	semidirect
14. #6862	C₂×C₃	semidirect
15. #6861	C₂×C₃	semidirect
17. #6858	C₂×C₃	semidirect
18. #6858	C₂×C₃	semidirect
132. C₂	#10624606	direct (N × H)

H ≅ G/N

type

1. #10624606

C₂

direct (N × H)

2. C₂×#6894

C₃

semidirect

4. C₂×#6887

C₃

semidirect

5. C₂×#6862

C₃

semidirect

6. C₂×#6861

C₃

semidirect

8. C₂×#6858

C₃

semidirect

9. C₂×#6858

C₃

semidirect

11. #6894

C₂×C₃

semidirect

13. #6887

C₂×C₃

semidirect

14. #6862

C₂×C₃

semidirect

15. #6861

C₂×C₃

semidirect

17. #6858

C₂×C₃

semidirect

18. #6858

C₂×C₃

semidirect

132. C₂

#10624606

direct (N × H)

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

subgroup	index
#10624606	2
C₂×#6894	3
C₂×C₉×H(3,𝔽₃)	3
C₂×#6887	3
C₂×#6862	3
C₂×#6861	3
C₂×#6859	3
C₂×#6858	3
C₂×#6857	3

subgroup

index

C₂×C₉×H(3,𝔽₃)

Subgroups — C₂×#10624606 #13090092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#10624606 #13090092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10624606 #13090092