Group #13142442 — subgroups

Subgroups — #13142442

Order 1728.

612 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 292 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	C₂×C₄²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(3₋¹⁺²⋊C₄)	C₂	1	6	yes	maximal
3	C₃×#4935	—	3	1	no	maximal
4	#10588615	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×3₋¹⁺²	C₂²	1	3	yes	—
6	C₂²×3₋¹⁺²⋊C₄	C₂²	1	4	yes	—
7	C₄²×3₋¹⁺²	C₄	1	4	yes	complemented
8	C₄⋊(3₋¹⁺²⋊C₄)	—	2	12	no	—
9	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
10	C₂×C₃×C₄⋊Dic₃	—	3	6	no	—
11	C₂×C₄²×C₉	—	2	1	no	—
12	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
13	C₂×C₄⋊Dic₉	C₂×C₃	1	6	yes	—
14	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
15	C₂×C₄×3₋¹⁺²	C₂×C₄	1	6	yes	—
16	C₂×C₄×3₋¹⁺²	D₄	1	6	yes	—
17	C₂×C₄×3₋¹⁺²	Q₈	1	6	yes	—
18	C₂×3₋¹⁺²⋊C₄	—	2	12	no	—
19	C₃×#384	—	9	1	no	—
20	#4935	—	3	1	no	—
21	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
22	C₂²×C₃×Dic₃	—	3	4	no	—
23	C₃²×C₄²	Dic₃	1	4	yes	—
24	C₃×C₄⋊Dic₃	—	6	12	no	—
25	C₂²×C₄×C₉	—	2	3	no	—
26	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
27	C₂²×Dic₉	C₂²×C₃	1	4	yes	—
28	C₄²×C₉	—	2	4	no	—
29	C₄²×C₉	C₃×C₄	1	4	yes	complemented
30	C₄⋊Dic₉	—	2	12	no	—
31	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
32	C₂²×3₋¹⁺²	C₂×D₄	1	3	yes	—
33	C₂²×3₋¹⁺²	C₂×Q₈	1	3	yes	—
34	C₄×3₋¹⁺²	C₄⋊C₄	1	12	yes	complemented
35	3₋¹⁺²⋊C₄	—	4	4	no	—
36	C₂×C₃×C₄⋊C₄	—	9	6	no	—
37	C₂×C₃×C₄²	—	3	1	no	—
38	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
39	C₂×C₄⋊Dic₃	—	3	6	no	—
40	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
41	C₂×C₃²×C₄	Dic₆	1	6	yes	—
42	C₂×C₃²×C₄	D₁₂	1	6	yes	—
43	C₂×C₃²×C₄	C₂×Dic₃	1	6	yes	—
44	C₂×C₃×Dic₃	—	6	12	no	—
45	C₂³×C₉	—	2	1	no	—
46	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
47	C₂×C₄×C₉	—	2	6	no	—
48	C₂×C₄×C₉	—	2	6	no	—
49	C₂×C₄×C₉	—	2	6	no	—
50	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
51	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
52	C₂×C₄×C₉	C₃×Q₈	1	6	yes	—
53	C₂×Dic₉	—	2	12	no	—
54	#384	—	9	1	no	Sylow
55	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	3	yes	—
56	C₂×3₋¹⁺²	#127	1	1	yes	characteristic
57	C₂×3₋¹⁺²	C₄⋊Q₈	1	3	yes	—
58	C₂²×C₃×C₄	—	3	3	no	—
59	C₂²×C₃×C₄	—	9	4	no	—
60	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
61	C₂²×Dic₃	—	3	4	no	—
62	C₃×C₄⋊C₄	—	18	12	no	—
63	C₃×C₄²	—	3	4	no	—
64	C₃×C₄²	C₃×Dic₃	1	4	yes	—
65	C₄⋊Dic₃	—	6	12	no	—
66	C₂²×C₃²	C₂×Dic₆	1	3	yes	—
67	C₂²×C₃²	C₂×D₁₂	1	3	yes	—
68	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
69	C₃²×C₄	C₄⋊Dic₃	1	12	yes	—
70	C₃×Dic₃	—	12	4	no	—
71	C₂²×C₉	—	2	3	no	—
72	C₂²×C₉	—	2	3	no	—
73	C₂²×C₉	—	2	1	no	—
74	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
75	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
76	C₂²×C₉	C₂×C₃×Q₈	1	3	yes	—
77	C₄×C₉	—	2	12	no	—
78	C₄×C₉	C₃×C₄⋊C₄	1	12	yes	complemented
79	Dic₉	—	4	4	no	—
80	C₂×C₄⋊C₄	—	9	6	no	—
81	C₂×C₄²	AGL(1,ℤ/9)	1	1	yes	characteristic
82	3₋¹⁺²	#384	1	1	yes	characteristic, complemented, Sylow
83	C₂³×C₃	—	3	1	no	—
84	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
85	C₂×C₃×C₄	—	3	6	no	—
86	C₂×C₃×C₄	—	3	6	no	—
87	C₂×C₃×C₄	—	3	6	no	—
88	C₂×C₃×C₄	—	18	12	no	—
89	C₂×C₃×C₄	C₃×Dic₆	1	6	yes	—
90	C₂×C₃×C₄	C₃×D₁₂	1	6	yes	—
91	C₂×C₃×C₄	C₂×C₃×Dic₃	1	6	yes	—
92	C₂×Dic₃	—	6	12	no	—
93	C₂×C₃²	#869	1	3	yes	—
94	C₂×C₃²	#874	1	1	yes	characteristic
95	C₂×C₃²	C₂×C₄⋊Dic₃	1	3	yes	—
96	C₂×C₉	—	2	3	no	—
97	C₂×C₉	—	2	3	no	—
98	C₂×C₉	—	2	1	no	—
99	C₂×C₉	C₂×C₃×C₄⋊C₄	1	3	yes	—
100	C₂×C₉	C₃×#127	1	1	yes	characteristic
101	C₂×C₉	C₃×C₄⋊Q₈	1	3	yes	—
102	C₂²×C₄	—	9	4	no	—
103	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
104	C₄⋊C₄	—	18	12	no	—
105	C₄²	3₋¹⁺²⋊C₄	1	4	yes	complemented
106	C₂²×C₃	—	3	3	no	—
107	C₂²×C₃	—	3	3	no	—
108	C₂²×C₃	—	3	1	no	—
109	C₂²×C₃	C₂×C₃×Dic₆	1	3	yes	—
110	C₂²×C₃	C₂×C₃×D₁₂	1	3	yes	—
111	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
112	C₃×C₄	—	3	12	no	—
113	C₃×C₄	—	36	4	no	—
114	C₃×C₄	C₃×C₄⋊Dic₃	1	12	yes	—
115	Dic₃	—	12	4	no	—
116	C₃²	#4935	1	1	yes	characteristic
117	C₉	—	2	1	no	—
118	C₉	C₃×#384	1	1	yes	characteristic, complemented
119	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
120	C₂×C₄	—	18	12	no	—
121	C₂×C₄	3₋¹⁺²⋊Q₈	1	6	yes	—
122	C₂×C₄	C₄⋊AGL(1,ℤ/9)	1	6	yes	—
123	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	6	yes	—
124	C₂×C₃	—	3	3	no	—
125	C₂×C₃	—	3	3	no	—
126	C₂×C₃	—	3	1	no	—
127	C₂×C₃	C₃×#869	1	3	yes	—
128	C₂×C₃	C₃×#874	1	1	yes	characteristic
129	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	3	yes	—
130	C₂²	C₂×3₋¹⁺²⋊Q₈	1	3	yes	—
131	C₂²	C₂×C₄⋊AGL(1,ℤ/9)	1	3	yes	—
132	C₂²	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
133	C₄	—	36	4	no	—
134	C₄	C₄⋊(3₋¹⁺²⋊C₄)	1	12	yes	complemented
135	C₃	—	3	1	no	—
136	C₃	C₃×#4935	1	1	yes	characteristic
137	C₂	#63431⋊C₃	1	3	yes	—
138	C₂	#11721394	1	1	yes	characteristic
139	C₂	C₂×C₄⋊(3₋¹⁺²⋊C₄)	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×3₋¹⁺²

C₂

yes

maximal, characteristic

C₂×C₄⋊(3₋¹⁺²⋊C₄)

C₂

yes

—

C₃

yes

maximal, characteristic, complemented

C₂²×C₄×3₋¹⁺²

C₂²

yes

—

C₂²×3₋¹⁺²⋊C₄

C₂²

yes

—

C₄²×3₋¹⁺²

C₄

yes

complemented

C₄⋊(3₋¹⁺²⋊C₄)

—

C₂×C₃²×C₄²

D₃

yes

characteristic

C₂×C₃×C₄⋊Dic₃

—

C₂×C₄²×C₉

—

C₂×C₄²×C₉

C₂×C₃

yes

characteristic

C₂×C₄⋊Dic₉

C₂×C₃

yes

—

C₂³×3₋¹⁺²

C₂³

yes

characteristic

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

—

C₂×C₄×3₋¹⁺²

D₄

yes

—

C₂×C₄×3₋¹⁺²

Q₈

yes

—

C₂×3₋¹⁺²⋊C₄

—

C₃×#384

—

#4935

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃×Dic₃

—

C₃²×C₄²

Dic₃

yes

—

C₃×C₄⋊Dic₃

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂²×C₃

yes

—

C₂²×Dic₉

C₂²×C₃

yes

—

C₄²×C₉

—

C₄²×C₉

C₃×C₄

yes

complemented

C₄⋊Dic₉

—

C₂²×3₋¹⁺²

C₂²×C₄

yes

characteristic

C₂²×3₋¹⁺²

C₂×D₄

yes

—

C₂²×3₋¹⁺²

C₂×Q₈

yes

—

C₄×3₋¹⁺²

C₄⋊C₄

yes

complemented

3₋¹⁺²⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₃×D₃

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂³×C₃²

C₂²×D₃

yes

characteristic

C₂×C₃²×C₄

Dic₆

yes

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃×Dic₃

—

C₂³×C₉

—

C₂³×C₉

C₂³×C₃

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₂×C₄×C₉

C₃×D₄

yes

—

C₂×C₄×C₉

C₃×Q₈

yes

—

C₂×Dic₉

—

#384

—

Sylow

C₂×3₋¹⁺²

C₂×C₄⋊C₄

yes

—

C₂×3₋¹⁺²

#127

yes

characteristic

C₂×3₋¹⁺²

C₄⋊Q₈

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

—

C₂²×Dic₃

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₃×Dic₃

yes

—

C₄⋊Dic₃

—

C₂²×C₃²

C₂×Dic₆

yes

—

C₂²×C₃²

C₂×D₁₂

yes

—

C₂²×C₃²

C₂²×Dic₃

yes

characteristic

C₃²×C₄

C₄⋊Dic₃

yes

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂²×C₃×C₄

yes

characteristic

C₂²×C₉

C₂×C₃×D₄

yes

—

C₂²×C₉

C₂×C₃×Q₈

yes

—

C₄×C₉

—

C₄×C₉

C₃×C₄⋊C₄

yes

complemented

Dic₉

—

C₂×C₄⋊C₄

—

C₂×C₄²

AGL(1,ℤ/9)

yes

characteristic

3₋¹⁺²

#384

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

C₂²×C₃×D₃

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃×Dic₆

yes

—

C₂×C₃×C₄

C₃×D₁₂

yes

—

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

—

C₂×Dic₃

—

C₂×C₃²

#869

yes

—

C₂×C₃²

#874

yes

characteristic

C₂×C₃²

C₂×C₄⋊Dic₃

yes

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂×C₃×C₄⋊C₄

yes

—

100

C₂×C₉

C₃×#127

yes

characteristic

101

C₂×C₉

C₃×C₄⋊Q₈

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂×AGL(1,ℤ/9)

yes

—

104

C₄⋊C₄

—

105

C₄²

3₋¹⁺²⋊C₄

yes

complemented

106

C₂²×C₃

—

107

C₂²×C₃

—

108

C₂²×C₃

—

109

C₂²×C₃

C₂×C₃×Dic₆

yes

—

110

C₂²×C₃

C₂×C₃×D₁₂

yes

—

111

C₂²×C₃

C₂²×C₃×Dic₃

yes

characteristic

112

C₃×C₄

—

113

C₃×C₄

—

114

C₃×C₄

C₃×C₄⋊Dic₃

yes

—

115

Dic₃

—

116

C₃²

#4935

yes

characteristic

117

C₉

—

118

C₉

C₃×#384

yes

characteristic, complemented

119

C₂³

C₂²×AGL(1,ℤ/9)

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

3₋¹⁺²⋊Q₈

yes

—

122

C₂×C₄

C₄⋊AGL(1,ℤ/9)

yes

—

123

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

—

124

C₂×C₃

—

125

C₂×C₃

—

126

C₂×C₃

—

127

C₂×C₃

C₃×#869

yes

—

128

C₂×C₃

C₃×#874

yes

characteristic

129

C₂×C₃

C₂×C₃×C₄⋊Dic₃

yes

—

130

C₂²

C₂×3₋¹⁺²⋊Q₈

yes

—

131

C₂²

C₂×C₄⋊AGL(1,ℤ/9)

yes

—

132

C₂²

C₂²×3₋¹⁺²⋊C₄

yes

characteristic

133

C₄

—

134

C₄

C₄⋊(3₋¹⁺²⋊C₄)

yes

complemented

135

C₃

—

136

C₃

C₃×#4935

yes

characteristic

137

C₂

#63431⋊C₃

yes

—

138

C₂

#11721394

yes

characteristic

139

C₂

C₂×C₄⋊(3₋¹⁺²⋊C₄)

yes

—

	subgroup	G/H
center	C₂³	C₂²×AGL(1,ℤ/9)
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini, socle	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₄²×3₋¹⁺²	C₂
prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₉	C₂³×C₃
hypercenter	C₂×C₄²	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#384
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center

C₂³

C₂²×AGL(1,ℤ/9)

commutator

C₂²×C₉

C₂²×C₃×C₄

Frattini, socle

C₂³×C₃

C₂²×C₃×D₃

Fitting

C₂×C₄²×3₋¹⁺²

C₂

prime radical

C₂³×C₃²

C₂²×D₃

cosocle

C₂³×C₉

C₂³×C₃

C₂×C₄²

C₉

G ▷ C₂²×C₉ ▷ C₉

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
4. #10588615	C₃	semidirect
7. C₄²×3₋¹⁺²	C₄	semidirect
29. C₄²×C₉	C₃×C₄	semidirect
34. C₄×3₋¹⁺²	C₄⋊C₄	semidirect
78. C₄×C₉	C₃×C₄⋊C₄	semidirect
82. 3₋¹⁺²	#384	semidirect
105. C₄²	3₋¹⁺²⋊C₄	semidirect
118. C₉	C₃×#384	semidirect
134. C₄	C₄⋊(3₋¹⁺²⋊C₄)	semidirect

H ≅ G/N

type

4. #10588615

C₃

semidirect

7. C₄²×3₋¹⁺²

C₄

semidirect

29. C₄²×C₉

C₃×C₄

semidirect

34. C₄×3₋¹⁺²

C₄⋊C₄

semidirect

78. C₄×C₉

C₃×C₄⋊C₄

semidirect

82. 3₋¹⁺²

#384

semidirect

105. C₄²

3₋¹⁺²⋊C₄

semidirect

118. C₉

C₃×#384

semidirect

134. C₄

C₄⋊(3₋¹⁺²⋊C₄)

semidirect

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

subgroup	index
C₂×C₄²×3₋¹⁺²	2
C₂×C₄⋊(3₋¹⁺²⋊C₄)	2
C₃×#4935	3
#10588615	3

subgroup

index

C₂×C₄²×3₋¹⁺²

C₂×C₄⋊(3₋¹⁺²⋊C₄)

C₃×#4935

#10588615

Subgroups — #13142442

Conjugacy classes of subgroups

Complements

Maximal subgroups