Group #13141367 — subgroups

Subgroups — #13141367

Order 1728.

572 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 208 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×C₄×3₋¹⁺²⋊C₄	C₂	1	3	yes	maximal
2	3₋¹⁺²×#95	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²⋊#95	C₂	1	3	yes	maximal
4	C₃×#4648	—	3	1	no	maximal
5	#10588328	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×3₋¹⁺²	C₂²	1	3	yes	—
7	C₂²×3₋¹⁺²⋊C₄	C₂²	1	3	yes	—
8	C₂²×3₋¹⁺²⋊C₄	C₂²	1	1	yes	characteristic
9	C₄×3₋¹⁺²⋊C₄	C₄	1	12	yes	complemented
10	C₂×C₃×C₄×Dic₃	—	3	3	no	—
11	C₃²×#95	D₃	1	1	yes	characteristic
12	C₃×C₃⋊#95	—	3	3	no	—
13	C₂×C₄×Dic₉	C₂×C₃	1	3	yes	—
14	C₉×#95	—	2	1	no	—
15	C₉×#95	C₂×C₃	1	1	yes	characteristic
16	C₉⋊#95	C₂×C₃	1	3	yes	—
17	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
18	C₂×C₄×3₋¹⁺²	—	2	6	no	—
19	C₂×C₄×3₋¹⁺²	C₂×C₄	1	6	yes	—
20	C₂×3₋¹⁺²⋊C₄	—	2	3	no	—
21	C₂×3₋¹⁺²⋊C₄	—	2	3	no	—
22	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	6	yes	—
23	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	6	yes	—
24	C₃×#376	—	9	1	no	—
25	#4648	—	3	1	no	—
26	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
27	C₂²×C₃×Dic₃	—	3	1	no	—
28	C₂²×C₃×Dic₃	—	3	3	no	—
29	C₃×C₄×Dic₃	—	3	12	no	—
30	C₂²×C₄×C₉	—	2	3	no	—
31	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
32	C₂²×Dic₉	C₂²×C₃	1	3	yes	—
33	C₂²×Dic₉	C₂²×C₃	1	1	yes	characteristic
34	C₄×Dic₉	C₃×C₄	1	12	yes	complemented
35	C₂²×3₋¹⁺²	C₂²×C₄	1	3	yes	—
36	C₂²×3₋¹⁺²	#41	1	3	yes	—
37	C₂²×3₋¹⁺²	#41	1	1	yes	characteristic
38	C₄×3₋¹⁺²	—	2	6	no	—
39	3₋¹⁺²⋊C₄	—	2	6	no	—
40	3₋¹⁺²⋊C₄	C₄²	1	4	yes	—
41	C₂×C₃×C₄²	—	9	3	no	—
42	C₂×C₄×Dic₃	—	3	3	no	—
43	C₃×#95	—	3	1	no	—
44	C₃×#95	—	9	3	no	—
45	C₃×#95	C₃×D₃	1	1	yes	characteristic
46	C₃⋊#95	—	3	3	no	—
47	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
48	C₂×C₃²×C₄	—	2	6	no	—
49	C₂×C₃²×C₄	C₄×D₃	1	6	yes	—
50	C₂×C₃×Dic₃	—	3	6	no	—
51	C₂×C₃×Dic₃	—	3	6	no	—
52	C₂×C₃×Dic₃	—	6	3	no	—
53	C₂×C₃×Dic₃	—	6	3	no	—
54	C₂³×C₉	—	2	1	no	—
55	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
56	C₂×C₄×C₉	—	2	6	no	—
57	C₂×C₄×C₉	—	2	6	no	—
58	C₂×C₄×C₉	—	4	6	no	—
59	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
60	C₂×Dic₉	—	2	3	no	—
61	C₂×Dic₉	—	2	3	no	—
62	C₂×Dic₉	C₂×C₃×C₄	1	6	yes	—
63	C₂×Dic₉	C₂×C₃×C₄	1	6	yes	—
64	#376	—	9	1	no	Sylow
65	C₂×3₋¹⁺²	C₂×C₄²	1	1	yes	characteristic
66	C₂×3₋¹⁺²	#117	1	3	yes	—
67	C₂×3₋¹⁺²	#117	1	3	yes	—
68	C₂²×C₃×C₄	—	3	3	no	—
69	C₂²×C₃×C₄	—	9	1	no	—
70	C₂²×C₃×C₄	—	9	3	no	—
71	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
72	C₂²×Dic₃	—	3	1	no	—
73	C₂²×Dic₃	—	3	3	no	—
74	C₃×C₄²	—	9	12	no	—
75	C₄×Dic₃	—	3	12	no	—
76	C₂²×C₃²	C₂×C₄×D₃	1	3	yes	—
77	C₂²×C₃²	#237	1	3	yes	—
78	C₂²×C₃²	#239	1	1	yes	characteristic
79	C₃²×C₄	—	2	6	no	—
80	C₃×Dic₃	—	3	4	no	—
81	C₃×Dic₃	—	6	6	no	—
82	C₂²×C₉	—	2	3	no	—
83	C₂²×C₉	—	2	3	no	—
84	C₂²×C₉	—	2	1	no	—
85	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
86	C₂²×C₉	C₃×#41	1	3	yes	—
87	C₂²×C₉	C₃×#41	1	1	yes	characteristic
88	C₄×C₉	—	2	6	no	—
89	C₄×C₉	—	4	6	no	—
90	Dic₉	—	2	6	no	—
91	Dic₉	C₃×C₄²	1	4	yes	—
92	C₂×C₄²	—	9	3	no	—
93	#95	—	9	3	no	—
94	#95	AGL(1,ℤ/9)	1	1	yes	characteristic
95	3₋¹⁺²	#376	1	1	yes	characteristic, complemented, Sylow
96	C₂³×C₃	—	3	1	no	—
97	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
98	C₂×C₃×C₄	—	2	6	no	—
99	C₂×C₃×C₄	—	3	6	no	—
100	C₂×C₃×C₄	—	6	6	no	—
101	C₂×C₃×C₄	—	9	6	no	—
102	C₂×C₃×C₄	—	9	6	no	—
103	C₂×C₃×C₄	—	18	3	no	—
104	C₂×C₃×C₄	—	18	3	no	—
105	C₂×C₃×C₄	C₃×C₄×D₃	1	6	yes	—
106	C₂×Dic₃	—	3	6	no	—
107	C₂×Dic₃	—	3	6	no	—
108	C₂×Dic₃	—	6	3	no	—
109	C₂×Dic₃	—	6	3	no	—
110	C₂×C₃²	C₄²×D₃	1	1	yes	characteristic
111	C₂×C₃²	#877	1	3	yes	—
112	C₂×C₃²	#892	1	3	yes	—
113	C₂×C₉	—	2	3	no	—
114	C₂×C₉	—	2	3	no	—
115	C₂×C₉	—	2	1	no	—
116	C₂×C₉	C₂×C₃×C₄²	1	1	yes	characteristic
117	C₂×C₉	C₃×#117	1	3	yes	—
118	C₂×C₉	C₃×#117	1	3	yes	—
119	C₂²×C₄	—	9	1	no	—
120	C₂²×C₄	—	9	3	no	—
121	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
122	C₄²	—	9	12	no	—
123	C₂²×C₃	—	3	3	no	—
124	C₂²×C₃	—	3	3	no	—
125	C₂²×C₃	—	3	1	no	—
126	C₂²×C₃	C₂×C₃×C₄×D₃	1	3	yes	—
127	C₂²×C₃	C₃×#237	1	3	yes	—
128	C₂²×C₃	C₃×#239	1	1	yes	characteristic
129	C₃×C₄	—	2	6	no	—
130	C₃×C₄	—	6	6	no	—
131	C₃×C₄	—	9	4	no	—
132	C₃×C₄	—	18	6	no	—
133	Dic₃	—	3	4	no	—
134	Dic₃	—	6	6	no	—
135	C₃²	#4648	1	1	yes	characteristic
136	C₉	—	2	1	no	—
137	C₉	C₃×#376	1	1	yes	characteristic, complemented
138	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	—	9	6	no	—
141	C₂×C₄	—	9	6	no	—
142	C₂×C₄	—	18	3	no	—
143	C₂×C₄	—	18	3	no	—
144	C₂×C₄	C₄×AGL(1,ℤ/9)	1	6	yes	—
145	C₂×C₃	—	3	3	no	—
146	C₂×C₃	—	3	1	no	—
147	C₂×C₃	—	3	3	no	—
148	C₂×C₃	C₃×C₄²×D₃	1	1	yes	characteristic
149	C₂×C₃	C₃×#877	1	3	yes	—
150	C₂×C₃	C₃×#892	1	3	yes	—
151	C₂²	C₂×C₄×AGL(1,ℤ/9)	1	3	yes	—
152	C₂²	#3699⋊C₃	1	3	yes	—
153	C₂²	#88724	1	1	yes	characteristic
154	C₄	—	2	6	no	—
155	C₄	—	9	4	no	—
156	C₄	—	18	6	no	—
157	C₃	—	3	1	no	—
158	C₃	C₃×#4648	1	1	yes	characteristic
159	C₂	C₄²×AGL(1,ℤ/9)	1	1	yes	characteristic
160	C₂	#63439⋊C₃	1	3	yes	—
161	C₂	#63454⋊C₃	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×3₋¹⁺²⋊C₄

C₂

yes

maximal

3₋¹⁺²×#95

C₂

yes

maximal, characteristic

3₋¹⁺²⋊#95

C₂

yes

—

C₃

yes

maximal, characteristic, complemented

C₂²×C₄×3₋¹⁺²

C₂²

yes

—

C₂²×3₋¹⁺²⋊C₄

C₂²

yes

—

C₂²×3₋¹⁺²⋊C₄

C₂²

yes

characteristic

C₄×3₋¹⁺²⋊C₄

C₄

yes

complemented

C₂×C₃×C₄×Dic₃

—

C₃²×#95

D₃

yes

characteristic

C₃×C₃⋊#95

—

C₂×C₄×Dic₉

C₂×C₃

yes

—

C₉×#95

—

C₉×#95

C₂×C₃

yes

characteristic

C₉⋊#95

C₂×C₃

yes

—

C₂³×3₋¹⁺²

C₂³

yes

characteristic

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

—

C₂×3₋¹⁺²⋊C₄

—

C₂×3₋¹⁺²⋊C₄

—

C₂×3₋¹⁺²⋊C₄

C₂×C₄

yes

—

C₂×3₋¹⁺²⋊C₄

C₂×C₄

yes

—

C₃×#376

—

#4648

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃×Dic₃

—

C₂²×C₃×Dic₃

—

C₃×C₄×Dic₃

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂²×C₃

yes

—

C₂²×Dic₉

C₂²×C₃

yes

—

C₂²×Dic₉

C₂²×C₃

yes

characteristic

C₄×Dic₉

C₃×C₄

yes

complemented

C₂²×3₋¹⁺²

C₂²×C₄

yes

—

C₂²×3₋¹⁺²

#41

yes

—

C₂²×3₋¹⁺²

#41

yes

characteristic

C₄×3₋¹⁺²

—

3₋¹⁺²⋊C₄

—

3₋¹⁺²⋊C₄

C₄²

yes

—

C₂×C₃×C₄²

—

C₂×C₄×Dic₃

—

C₃×#95

—

C₃×#95

—

C₃×#95

C₃×D₃

yes

characteristic

C₃⋊#95

—

C₂³×C₃²

C₂²×D₃

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₄×D₃

yes

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

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×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

C₂×C₃×C₄

yes

—

C₂×Dic₉

C₂×C₃×C₄

yes

—

#376

—

Sylow

C₂×3₋¹⁺²

C₂×C₄²

yes

characteristic

C₂×3₋¹⁺²

#117

yes

—

C₂×3₋¹⁺²

#117

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₃×C₄²

—

C₄×Dic₃

—

C₂²×C₃²

C₂×C₄×D₃

yes

—

C₂²×C₃²

#237

yes

—

C₂²×C₃²

#239

yes

characteristic

C₃²×C₄

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂²×C₃×C₄

yes

—

C₂²×C₉

C₃×#41

yes

—

C₂²×C₉

C₃×#41

yes

characteristic

C₄×C₉

—

C₄×C₉

—

Dic₉

—

Dic₉

C₃×C₄²

yes

—

C₂×C₄²

—

—

yes

3₋¹⁺²

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

C₂²×C₃×D₃

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

—

101

C₂×C₃×C₄

—

102

C₂×C₃×C₄

—

103

C₂×C₃×C₄

—

104

C₂×C₃×C₄

—

105

C₂×C₃×C₄

C₃×C₄×D₃

yes

—

106

C₂×Dic₃

—

107

C₂×Dic₃

—

108

C₂×Dic₃

—

109

C₂×Dic₃

—

110

C₂×C₃²

C₄²×D₃

yes

characteristic

111

C₂×C₃²

#877

yes

—

112

C₂×C₃²

#892

yes

—

113

C₂×C₉

—

114

C₂×C₉

—

115

C₂×C₉

—

116

C₂×C₉

C₂×C₃×C₄²

yes

characteristic

117

C₂×C₉

C₃×#117

yes

—

118

C₂×C₉

C₃×#117

yes

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

C₂×AGL(1,ℤ/9)

yes

—

122

C₄²

—

123

C₂²×C₃

—

124

C₂²×C₃

—

125

C₂²×C₃

—

126

C₂²×C₃

C₂×C₃×C₄×D₃

yes

—

127

C₂²×C₃

C₃×#237

yes

—

128

C₂²×C₃

C₃×#239

yes

characteristic

129

C₃×C₄

—

130

C₃×C₄

—

131

C₃×C₄

—

132

C₃×C₄

—

133

Dic₃

—

134

Dic₃

—

135

C₃²

#4648

yes

characteristic

136

C₉

—

137

C₉

C₃×#376

yes

characteristic, complemented

138

C₂³

C₂²×AGL(1,ℤ/9)

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

C₄×AGL(1,ℤ/9)

yes

—

145

C₂×C₃

—

146

C₂×C₃

—

147

C₂×C₃

—

148

C₂×C₃

C₃×C₄²×D₃

yes

characteristic

149

C₂×C₃

C₃×#877

yes

—

150

C₂×C₃

C₃×#892

yes

—

151

C₂²

C₂×C₄×AGL(1,ℤ/9)

yes

—

152

C₂²

#3699⋊C₃

yes

—

153

C₂²

#88724

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₃

—

158

C₃

C₃×#4648

yes

characteristic

159

C₂

C₄²×AGL(1,ℤ/9)

yes

characteristic

160

C₂

#63439⋊C₃

yes

—

161

C₂

#63454⋊C₃

yes

—

	subgroup	G/H
center	C₂³	C₂²×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂×C₃×C₄²
Frattini, socle	C₂³×C₃	C₂²×C₃×D₃
Fitting	3₋¹⁺²×#95	C₂
prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₉	C₂³×C₃
hypercenter	#95	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#376
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center

C₂³

C₂²×AGL(1,ℤ/9)

commutator

C₂×C₉

C₂×C₃×C₄²

Frattini, socle

C₂³×C₃

C₂²×C₃×D₃

Fitting

3₋¹⁺²×#95

C₂

prime radical

C₂³×C₃²

C₂²×D₃

cosocle

C₂³×C₉

C₂³×C₃

C₉

G ▷ C₂×C₉ ▷ C₉

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
5. #10588328	C₃	semidirect
9. C₄×3₋¹⁺²⋊C₄	C₄	semidirect
34. C₄×Dic₉	C₃×C₄	semidirect
95. 3₋¹⁺²	#376	semidirect
137. C₉	C₃×#376	semidirect

H ≅ G/N

type

5. #10588328

C₃

semidirect

9. C₄×3₋¹⁺²⋊C₄

C₄

semidirect

34. C₄×Dic₉

C₃×C₄

semidirect

95. 3₋¹⁺²

#376

semidirect

137. C₉

C₃×#376

semidirect

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

subgroup	index
C₂×C₄×3₋¹⁺²⋊C₄	2
3₋¹⁺²×#95	2
3₋¹⁺²⋊#95	2
C₃×#4648	3
#10588328	3

subgroup

index

C₂×C₄×3₋¹⁺²⋊C₄

3₋¹⁺²×#95

3₋¹⁺²⋊#95

C₃×#4648

#10588328

Subgroups — #13141367

Conjugacy classes of subgroups

Complements

Maximal subgroups