Group #13143675 — subgroups

Subgroups — #13143675

Order 1728.

1892 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 1892 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	C₂⁵×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×#3676⋊C₃	C₂	1	6	yes	maximal, complemented
3	C₃×#5247	—	3	1	no	maximal
4	#10588927	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×3₋¹⁺²	—	2	12	no	—
6	C₂⁴×3₋¹⁺²	C₄	1	4	yes	complemented
7	C₂⁴×3₋¹⁺²	C₂²	1	3	yes	—
8	C₂²×3₋¹⁺²⋊C₄	C₂²	1	4	yes	complemented
9	#3676⋊C₃	—	2	12	no	—
10	C₂⁵×C₃²	D₃	1	1	yes	characteristic
11	C₂⁵×C₉	—	2	1	no	—
12	C₂⁵×C₉	C₂×C₃	1	1	yes	characteristic
13	C₂×C₃×#217	—	3	6	no	—
14	C₂×#3676	C₂×C₃	1	6	yes	complemented
15	C₂³×3₋¹⁺²	—	2	32	no	—
16	C₂³×3₋¹⁺²	—	2	24	no	—
17	C₂³×3₋¹⁺²	—	2	12	no	—
18	C₂³×3₋¹⁺²	C₂×C₄	1	6	yes	—
19	C₂³×3₋¹⁺²	D₄	1	12	yes	—
20	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
21	C₂×3₋¹⁺²⋊C₄	—	2	12	no	—
22	C₃×#379	—	9	1	no	—
23	#5247	—	3	1	no	—
24	C₂⁴×C₃²	—	2	12	no	—
25	C₂⁴×C₃²	Dic₃	1	4	yes	—
26	C₂⁴×C₃²	C₂×D₃	1	3	yes	—
27	C₂²×C₃×Dic₃	—	3	4	no	—
28	C₂⁴×C₉	—	2	24	no	—
29	C₂⁴×C₉	—	2	4	no	—
30	C₂⁴×C₉	—	2	3	no	—
31	C₂⁴×C₉	—	2	12	no	—
32	C₂⁴×C₉	C₃×C₄	1	4	yes	complemented
33	C₂⁴×C₉	C₂²×C₃	1	3	yes	—
34	C₃×#217	—	6	12	no	—
35	C₂²×Dic₉	C₂²×C₃	1	4	yes	complemented
36	#3676	—	2	12	no	—
37	C₂²×3₋¹⁺²	—	2	32	no	—
38	C₂²×3₋¹⁺²	—	2	12	no	—
39	C₂²×3₋¹⁺²	—	2	24	no	—
40	C₂²×3₋¹⁺²	C₂²⋊C₄	1	12	yes	complemented
41	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
42	C₂²×3₋¹⁺²	C₂×D₄	1	6	yes	—
43	3₋¹⁺²⋊C₄	—	4	4	no	—
44	C₂⁵×C₃	—	3	1	no	—
45	C₂⁵×C₃	C₃×D₃	1	1	yes	characteristic
46	C₂×C₃×C₂²⋊C₄	—	9	6	no	—
47	C₂×#217	—	3	6	no	—
48	C₂³×C₃²	—	2	32	no	—
49	C₂³×C₃²	—	2	24	no	—
50	C₂³×C₃²	—	2	12	no	—
51	C₂³×C₃²	C₂×Dic₃	1	6	yes	—
52	C₂³×C₃²	#67	1	12	yes	—
53	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
54	C₂×C₃×Dic₃	—	6	12	no	—
55	C₂³×C₉	—	2	64	no	—
56	C₂³×C₉	—	2	32	no	—
57	C₂³×C₉	—	2	48	no	—
58	C₂³×C₉	—	2	24	no	—
59	C₂³×C₉	—	2	24	no	—
60	C₂³×C₉	—	2	12	no	—
61	C₂³×C₉	—	2	6	no	—
62	C₂³×C₉	—	2	12	no	—
63	C₂³×C₉	—	2	1	no	—
64	C₂³×C₉	C₂×C₃×C₄	1	6	yes	—
65	C₂³×C₉	C₃×D₄	1	12	yes	—
66	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
67	C₂×Dic₉	—	2	12	no	—
68	#379	—	9	1	no	Sylow
69	C₂×3₋¹⁺²	—	2	12	no	—
70	C₂×3₋¹⁺²	C₂×C₂²⋊C₄	1	3	yes	—
71	C₂×3₋¹⁺²	#120	1	4	yes	—
72	C₂⁴×C₃	—	2	12	no	—
73	C₂⁴×C₃	—	3	4	no	—
74	C₂⁴×C₃	—	3	3	no	—
75	C₂⁴×C₃	—	6	12	no	—
76	C₂⁴×C₃	C₃×Dic₃	1	4	yes	—
77	C₂⁴×C₃	C₂×C₃×D₃	1	3	yes	—
78	C₂²×C₃×C₄	—	9	4	no	—
79	C₂²×Dic₃	—	3	4	no	—
80	C₃×C₂²⋊C₄	—	18	12	no	—
81	#217	—	6	12	no	—
82	C₂²×C₃²	—	2	32	no	—
83	C₂²×C₃²	—	2	12	no	—
84	C₂²×C₃²	—	2	24	no	—
85	C₂²×C₃²	#217	1	12	yes	—
86	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
87	C₂²×C₃²	C₂×#67	1	6	yes	—
88	C₃×Dic₃	—	12	4	no	—
89	C₂²×C₉	—	2	64	no	—
90	C₂²×C₉	—	2	32	no	—
91	C₂²×C₉	—	2	24	no	—
92	C₂²×C₉	—	2	48	no	—
93	C₂²×C₉	—	2	12	no	—
94	C₂²×C₉	—	2	24	no	—
95	C₂²×C₉	—	2	12	no	—
96	C₂²×C₉	—	2	1	no	—
97	C₂²×C₉	—	2	6	no	—
98	C₂²×C₉	C₃×C₂²⋊C₄	1	12	yes	complemented
99	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
100	C₂²×C₉	C₂×C₃×D₄	1	6	yes	—
101	Dic₉	—	4	4	no	—
102	C₂⁵	AGL(1,ℤ/9)	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	—	9	6	no	—
104	3₋¹⁺²	#379	1	1	yes	characteristic, complemented, Sylow
105	C₂³×C₃	—	2	32	no	—
106	C₂³×C₃	—	2	24	no	—
107	C₂³×C₃	—	2	12	no	—
108	C₂³×C₃	—	3	6	no	—
109	C₂³×C₃	—	3	12	no	—
110	C₂³×C₃	—	3	1	no	—
111	C₂³×C₃	—	6	12	no	—
112	C₂³×C₃	—	6	24	no	—
113	C₂³×C₃	—	6	32	no	—
114	C₂³×C₃	C₂×C₃×Dic₃	1	6	yes	—
115	C₂³×C₃	C₃×#67	1	12	yes	—
116	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
117	C₂×C₃×C₄	—	18	12	no	—
118	C₂×Dic₃	—	6	12	no	—
119	C₂×C₃²	—	2	12	no	—
120	C₂×C₃²	C₂×#217	1	3	yes	—
121	C₂×C₃²	#953	1	4	yes	—
122	C₂×C₉	—	2	24	no	—
123	C₂×C₉	—	2	12	no	—
124	C₂×C₉	—	2	3	no	—
125	C₂×C₉	—	2	4	no	—
126	C₂×C₉	C₂×C₃×C₂²⋊C₄	1	3	yes	—
127	C₂×C₉	C₃×#120	1	4	yes	—
128	C₂⁴	—	2	12	no	—
129	C₂⁴	3₋¹⁺²⋊C₄	1	4	yes	complemented
130	C₂⁴	C₂×AGL(1,ℤ/9)	1	3	yes	—
131	C₂²×C₄	—	9	4	no	—
132	C₂²⋊C₄	—	18	12	no	—
133	C₂²×C₃	—	2	32	no	—
134	C₂²×C₃	—	2	12	no	—
135	C₂²×C₃	—	2	24	no	—
136	C₂²×C₃	—	3	1	no	—
137	C₂²×C₃	—	3	6	no	—
138	C₂²×C₃	—	3	12	no	—
139	C₂²×C₃	—	6	12	no	—
140	C₂²×C₃	—	6	24	no	—
141	C₂²×C₃	—	6	32	no	—
142	C₂²×C₃	C₃×#217	1	12	yes	—
143	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
144	C₂²×C₃	C₂×C₃×#67	1	6	yes	—
145	C₃×C₄	—	36	4	no	—
146	Dic₃	—	12	4	no	—
147	C₃²	#5247	1	1	yes	characteristic
148	C₉	—	2	1	no	—
149	C₉	C₃×#379	1	1	yes	characteristic, complemented
150	C₂³	—	2	32	no	—
151	C₂³	—	2	12	no	—
152	C₂³	—	2	24	no	—
153	C₂³	C₂×3₋¹⁺²⋊C₄	1	6	yes	—
154	C₂³	#611⋊C₃	1	12	yes	—
155	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
156	C₂×C₄	—	18	12	no	—
157	C₂×C₃	—	2	12	no	—
158	C₂×C₃	—	3	3	no	—
159	C₂×C₃	—	3	4	no	—
160	C₂×C₃	—	6	12	no	—
161	C₂×C₃	C₂×C₃×#217	1	3	yes	—
162	C₂×C₃	C₃×#953	1	4	yes	—
163	C₂²	—	2	32	no	—
164	C₂²	—	2	12	no	—
165	C₂²	—	2	24	no	—
166	C₂²	#3676⋊C₃	1	12	yes	complemented
167	C₂²	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
168	C₂²	C₂×#611⋊C₃	1	6	yes	—
169	C₄	—	36	4	no	—
170	C₃	—	3	1	no	—
171	C₃	C₃×#5247	1	1	yes	characteristic
172	C₂	—	2	12	no	—
173	C₂	C₂×#3676⋊C₃	1	3	yes	—
174	C₂	#11721704	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁵×3₋¹⁺²

C₂

yes

maximal, characteristic

C₂×#3676⋊C₃

C₂

yes

maximal, complemented

C₃×#5247

—

maximal

#10588927

C₃

yes

maximal, characteristic, complemented

C₂⁴×3₋¹⁺²

—

C₂⁴×3₋¹⁺²

C₄

yes

complemented

C₂⁴×3₋¹⁺²

C₂²

yes

—

C₂²×3₋¹⁺²⋊C₄

C₂²

yes

complemented

#3676⋊C₃

—

C₂⁵×C₃²

D₃

yes

characteristic

C₂⁵×C₉

—

C₂⁵×C₉

C₂×C₃

yes

characteristic

C₂×C₃×#217

—

C₂×#3676

C₂×C₃

yes

complemented

C₂³×3₋¹⁺²

—

C₂³×3₋¹⁺²

—

C₂³×3₋¹⁺²

—

C₂³×3₋¹⁺²

C₂×C₄

yes

—

C₂³×3₋¹⁺²

D₄

yes

—

C₂³×3₋¹⁺²

C₂³

yes

characteristic

C₂×3₋¹⁺²⋊C₄

—

C₃×#379

—

#5247

—

C₂⁴×C₃²

—

C₂⁴×C₃²

Dic₃

yes

—

C₂⁴×C₃²

C₂×D₃

yes

—

C₂²×C₃×Dic₃

—

C₂⁴×C₉

—

C₂⁴×C₉

—

C₂⁴×C₉

—

C₂⁴×C₉

—

C₂⁴×C₉

C₃×C₄

yes

complemented

C₂⁴×C₉

C₂²×C₃

yes

—

C₃×#217

—

C₂²×Dic₉

C₂²×C₃

yes

complemented

#3676

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

C₂²⋊C₄

yes

complemented

C₂²×3₋¹⁺²

C₂²×C₄

yes

characteristic

C₂²×3₋¹⁺²

C₂×D₄

yes

—

3₋¹⁺²⋊C₄

—

C₂⁵×C₃

—

C₂⁵×C₃

C₃×D₃

yes

characteristic

C₂×C₃×C₂²⋊C₄

—

C₂×#217

—

C₂³×C₃²

—

C₂³×C₃²

—

C₂³×C₃²

—

C₂³×C₃²

C₂×Dic₃

yes

—

C₂³×C₃²

#67

yes

—

C₂³×C₃²

C₂²×D₃

yes

characteristic

C₂×C₃×Dic₃

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

C₂×C₃×C₄

yes

—

C₂³×C₉

C₃×D₄

yes

—

C₂³×C₉

C₂³×C₃

yes

characteristic

C₂×Dic₉

—

#379

—

Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂×C₂²⋊C₄

yes

—

C₂×3₋¹⁺²

#120

yes

—

C₂⁴×C₃

—

C₂⁴×C₃

—

C₂⁴×C₃

—

C₂⁴×C₃

—

C₂⁴×C₃

C₃×Dic₃

yes

—

C₂⁴×C₃

C₂×C₃×D₃

yes

—

C₂²×C₃×C₄

—

C₂²×Dic₃

—

C₃×C₂²⋊C₄

—

#217

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

#217

yes

—

C₂²×C₃²

C₂²×Dic₃

yes

characteristic

C₂²×C₃²

C₂×#67

yes

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₃×C₂²⋊C₄

yes

complemented

C₂²×C₉

C₂²×C₃×C₄

yes

characteristic

100

C₂²×C₉

C₂×C₃×D₄

yes

—

101

Dic₉

—

102

C₂⁵

AGL(1,ℤ/9)

yes

characteristic

103

C₂×C₂²⋊C₄

—

104

3₋¹⁺²

#379

yes

characteristic, complemented, Sylow

105

C₂³×C₃

—

106

C₂³×C₃

—

107

C₂³×C₃

—

108

C₂³×C₃

—

109

C₂³×C₃

—

110

C₂³×C₃

—

111

C₂³×C₃

—

112

C₂³×C₃

—

113

C₂³×C₃

—

114

C₂³×C₃

C₂×C₃×Dic₃

yes

—

115

C₂³×C₃

C₃×#67

yes

—

116

C₂³×C₃

C₂²×C₃×D₃

yes

characteristic

117

C₂×C₃×C₄

—

118

C₂×Dic₃

—

119

C₂×C₃²

—

120

C₂×C₃²

C₂×#217

yes

—

121

C₂×C₃²

#953

yes

—

122

C₂×C₉

—

123

C₂×C₉

—

124

C₂×C₉

—

125

C₂×C₉

—

126

C₂×C₉

C₂×C₃×C₂²⋊C₄

yes

—

127

C₂×C₉

C₃×#120

yes

—

128

C₂⁴

—

129

C₂⁴

3₋¹⁺²⋊C₄

yes

complemented

130

C₂⁴

C₂×AGL(1,ℤ/9)

yes

—

131

C₂²×C₄

—

132

C₂²⋊C₄

—

133

C₂²×C₃

—

134

C₂²×C₃

—

135

C₂²×C₃

—

136

C₂²×C₃

—

137

C₂²×C₃

—

138

C₂²×C₃

—

139

C₂²×C₃

—

140

C₂²×C₃

—

141

C₂²×C₃

—

142

C₂²×C₃

C₃×#217

yes

—

143

C₂²×C₃

C₂²×C₃×Dic₃

yes

characteristic

144

C₂²×C₃

C₂×C₃×#67

yes

—

145

C₃×C₄

—

146

Dic₃

—

147

C₃²

#5247

yes

characteristic

148

C₉

—

149

C₉

C₃×#379

yes

characteristic, complemented

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

C₂×3₋¹⁺²⋊C₄

yes

—

154

C₂³

#611⋊C₃

yes

—

155

C₂³

C₂²×AGL(1,ℤ/9)

yes

characteristic

156

C₂×C₄

—

157

C₂×C₃

—

158

C₂×C₃

—

159

C₂×C₃

—

160

C₂×C₃

—

161

C₂×C₃

C₂×C₃×#217

yes

—

162

C₂×C₃

C₃×#953

yes

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

#3676⋊C₃

yes

complemented

167

C₂²

C₂²×3₋¹⁺²⋊C₄

yes

characteristic

168

C₂²

C₂×#611⋊C₃

yes

—

169

C₄

—

170

C₃

—

171

C₃

C₃×#5247

yes

characteristic

172

C₂

—

173

C₂

C₂×#3676⋊C₃

yes

—

174

C₂

#11721704

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	C₂⁵×3₋¹⁺²	C₂
prime radical	C₂⁵×C₃²	D₃
cosocle	C₂³×C₉	C₂³×C₃
hypercenter	C₂⁵	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#379
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂⁵

subgroup

G/H

center

C₂³

C₂²×AGL(1,ℤ/9)

commutator

C₂²×C₉

C₂²×C₃×C₄

Frattini, socle

C₂³×C₃

C₂²×C₃×D₃

Fitting

C₂⁵×3₋¹⁺²

C₂

prime radical

C₂⁵×C₃²

D₃

cosocle

C₂³×C₉

C₂³×C₃

C₂⁵

C₉

G ▷ C₂²×C₉ ▷ C₉

C₁ ◁ C₂³ ◁ C₂⁵

N	H ≅ G/N	type
2. C₂×#3676⋊C₃	C₂	semidirect
4. #10588927	C₃	semidirect
6. C₂⁴×3₋¹⁺²	C₄	semidirect
8. C₂²×3₋¹⁺²⋊C₄	C₂²	semidirect
14. C₂×#3676	C₂×C₃	semidirect
32. C₂⁴×C₉	C₃×C₄	semidirect
35. C₂²×Dic₉	C₂²×C₃	semidirect
40. C₂²×3₋¹⁺²	C₂²⋊C₄	semidirect
98. C₂²×C₉	C₃×C₂²⋊C₄	semidirect
104. 3₋¹⁺²	#379	semidirect
129. C₂⁴	3₋¹⁺²⋊C₄	semidirect
149. C₉	C₃×#379	semidirect
166. C₂²	#3676⋊C₃	semidirect

H ≅ G/N

type

2. C₂×#3676⋊C₃

C₂

semidirect

4. #10588927

C₃

semidirect

6. C₂⁴×3₋¹⁺²

C₄

semidirect

8. C₂²×3₋¹⁺²⋊C₄

C₂²

semidirect

14. C₂×#3676

C₂×C₃

semidirect

32. C₂⁴×C₉

C₃×C₄

semidirect

35. C₂²×Dic₉

C₂²×C₃

semidirect

40. C₂²×3₋¹⁺²

C₂²⋊C₄

semidirect

98. C₂²×C₉

C₃×C₂²⋊C₄

semidirect

104. 3₋¹⁺²

#379

semidirect

129. C₂⁴

3₋¹⁺²⋊C₄

semidirect

149. C₉

C₃×#379

semidirect

166. C₂²

#3676⋊C₃

semidirect

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

subgroup	index
C₂⁵×3₋¹⁺²	2
C₂×#3676⋊C₃	2
C₃×#5247	3
#10588927	3

subgroup

index

C₂⁵×3₋¹⁺²

C₂×#3676⋊C₃

C₃×#5247

#10588927

Subgroups — #13143675

Conjugacy classes of subgroups

Complements

Maximal subgroups