Group #18003227 — subgroups

Subgroups — #18003227

Order 4736.

484 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 197 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×D₃₇	C₂	1	1	yes	maximal, characteristic
2	#14557588	C₂	1	4	yes	maximal, complemented
3	#14557584	C₂	1	4	yes	maximal, complemented
4	#14557556	C₂	1	2	yes	maximal, complemented
5	#14557496	C₂	1	2	yes	maximal
6	C₄×#10596934	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₃₇	C₄	1	2	yes	—
9	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
10	C₂×#10596934	C₂²	1	2	yes	—
11	C₂×C₄×Dic₃₇	C₄	1	2	yes	—
12	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
13	C₄²×D₃₇	C₄	1	2	yes	—
14	C₄²×D₃₇	C₄	1	2	yes	—
15	C₄²×D₃₇	C₂²	1	2	yes	—
16	C₄²×D₃₇	C₂²	1	2	yes	—
17	#11924465	C₂²	1	8	yes	complemented
18	#11924448	C₂²	1	8	yes	—
19	#11924447	C₂²	1	2	yes	complemented
20	C₄×#64436	C₂²	1	2	yes	complemented
21	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
23	#97×C₃₇	C₂²	1	2	yes	complemented
24	C₄×C₈×C₃₇	C₂²	1	2	yes	complemented
25	C₂³×D₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
28	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
30	C₂×C₄×D₃₇	—	2	4	no	—
31	C₂×C₄×D₃₇	—	2	8	no	—
32	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₃₇	C₂³	1	2	yes	—
38	M₁₆×C₃₇	—	2	4	no	—
39	C₂×C₈×C₃₇	C₂³	1	4	yes	—
40	C₄²×C₃₇	C₂×C₄	1	2	yes	—
41	C₄²×C₃₇	C₂³	1	1	yes	characteristic
42	C₄²×C₃₇	C₂³	1	1	yes	characteristic
43	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
45	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
46	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
47	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
48	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
49	C₄×Dic₃₇	C₂³	1	2	yes	—
50	#10596934	—	2	4	no	—
51	C₂×#64436	C₂³	1	4	yes	—
52	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×D₃₇	—	2	2	no	—
54	C₂²×D₃₇	—	2	4	no	—
55	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₃₇	—	2	4	no	—
57	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₃₇	#41	1	4	yes	—
61	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₃₇	—	2	2	no	—
63	C₂×Dic₃₇	—	2	4	no	—
64	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
65	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
66	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
67	C₄×D₃₇	—	2	4	no	—
68	C₄×D₃₇	—	2	4	no	—
69	C₄×D₃₇	—	2	4	no	—
70	C₄×D₃₇	—	2	4	no	—
71	C₄×D₃₇	—	2	4	no	—
72	C₄×D₃₇	M₁₆	1	8	yes	complemented
73	C₈×C₃₇	—	2	4	no	—
74	#64436	—	2	4	no	—
75	C₂²×C₃₇	—	2	2	no	—
76	C₂²×C₃₇	#117	1	2	yes	—
77	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
78	C₂×D₃₇	—	2	8	no	—
79	C₂×D₃₇	—	2	2	no	—
80	C₂×D₃₇	—	2	2	no	—
81	C₂×D₃₇	C₂×M₁₆	1	4	yes	—
82	C₄×C₃₇	—	2	2	no	—
83	C₄×C₃₇	#117	1	2	yes	—
84	C₄×C₃₇	C₂×M₁₆	1	4	yes	—
85	C₄×C₃₇	C₂×#41	1	2	yes	—
86	Dic₃₇	—	2	2	no	—
87	Dic₃₇	—	2	2	no	—
88	Dic₃₇	C₂×M₁₆	1	4	yes	—
89	#2920	—	37	1	no	maximal, Sylow
90	C₂×C₃₇	—	2	2	no	—
91	C₂×C₃₇	C₂×#117	1	1	yes	characteristic
92	C₂×C₃₇	C₂²×M₁₆	1	2	yes	—
93	D₃₇	—	2	4	no	—
94	C₂²×C₄²	—	37	1	no	—
95	#432	—	37	4	no	—
96	#431	—	37	4	no	—
97	#423	—	37	2	no	—
98	#407	—	37	2	no	—
99	C₄×M₁₆	—	37	1	no	—
100	C₄×M₁₆	D₃₇	1	1	yes	characteristic, complemented
101	C₃₇	#2920	1	1	yes	characteristic, complemented, Sylow
102	C₂³×C₄	—	37	2	no	—
103	C₂³×C₄	—	37	1	no	—
104	C₂×M₁₆	—	37	2	no	—
105	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
106	C₂×C₄²	—	37	2	no	—
107	C₂×C₄²	—	37	1	no	—
108	C₂×C₄²	—	37	2	no	—
109	C₂×C₄²	—	37	2	no	—
110	C₂×C₄²	—	37	2	no	—
111	C₂×C₄²	—	37	2	no	—
112	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
113	C₄⋊C₈	—	37	8	no	—
114	C₂²⋊C₈	—	37	8	no	—
115	#97	—	37	2	no	—
116	#97	C₂×D₃₇	1	2	yes	complemented
117	C₄×C₈	—	37	2	no	—
118	C₄×C₈	C₂×D₃₇	1	2	yes	complemented
119	C₂⁴	—	37	1	no	—
120	C₂²×C₄	—	37	2	no	—
121	C₂²×C₄	—	37	2	no	—
122	C₂²×C₄	—	37	1	no	—
123	C₂²×C₄	—	37	2	no	—
124	C₂²×C₄	—	37	2	no	—
125	C₂²×C₄	—	37	2	no	—
126	C₂²×C₄	—	37	2	no	—
127	C₂²×C₄	—	37	2	no	—
128	C₂²×C₄	—	74	8	no	—
129	C₂²×C₄	—	74	4	no	—
130	C₂²×C₄	C₄×D₃₇	1	2	yes	—
131	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
132	M₁₆	—	2	4	no	—
133	M₁₆	—	74	4	no	—
134	C₂×C₈	—	37	4	no	—
135	C₂×C₈	C₂²×D₃₇	1	4	yes	—
136	C₄²	—	37	1	no	—
137	C₄²	—	37	2	no	—
138	C₄²	—	37	2	no	—
139	C₄²	—	37	2	no	—
140	C₄²	—	37	1	no	—
141	C₄²	—	37	2	no	—
142	C₄²	—	37	2	no	—
143	C₄²	C₄×D₃₇	1	2	yes	—
144	C₄²	C₂²×D₃₇	1	1	yes	characteristic
145	C₄²	C₂²×D₃₇	1	1	yes	characteristic
146	C₂³	—	37	2	no	—
147	C₂³	—	74	4	no	—
148	C₂³	—	74	2	no	—
149	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	37	2	no	—
152	C₂×C₄	—	37	8	no	—
153	C₂×C₄	—	37	2	no	—
154	C₂×C₄	—	37	2	no	—
155	C₂×C₄	—	74	4	no	—
156	C₂×C₄	—	74	4	no	—
157	C₂×C₄	—	74	4	no	—
158	C₂×C₄	—	74	2	no	—
159	C₂×C₄	—	74	4	no	—
160	C₂×C₄	—	74	4	no	—
161	C₂×C₄	—	74	4	no	—
162	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
163	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
165	C₂×C₄	#10596962	1	4	yes	—
166	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
167	C₈	—	2	4	no	—
168	C₈	—	74	4	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	37	4	no	—
171	C₂²	—	74	8	no	—
172	C₂²	—	74	2	no	—
173	C₂²	—	74	2	no	—
174	C₂²	#11924493	1	2	yes	—
175	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	37	4	no	—
178	C₄	—	74	2	no	—
179	C₄	—	74	2	no	—
180	C₄	#11924493	1	2	yes	—
181	C₄	M₁₆×D₃₇	1	4	yes	—
182	C₄	C₂×#10596962	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	74	4	no	—
185	C₂	C₂×#11924493	1	1	yes	characteristic
186	C₂	C₂×M₁₆×D₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×D₃₇

C₂

yes

maximal, characteristic

#14557588

C₂

yes

maximal, complemented

#14557584

C₂

yes

maximal, complemented

#14557556

C₂

yes

maximal, complemented

#14557496

C₂

yes

maximal

C₄×#10596934

C₂

yes

maximal, characteristic, complemented

C₄×M₁₆×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₃₇

C₄

yes

—

C₂²×C₄×D₃₇

C₂²

yes

characteristic

C₂×#10596934

C₂²

yes

—

C₂×C₄×Dic₃₇

C₄

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₄²×D₃₇

C₄

yes

—

C₄²×D₃₇

C₄

yes

—

C₄²×D₃₇

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

—

#11924465

C₂²

yes

complemented

#11924448

C₂²

yes

—

#11924447

C₂²

yes

complemented

C₄×#64436

C₂²

yes

complemented

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#97×C₃₇

C₂²

yes

complemented

C₄×C₈×C₃₇

C₂²

yes

complemented

C₂³×D₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂³

yes

characteristic

C₄²×C₃₇

C₂³

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂³

yes

—

#10596934

—

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

M₁₆

yes

complemented

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

#117

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

C₂×M₁₆

yes

—

C₄×C₃₇

—

C₄×C₃₇

#117

yes

—

C₄×C₃₇

C₂×M₁₆

yes

—

C₄×C₃₇

C₂×#41

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

C₂×M₁₆

yes

—

#2920

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×#117

yes

characteristic

C₂×C₃₇

C₂²×M₁₆

yes

—

D₃₇

—

C₂²×C₄²

—

—

—

—

—

C₄×M₁₆

—

100

C₄×M₁₆

D₃₇

yes

characteristic, complemented

101

C₃₇

#2920

yes

characteristic, complemented, Sylow

102

C₂³×C₄

—

103

C₂³×C₄

—

104

C₂×M₁₆

—

105

C₂×M₁₆

C₂×D₃₇

yes

—

106

C₂×C₄²

—

107

C₂×C₄²

—

108

C₂×C₄²

—

109

C₂×C₄²

—

110

C₂×C₄²

—

111

C₂×C₄²

—

112

C₂×C₄²

C₂×D₃₇

yes

characteristic

113

C₄⋊C₈

—

114

C₂²⋊C₈

—

115

#97

—

116

#97

C₂×D₃₇

yes

complemented

117

C₄×C₈

—

118

C₄×C₈

C₂×D₃₇

yes

complemented

119

C₂⁴

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₄×D₃₇

yes

—

131

C₂²×C₄

C₂²×D₃₇

yes

characteristic

132

M₁₆

—

133

M₁₆

—

134

C₂×C₈

—

135

C₂×C₈

C₂²×D₃₇

yes

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

C₄×D₃₇

yes

—

144

C₄²

C₂²×D₃₇

yes

characteristic

145

C₄²

C₂²×D₃₇

yes

characteristic

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

C₂×C₄×D₃₇

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

C₂×C₄×D₃₇

yes

—

163

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

164

C₂×C₄

C₂×C₄×D₃₇

yes

—

165

C₂×C₄

#10596962

yes

—

166

C₂×C₄

C₂³×D₃₇

yes

characteristic

167

C₈

—

168

C₈

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#11924493

yes

—

175

C₂²

C₂²×C₄×D₃₇

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#11924493

yes

—

181

C₄

M₁₆×D₃₇

yes

—

182

C₄

C₂×#10596962

yes

—

183

C₂

—

184

C₂

—

185

C₂

C₂×#11924493

yes

characteristic

186

C₂

C₂×M₁₆×D₃₇

yes

—

	subgroup	G/H
center	C₄²	C₂²×D₃₇
commutator, socle	C₂²×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₄×M₁₆×C₃₇	C₂
prime radical	C₂³×D₃₇	C₂×C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₄×M₁₆	D₃₇
hypocenter	C₃₇	#2920
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₄²

C₂²×D₃₇

commutator, socle

C₂²×C₃₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₄×M₁₆×C₃₇

C₂

prime radical

C₂³×D₃₇

C₂×C₄

cosocle

C₂×C₄×C₃₇

C₂⁴

hypercenter

C₄×M₁₆

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #14557588	C₂	semidirect
3. #14557584	C₂	semidirect
4. #14557556	C₂	semidirect
6. C₄×#10596934	C₂	semidirect
7. C₄×M₁₆×C₃₇	C₂	semidirect
17. #11924465	C₂²	semidirect
19. #11924447	C₂²	semidirect
20. C₄×#64436	C₂²	semidirect
23. #97×C₃₇	C₂²	semidirect
24. C₄×C₈×C₃₇	C₂²	semidirect
72. C₄×D₃₇	M₁₆	semidirect
100. C₄×M₁₆	D₃₇	semidirect
101. C₃₇	#2920	semidirect
116. #97	C₂×D₃₇	semidirect
118. C₄×C₈	C₂×D₃₇	semidirect

H ≅ G/N

type

2. #14557588

C₂

semidirect

3. #14557584

C₂

semidirect

4. #14557556

C₂

semidirect

6. C₄×#10596934

C₂

semidirect

7. C₄×M₁₆×C₃₇

C₂

semidirect

17. #11924465

C₂²

semidirect

19. #11924447

C₂²

semidirect

20. C₄×#64436

C₂²

semidirect

23. #97×C₃₇

C₂²

semidirect

24. C₄×C₈×C₃₇

C₂²

semidirect

72. C₄×D₃₇

M₁₆

semidirect

100. C₄×M₁₆

D₃₇

semidirect

101. C₃₇

#2920

semidirect

116. #97

C₂×D₃₇

semidirect

118. C₄×C₈

C₂×D₃₇

semidirect

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

subgroup	index
C₂×C₄²×D₃₇	2
#14557588	2
#14557584	2
#14557556	2
#14557496	2
C₄×#10596934	2
C₄×M₁₆×C₃₇	2
#2920	37

subgroup

index

C₂×C₄²×D₃₇

C₄×M₁₆×C₃₇

Subgroups — #18003227

Conjugacy classes of subgroups

Complements

Maximal subgroups