Group #18003215 — subgroups

Subgroups — C₂×#14557588 #18003215

Order 4736.

488 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 205 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	8	yes	maximal, direct product factor
3	C₂×#11924465	C₂	1	2	yes	maximal, complemented
4	C₂×#11924448	C₂	1	2	yes	maximal
5	C₂×#11924447	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#97×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₃₇	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₃₇	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
10	C₂²×#64436	C₂²	1	2	yes	—
11	C₂×C₄×Dic₃₇	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₃₇	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
14	C₄²×D₃₇	C₄	1	4	yes	—
15	C₄²×D₃₇	C₂²	1	4	yes	—
16	#11924465	C₂²	1	8	yes	complemented
17	#11924448	C₂²	1	8	yes	—
18	#11924447	C₂²	1	4	yes	complemented
19	C₂²×C₈×C₃₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
21	#97×C₃₇	C₂²	1	4	yes	complemented
22	C₂³×D₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₃₇	—	2	4	no	—
30	C₂×C₄×D₃₇	—	2	4	no	—
31	C₂×C₄×D₃₇	—	2	4	no	—
32	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₃₇	C₂×C₄	1	4	yes	—
34	C₂×C₄×D₃₇	C₂×C₄	1	4	yes	—
35	C₂×C₄×D₃₇	C₂³	1	2	yes	—
36	C₂×C₈×C₃₇	—	2	4	no	—
37	C₂×C₈×C₃₇	C₂³	1	4	yes	—
38	C₄²×C₃₇	C₂×C₄	1	2	yes	—
39	C₄²×C₃₇	C₂³	1	2	yes	—
40	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
41	C₄×Dic₃₇	C₂×C₄	1	4	yes	—
42	C₄×Dic₃₇	C₂×C₄	1	4	yes	—
43	C₄×Dic₃₇	C₂³	1	2	yes	—
44	C₂×#64436	—	2	4	no	—
45	C₂×#64436	C₂³	1	4	yes	—
46	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×D₃₇	—	2	2	no	—
48	C₂²×D₃₇	—	2	2	no	—
49	C₂²×D₃₇	M₁₆	1	4	yes	—
50	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₃₇	—	2	2	no	—
52	C₂×C₄×C₃₇	—	2	2	no	—
53	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₃₇	#41	1	4	yes	—
57	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₃₇	—	2	2	no	—
59	C₂×Dic₃₇	—	2	2	no	—
60	C₂×Dic₃₇	M₁₆	1	4	yes	—
61	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
62	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
63	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
64	C₄×D₃₇	—	2	8	no	—
65	C₄×D₃₇	—	2	4	no	—
66	C₄×D₃₇	—	2	4	no	—
67	C₄×D₃₇	—	2	8	no	—
68	C₈×C₃₇	—	2	4	no	—
69	#64436	—	2	4	no	—
70	C₂²×C₃₇	#117	1	2	yes	—
71	C₂²×C₃₇	C₂×M₁₆	1	2	yes	—
72	C₂²×C₃₇	C₂×M₁₆	1	2	yes	—
73	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×D₃₇	—	2	2	no	—
75	C₂×D₃₇	—	2	2	no	—
76	C₂×D₃₇	—	2	8	no	—
77	C₂×D₃₇	C₂×M₁₆	1	4	yes	—
78	C₄×C₃₇	—	2	2	no	—
79	C₄×C₃₇	—	2	2	no	—
80	C₄×C₃₇	#117	1	2	yes	—
81	C₄×C₃₇	C₂×#41	1	2	yes	—
82	Dic₃₇	—	2	2	no	—
83	Dic₃₇	—	2	2	no	—
84	Dic₃₇	C₂×M₁₆	1	4	yes	—
85	C₂×#432	—	37	1	no	maximal, Sylow
86	C₂×C₃₇	#432	1	4	yes	complemented
87	C₂×C₃₇	C₂×#117	1	1	yes	characteristic
88	C₂×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
89	C₂×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
90	D₃₇	—	2	4	no	—
91	C₂²×C₄²	—	37	1	no	—
92	#432	—	37	8	no	—
93	C₂×C₄⋊C₈	—	37	2	no	—
94	C₂×C₂²⋊C₈	—	37	2	no	—
95	C₂×#97	—	37	1	no	—
96	C₂×#97	D₃₇	1	1	yes	characteristic, complemented
97	C₃₇	C₂×#432	1	1	yes	characteristic, complemented, Sylow
98	C₂³×C₄	—	37	1	no	—
99	C₂³×C₄	—	37	1	no	—
100	C₂³×C₄	—	37	1	no	—
101	C₂²×C₈	—	37	2	no	—
102	C₂²×C₈	C₂×D₃₇	1	2	yes	—
103	C₂×C₄²	—	37	1	no	—
104	C₂×C₄²	—	37	1	no	—
105	C₂×C₄²	—	37	1	no	—
106	C₂×C₄²	—	37	4	no	—
107	C₂×C₄²	—	37	4	no	—
108	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
109	C₄⋊C₈	—	37	8	no	—
110	C₂²⋊C₈	—	37	8	no	—
111	#97	—	37	4	no	—
112	#97	C₂×D₃₇	1	4	yes	complemented
113	C₂⁴	—	37	1	no	—
114	C₂²×C₄	—	37	1	no	—
115	C₂²×C₄	—	37	1	no	—
116	C₂²×C₄	—	37	1	no	—
117	C₂²×C₄	—	37	4	no	—
118	C₂²×C₄	—	37	4	no	—
119	C₂²×C₄	—	37	2	no	—
120	C₂²×C₄	—	37	2	no	—
121	C₂²×C₄	—	74	4	no	—
122	C₂²×C₄	—	74	4	no	—
123	C₂²×C₄	—	74	4	no	—
124	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
125	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	—	37	4	no	—
129	C₂×C₈	—	74	4	no	—
130	C₂×C₈	C₂²×D₃₇	1	4	yes	—
131	C₄²	—	37	4	no	—
132	C₄²	—	37	2	no	—
133	C₄²	—	37	4	no	—
134	C₄²	—	37	2	no	—
135	C₄²	C₄×D₃₇	1	2	yes	—
136	C₄²	C₂²×D₃₇	1	2	yes	—
137	C₂³	—	37	4	no	—
138	C₂³	—	37	2	no	—
139	C₂³	—	74	2	no	—
140	C₂³	—	74	2	no	—
141	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	37	4	no	—
145	C₂×C₄	—	37	2	no	—
146	C₂×C₄	—	37	2	no	—
147	C₂×C₄	—	37	2	no	—
148	C₂×C₄	—	74	8	no	—
149	C₂×C₄	—	74	8	no	—
150	C₂×C₄	—	74	2	no	—
151	C₂×C₄	—	74	2	no	—
152	C₂×C₄	—	74	4	no	—
153	C₂×C₄	—	74	4	no	—
154	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
155	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
156	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
157	C₂×C₄	#10596962	1	4	yes	—
158	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
159	C₈	—	2	4	no	—
160	C₈	—	74	4	no	—
161	C₂²	—	37	4	no	—
162	C₂²	—	74	2	no	—
163	C₂²	—	74	2	no	—
164	C₂²	—	74	8	no	—
165	C₂²	#11924493	1	2	yes	—
166	C₂²	M₁₆×D₃₇	1	2	yes	—
167	C₂²	M₁₆×D₃₇	1	2	yes	—
168	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	2	2	no	—
171	C₄	—	37	4	no	—
172	C₄	—	74	2	no	—
173	C₄	—	74	2	no	—
174	C₄	#11924493	1	2	yes	—
175	C₄	C₂×#10596962	1	2	yes	—
176	C₂	—	74	4	no	—
177	C₂	#14557588	1	4	yes	direct product factor
178	C₂	C₂×#11924493	1	1	yes	characteristic
179	C₂	C₂×M₁₆×D₃₇	1	1	yes	characteristic
180	C₂	C₂×M₁₆×D₃₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×D₃₇

C₂

yes

maximal, characteristic

#14557588

C₂

yes

maximal, direct product factor

C₂×#11924465

C₂

yes

maximal, complemented

C₂×#11924448

C₂

yes

maximal

C₂×#11924447

C₂

yes

maximal, characteristic, complemented

C₂×#97×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₃₇

C₄

yes

characteristic

C₂²×C₄×D₃₇

C₄

yes

characteristic

C₂²×C₄×D₃₇

C₂²

yes

characteristic

C₂²×#64436

C₂²

yes

—

C₂×C₄×Dic₃₇

C₄

yes

characteristic

C₂×C₄×Dic₃₇

C₄

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₄²×D₃₇

C₄

yes

—

C₄²×D₃₇

C₂²

yes

—

#11924465

C₂²

yes

complemented

#11924448

C₂²

yes

—

#11924447

C₂²

yes

complemented

C₂²×C₈×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#97×C₃₇

C₂²

yes

complemented

C₂³×D₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

—

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

C₂³

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂³

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₂×#64436

—

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

M₁₆

yes

—

C₂²×D₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

M₁₆

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

#117

yes

—

C₂²×C₃₇

C₂×M₁₆

yes

—

C₂²×C₃₇

C₂×M₁₆

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

C₂×M₁₆

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#117

yes

—

C₄×C₃₇

C₂×#41

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

C₂×M₁₆

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

C₂²×M₁₆

yes

characteristic

C₂×C₃₇

C₂²×M₁₆

yes

characteristic

D₃₇

—

C₂²×C₄²

—

#432

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×#97

—

C₂×#97

D₃₇

yes

characteristic, complemented

C₃₇

C₂×#432

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

100

C₂³×C₄

—

101

C₂²×C₈

—

102

C₂²×C₈

C₂×D₃₇

yes

—

103

C₂×C₄²

—

104

C₂×C₄²

—

105

C₂×C₄²

—

106

C₂×C₄²

—

107

C₂×C₄²

—

108

C₂×C₄²

C₂×D₃₇

yes

characteristic

109

C₄⋊C₈

—

110

C₂²⋊C₈

—

111

#97

—

112

#97

C₂×D₃₇

yes

complemented

113

C₂⁴

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₄×D₃₇

yes

characteristic

125

C₂²×C₄

C₄×D₃₇

yes

characteristic

126

C₂²×C₄

C₂²×D₃₇

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₂²×D₃₇

yes

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

C₄×D₃₇

yes

—

136

C₄²

C₂²×D₃₇

yes

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

C₂×C₄×D₃₇

yes

characteristic

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

C₂×C₄×D₃₇

yes

—

155

C₂×C₄

C₂×C₄×D₃₇

yes

—

156

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

157

C₂×C₄

#10596962

yes

—

158

C₂×C₄

C₂³×D₃₇

yes

characteristic

159

C₈

—

160

C₈

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#11924493

yes

—

166

C₂²

M₁₆×D₃₇

yes

—

167

C₂²

M₁₆×D₃₇

yes

—

168

C₂²

C₂²×C₄×D₃₇

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

#11924493

yes

—

175

C₄

C₂×#10596962

yes

—

176

C₂

—

177

C₂

#14557588

yes

direct product factor

178

C₂

C₂×#11924493

yes

characteristic

179

C₂

C₂×M₁₆×D₃₇

yes

characteristic

180

C₂

C₂×M₁₆×D₃₇

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×#97×C₃₇	C₂
socle	C₂³×C₃₇	C₂²×C₄
prime radical	C₂³×D₃₇	C₂×C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₂×#97	D₃₇
hypocenter	C₃₇	C₂×#432
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center

C₂²×C₄

C₂²×D₃₇

commutator

C₂²×C₃₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂×#97×C₃₇

C₂

socle

C₂³×C₃₇

C₂²×C₄

prime radical

C₂³×D₃₇

C₂×C₄

cosocle

C₂×C₄×C₃₇

C₂⁴

hypercenter

C₂×#97

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
2. #14557588	C₂	direct (N × H)
3. C₂×#11924465	C₂	semidirect
5. C₂×#11924447	C₂	semidirect
6. C₂×#97×C₃₇	C₂	semidirect
16. #11924465	C₂²	semidirect
18. #11924447	C₂²	semidirect
21. #97×C₃₇	C₂²	semidirect
86. C₂×C₃₇	#432	semidirect
96. C₂×#97	D₃₇	semidirect
97. C₃₇	C₂×#432	semidirect
112. #97	C₂×D₃₇	semidirect
177. C₂	#14557588	direct (N × H)

H ≅ G/N

type

2. #14557588

C₂

direct (N × H)

3. C₂×#11924465

C₂

semidirect

5. C₂×#11924447

C₂

semidirect

6. C₂×#97×C₃₇

C₂

semidirect

16. #11924465

C₂²

semidirect

18. #11924447

C₂²

semidirect

21. #97×C₃₇

C₂²

semidirect

86. C₂×C₃₇

#432

semidirect

96. C₂×#97

D₃₇

semidirect

97. C₃₇

C₂×#432

semidirect

112. #97

C₂×D₃₇

semidirect

177. C₂

#14557588

direct (N × H)

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
C₂×#11924465	2
C₂×#11924448	2
C₂×#11924447	2
C₂×#97×C₃₇	2
C₂×#432	37

subgroup

index

C₂×C₄²×D₃₇

C₂×#97×C₃₇

Subgroups — C₂×#14557588 #18003215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14557588 #18003215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557588 #18003215