Group #18010270 — subgroups

Subgroups — D₃₇⋊#338 #18010270

Order 4736.

372 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 128 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×D₃₇⋊M₁₆	C₂	1	2	yes	maximal
2	#14558402	C₂	1	4	yes	maximal, complemented
3	C₂×M₁₆×D₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10596953	—	2	2	no	—
5	D₃₇⋊M₁₆	—	2	2	no	—
6	D₃₇⋊M₁₆	—	2	2	no	—
7	D₃₇⋊M₁₆	C₄	1	4	yes	—
8	D₃₇⋊M₁₆	C₂²	1	4	yes	—
9	C₂×#10596957	—	2	2	no	—
10	C₂×D₃₇⋊C₈	C₄	1	4	yes	—
11	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
12	C₂×#10596934	C₄	1	1	yes	characteristic
13	C₂×#10596929	C₂²	1	2	yes	—
14	M₁₆×D₃₇	—	2	2	no	—
15	M₁₆×D₃₇	—	2	2	no	—
16	C₂×C₈×D₃₇	C₄	1	2	yes	—
17	C₂×M₁₆×C₃₇	C₄	1	1	yes	characteristic
18	C₂³×D₃₇	Q₈	1	1	yes	characteristic
19	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
21	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
23	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
24	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
25	C₂×C₄×D₃₇	D₄	1	2	yes	—
26	C₂×C₄×D₃₇	D₄	1	2	yes	—
27	C₂×C₄×D₃₇	Q₈	1	1	yes	characteristic
28	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
29	#10596957	—	2	4	no	—
30	#10596957	—	2	4	no	—
31	C₂×#64438	—	2	4	no	—
32	#10596953	—	2	4	no	—
33	#10596953	—	2	4	no	—
34	D₃₇⋊C₈	—	2	4	no	—
35	D₃₇⋊C₈	C₂×C₄	1	8	yes	—
36	M₁₆×C₃₇	—	2	2	no	—
37	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
38	#10596934	—	2	2	no	—
39	C₂×#64436	C₂×C₄	1	2	yes	—
40	#10596929	—	2	4	no	—
41	C₈×D₃₇	—	2	2	no	—
42	C₈×D₃₇	—	2	2	no	—
43	C₂³×C₃₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×D₃₇	—	4	2	no	—
45	C₂²×D₃₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×D₃₇	C₄⋊C₄	1	2	yes	—
47	C₂²×D₃₇	C₄⋊C₄	1	2	yes	—
48	C₂²×D₃₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₃₇	C₄²	1	2	yes	—
50	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
54	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
55	C₂×Dic₃₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
57	C₄×D₃₇	C₄²	1	1	yes	characteristic
58	C₄×D₃₇	C₄²	1	1	yes	characteristic
59	C₄×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₄×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₄×D₃₇	C₂²⋊C₄	1	2	yes	—
62	C₄×D₃₇	C₂²⋊C₄	1	2	yes	—
63	C₄×D₃₇	C₄⋊C₄	1	2	yes	—
64	C₄×D₃₇	C₄⋊C₄	1	2	yes	—
65	C₄×D₃₇	C₂²×C₄	1	2	yes	—
66	C₄×D₃₇	C₂×D₄	1	2	yes	—
67	#64438	—	2	8	no	—
68	C₈×C₃₇	—	2	2	no	—
69	#64436	—	2	2	no	—
70	C₂²×C₃₇	—	4	1	no	—
71	C₂²×C₃₇	#95	1	2	yes	—
72	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×D₃₇	—	2	2	no	—
74	C₂×D₃₇	—	2	4	no	—
75	C₂×D₃₇	—	4	2	no	—
76	C₂×D₃₇	—	4	1	no	—
77	C₂×D₃₇	#95	1	1	yes	characteristic
78	C₂×D₃₇	#95	1	1	yes	characteristic
79	C₂×D₃₇	C₂×C₄⋊C₄	1	2	yes	—
80	C₄×C₃₇	#95	1	2	yes	—
81	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
82	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	Dic₃₇	#95	1	1	yes	characteristic
84	Dic₃₇	#95	1	1	yes	characteristic
85	Dic₃₇	C₂×C₂²⋊C₄	1	2	yes	—
86	C₂×#338	—	37	1	no	maximal, Sylow
87	C₂×C₃₇	—	2	2	no	—
88	C₂×C₃₇	—	2	1	no	—
89	C₂×C₃₇	C₂×#95	1	1	yes	characteristic
90	D₃₇	—	2	1	no	—
91	D₃₇	—	2	2	no	—
92	D₃₇	#338	1	2	yes	complemented
93	C₂²×M₁₆	—	37	2	no	—
94	C₂²×M₁₆	—	37	1	no	—
95	#338	—	37	4	no	—
96	C₃₇	C₂×#338	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₄	—	37	1	no	—
98	C₂×M₁₆	—	37	4	no	—
99	C₂×M₁₆	—	37	4	no	—
100	C₂×M₁₆	—	37	1	no	—
101	C₂×M₁₆	—	37	2	no	—
102	C₂×M₁₆	—	74	2	no	—
103	C₂×M₁₆	—	74	2	no	—
104	C₂×M₁₆	—	74	2	no	—
105	C₂×M₁₆	—	74	2	no	—
106	C₂×M₁₆	—	74	2	no	—
107	C₂×M₁₆	—	74	2	no	—
108	C₂×M₁₆	#3866	1	1	yes	characteristic
109	C₂²×C₈	—	37	4	no	—
110	C₂²×C₈	—	37	2	no	—
111	C₂⁴	—	37	1	no	—
112	C₂²×C₄	—	37	1	no	—
113	C₂²×C₄	—	37	2	no	—
114	C₂²×C₄	—	37	2	no	—
115	C₂²×C₄	—	37	1	no	—
116	C₂²×C₄	—	37	1	no	—
117	C₂²×C₄	—	37	2	no	—
118	C₂²×C₄	—	37	2	no	—
119	C₂²×C₄	—	37	1	no	—
120	C₂²×C₄	—	37	1	no	—
121	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
122	M₁₆	—	2	2	no	—
123	M₁₆	—	74	4	no	—
124	M₁₆	—	74	4	no	—
125	M₁₆	—	74	4	no	—
126	M₁₆	—	74	4	no	—
127	M₁₆	—	74	4	no	—
128	M₁₆	—	74	2	no	—
129	C₂×C₈	—	37	8	no	—
130	C₂×C₈	—	37	2	no	—
131	C₂×C₈	—	74	2	no	—
132	C₂×C₈	—	74	2	no	—
133	C₂×C₈	—	74	4	no	—
134	C₂×C₈	—	74	4	no	—
135	C₂×C₈	C₂×#3866	1	2	yes	—
136	C₂³	—	37	2	no	—
137	C₂³	—	37	1	no	—
138	C₂³	—	37	2	no	—
139	C₂³	—	37	1	no	—
140	C₂³	—	148	2	no	—
141	C₂³	C₄⋊#3866	1	1	yes	characteristic
142	C₂×C₄	—	37	2	no	—
143	C₂×C₄	—	37	2	no	—
144	C₂×C₄	—	37	1	no	—
145	C₂×C₄	—	37	1	no	—
146	C₂×C₄	—	37	2	no	—
147	C₂×C₄	—	37	2	no	—
148	C₂×C₄	—	37	2	no	—
149	C₂×C₄	—	37	2	no	—
150	C₂×C₄	—	37	2	no	—
151	C₂×C₄	—	37	1	no	—
152	C₂×C₄	—	37	1	no	—
153	C₂×C₄	—	37	1	no	—
154	C₂×C₄	—	37	1	no	—
155	C₂×C₄	—	37	2	no	—
156	C₂×C₄	C₄×#3866	1	2	yes	—
157	C₂×C₄	C₄⋊#3866	1	1	yes	characteristic
158	C₂×C₄	C₂²⋊#3866	1	2	yes	—
159	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	74	8	no	—
162	C₈	—	74	2	no	—
163	C₂²	—	4	1	no	—
164	C₂²	—	37	1	no	—
165	C₂²	—	37	1	no	—
166	C₂²	—	37	2	no	—
167	C₂²	—	74	4	no	—
168	C₂²	—	74	2	no	—
169	C₂²	—	148	2	no	—
170	C₂²	—	148	1	no	—
171	C₂²	(C₂×#3866)⋊C₄	1	2	yes	—
172	C₂²	C₂×C₄⋊#3866	1	1	yes	characteristic
173	C₄	—	37	2	no	—
174	C₄	—	37	1	no	—
175	C₄	—	37	1	no	—
176	C₄	(C₂×#3866)⋊C₄	1	2	yes	—
177	C₄	C₂×C₄×#3866	1	1	yes	characteristic
178	C₄	C₂×C₂²⋊#3866	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	—	2	1	no	—
181	C₂	—	37	2	no	—
182	C₂	—	74	1	no	—
183	C₂	—	74	2	no	—
184	C₂	C₂×(C₂×#3866)⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₃₇⋊M₁₆

C₂

yes

maximal

#14558402

C₂

yes

maximal, complemented

C₂×M₁₆×D₃₇

C₂

yes

maximal, characteristic

C₂×#10596953

—

D₃₇⋊M₁₆

—

D₃₇⋊M₁₆

—

D₃₇⋊M₁₆

C₄

yes

—

D₃₇⋊M₁₆

C₂²

yes

—

C₂×#10596957

—

C₂×D₃₇⋊C₈

C₄

yes

—

C₂²×C₄×D₃₇

C₂²

yes

characteristic

C₂×#10596934

C₄

yes

characteristic

C₂×#10596929

C₂²

yes

—

M₁₆×D₃₇

—

M₁₆×D₃₇

—

C₂×C₈×D₃₇

C₄

yes

—

C₂×M₁₆×C₃₇

C₄

yes

characteristic

C₂³×D₃₇

Q₈

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

D₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

D₄

yes

characteristic

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₄×D₃₇

Q₈

yes

characteristic

C₂×C₄×D₃₇

C₂³

yes

—

—

—

—

—

D₃₇⋊C₈

—

D₃₇⋊C₈

C₂×C₄

yes

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

#10596934

—

C₂×#64436

C₂×C₄

yes

—

#10596929

—

C₈×D₃₇

—

C₈×D₃₇

—

C₂³×C₃₇

C₄⋊C₄

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

C₄⋊C₄

yes

characteristic

C₂²×D₃₇

C₄⋊C₄

yes

—

C₂²×D₃₇

C₄⋊C₄

yes

—

C₂²×D₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

C₄²

yes

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

C₄⋊C₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

C₄²

yes

characteristic

C₄×D₃₇

C₄²

yes

characteristic

C₄×D₃₇

C₂²⋊C₄

yes

characteristic

C₄×D₃₇

C₂²⋊C₄

yes

characteristic

C₄×D₃₇

C₂²⋊C₄

yes

—

C₄×D₃₇

C₂²⋊C₄

yes

—

C₄×D₃₇

C₄⋊C₄

yes

—

C₄×D₃₇

C₄⋊C₄

yes

—

C₄×D₃₇

C₂²×C₄

yes

—

C₄×D₃₇

C₂×D₄

yes

—

#64438

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

#95

yes

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

yes

C₂×D₃₇

yes

C₂×D₃₇

C₂×C₄⋊C₄

yes

—

C₄×C₃₇

#95

yes

—

C₄×C₃₇

C₂×C₄²

yes

characteristic

C₄×C₃₇

C₂×C₂²⋊C₄

yes

Dic₃₇

yes

Dic₃₇

yes

Dic₃₇

C₂×C₂²⋊C₄

yes

—

C₂×#338

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

C₂×#95

yes

characteristic

D₃₇

—

D₃₇

—

D₃₇

#338

yes

complemented

C₂²×M₁₆

—

C₂²×M₁₆

—

#338

—

C₃₇

C₂×#338

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂×M₁₆

—

101

C₂×M₁₆

—

102

C₂×M₁₆

—

103

C₂×M₁₆

—

104

C₂×M₁₆

—

105

C₂×M₁₆

—

106

C₂×M₁₆

—

107

C₂×M₁₆

—

108

C₂×M₁₆

#3866

yes

characteristic

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂⁴

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

C₂×#3866

yes

characteristic

122

M₁₆

—

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

127

M₁₆

—

128

M₁₆

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

C₂×#3866

yes

—

136

C₂³

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

148

—

141

C₂³

C₄⋊#3866

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

—

155

C₂×C₄

—

156

C₂×C₄

C₄×#3866

yes

—

157

C₂×C₄

C₄⋊#3866

yes

characteristic

158

C₂×C₄

C₂²⋊#3866

yes

—

159

C₂×C₄

C₂²×#3866

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

148

—

170

C₂²

—

148

—

171

C₂²

(C₂×#3866)⋊C₄

yes

—

172

C₂²

C₂×C₄⋊#3866

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

(C₂×#3866)⋊C₄

yes

—

177

C₄

C₂×C₄×#3866

yes

characteristic

178

C₄

C₂×C₂²⋊#3866

yes

characteristic

179

C₂

—

180

C₂

—

181

C₂

—

182

C₂

—

183

C₂

—

184

C₂

C₂×(C₂×#3866)⋊C₄

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₄×#3866
commutator	C₄×C₃₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₂×M₁₆×C₃₇	C₄
socle	C₂×C₃₇	C₂×#95
prime radical	C₂³×D₃₇	Q₈
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	C₂×M₁₆	#3866
hypocenter	C₃₇	C₂×#338
LCS	G ▷ C₄×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

C₂×C₄×#3866

commutator

C₄×C₃₇

C₂×C₄²

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₂×M₁₆×C₃₇

C₄

socle

C₂×C₃₇

C₂×#95

prime radical

C₂³×D₃₇

Q₈

cosocle

C₂×C₄×D₃₇

C₂³

C₂×M₁₆

C₃₇

G ▷ C₄×C₃₇ ▷ C₃₇

C₁ ◁ C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
2. #14558402	C₂	semidirect
92. D₃₇	#338	semidirect
96. C₃₇	C₂×#338	semidirect

H ≅ G/N

type

2. #14558402

C₂

semidirect

92. D₃₇

#338

semidirect

96. C₃₇

C₂×#338

semidirect

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

subgroup	index
C₂×D₃₇⋊M₁₆	2
#14558402	2
C₂×M₁₆×D₃₇	2
C₂×#338	37

subgroup

index

C₂×D₃₇⋊M₁₆

#14558402

C₂×M₁₆×D₃₇

C₂×#338

Subgroups — D₃₇⋊#338 #18010270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D37⋊#338 #18010270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₇⋊#338 #18010270