Group #17991439 — subgroups

Subgroups — C₃₇×#2621 #17991439

Order 4736.

528 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 216 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⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Q₈×C₃₇	C₂	1	2	yes	maximal
3	C₂×#117×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#398	C₂	1	4	yes	maximal, complemented
5	C₃₇×#397	C₂	1	4	yes	maximal
6	C₃₇×#391	C₂	1	2	yes	maximal, complemented
7	C₃₇×#382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₃₇	C₂²	1	2	yes	—
9	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈×C₃₇	—	2	4	no	—
11	C₂²⋊Q₈×C₃₇	—	2	8	no	—
12	#117×C₃₇	—	2	4	no	—
13	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₃₇	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₃₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₃₇	C₂²	1	4	yes	—
18	C₂×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
21	#95×C₃₇	C₂²	1	4	yes	—
22	#95×C₃₇	C₂²	1	8	yes	—
23	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
24	C₂×Q₈×C₃₇	—	2	4	no	—
25	C₂×Q₈×C₃₇	—	2	4	no	—
26	C₂×Q₈×C₃₇	—	2	4	no	—
27	C₂²×C₄×C₃₇	—	2	2	no	—
28	C₂²×C₄×C₃₇	—	2	2	no	—
29	C₂²×C₄×C₃₇	D₄	1	4	yes	—
30	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
32	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₃₇	—	2	4	no	—
37	C₄⋊C₄×C₃₇	—	2	4	no	—
38	C₄⋊C₄×C₃₇	—	2	8	no	—
39	C₄⋊C₄×C₃₇	—	2	8	no	—
40	C₂²⋊C₄×C₃₇	—	2	8	no	—
41	C₂²⋊C₄×C₃₇	—	2	4	no	—
42	C₄²×C₃₇	—	2	2	no	—
43	C₄²×C₃₇	D₄	1	4	yes	—
44	C₂³×C₃₇	—	2	1	no	—
45	C₂³×C₃₇	—	2	1	no	—
46	C₂³×C₃₇	—	2	4	no	—
47	C₂³×C₃₇	C₂×D₄	1	2	yes	—
48	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
49	Q₈×C₃₇	—	4	8	no	—
50	C₂×C₄×C₃₇	—	2	1	no	—
51	C₂×C₄×C₃₇	—	2	1	no	—
52	C₂×C₄×C₃₇	—	2	4	no	—
53	C₂×C₄×C₃₇	—	2	8	no	—
54	C₂×C₄×C₃₇	—	2	2	no	—
55	C₂×C₄×C₃₇	—	2	4	no	—
56	C₂×C₄×C₃₇	—	2	4	no	—
57	C₂×C₄×C₃₇	—	2	4	no	—
58	C₂×C₄×C₃₇	—	2	4	no	—
59	C₂×C₄×C₃₇	—	2	4	no	—
60	C₂×C₄×C₃₇	—	2	8	no	—
61	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
62	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₃₇	#41	1	2	yes	—
66	C₂²×C₃₇	—	2	2	no	—
67	C₂²×C₃₇	—	2	4	no	—
68	C₂²×C₃₇	—	4	2	no	—
69	C₂²×C₃₇	—	4	2	no	—
70	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
71	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
72	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
73	C₂²×C₃₇	2₋¹⁺⁴	1	4	yes	—
74	C₄×C₃₇	—	2	2	no	—
75	C₄×C₃₇	—	2	4	no	—
76	C₄×C₃₇	—	4	4	no	—
77	C₄×C₃₇	—	4	2	no	—
78	C₄×C₃₇	—	4	4	no	—
79	C₄×C₃₇	#121	1	4	yes	—
80	#2621	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂×C₃₇	—	4	2	no	—
82	C₂×C₃₇	C₂×#121	1	1	yes	characteristic
83	C₂×C₃₇	#536	1	2	yes	—
84	C₂×C₃₇	#536	1	2	yes	—
85	C₂×C₃₇	#541	1	2	yes	—
86	C₂×C₄⋊Q₈	C₂×C₃₇	1	1	yes	characteristic, complemented
87	C₂×C₂²⋊Q₈	C₂×C₃₇	1	2	yes	—
88	C₂×#117	C₂×C₃₇	1	1	yes	characteristic
89	#398	C₂×C₃₇	1	4	yes	complemented
90	#397	C₂×C₃₇	1	4	yes	—
91	#391	C₂×C₃₇	1	2	yes	complemented
92	#382	C₂×C₃₇	1	1	yes	characteristic, complemented
93	C₃₇	#2621	1	1	yes	characteristic, direct product factor, Sylow
94	C₂²×Q₈	C₂²×C₃₇	1	2	yes	—
95	C₂³×C₄	C₂²×C₃₇	1	1	yes	characteristic
96	C₄⋊Q₈	—	2	4	no	—
97	C₂²⋊Q₈	—	2	8	no	—
98	#117	—	2	4	no	—
99	C₂×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
100	C₂×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
101	C₂×C₄⋊C₄	C₂²×C₃₇	1	4	yes	—
102	C₂×C₄⋊C₄	C₂²×C₃₇	1	4	yes	—
103	C₂×C₂²⋊C₄	C₂²×C₃₇	1	2	yes	—
104	C₂×C₂²⋊C₄	C₂²×C₃₇	1	4	yes	—
105	C₂×C₄²	C₂²×C₃₇	1	1	yes	characteristic
106	C₂×C₄²	C₂²×C₃₇	1	1	yes	characteristic
107	#95	C₂²×C₃₇	1	4	yes	—
108	#95	C₂²×C₃₇	1	8	yes	—
109	C₂⁴	C₂³×C₃₇	1	1	yes	characteristic
110	C₂×Q₈	—	2	4	no	—
111	C₂×Q₈	—	2	4	no	—
112	C₂×Q₈	—	2	4	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	D₄×C₃₇	1	4	yes	—
116	C₂²×C₄	C₂³×C₃₇	1	2	yes	—
117	C₂²×C₄	C₂³×C₃₇	1	4	yes	—
118	C₂²×C₄	C₂³×C₃₇	1	4	yes	—
119	C₂²×C₄	C₂³×C₃₇	1	2	yes	—
120	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
121	C₂²×C₄	C₂³×C₃₇	1	1	yes	characteristic
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	4	no	—
124	C₄⋊C₄	—	2	8	no	—
125	C₄⋊C₄	—	2	8	no	—
126	C₂²⋊C₄	—	2	8	no	—
127	C₂²⋊C₄	—	2	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	D₄×C₃₇	1	4	yes	—
130	C₂³	—	2	1	no	—
131	C₂³	—	2	1	no	—
132	C₂³	—	2	4	no	—
133	C₂³	C₂×D₄×C₃₇	1	2	yes	—
134	C₂³	C₂⁴×C₃₇	1	1	yes	characteristic
135	Q₈	—	4	8	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	C₂×D₄×C₃₇	1	4	yes	—
148	C₂×C₄	C₂×D₄×C₃₇	1	2	yes	—
149	C₂×C₄	C₂×D₄×C₃₇	1	2	yes	—
150	C₂×C₄	C₂×D₄×C₃₇	1	2	yes	—
151	C₂×C₄	#41×C₃₇	1	2	yes	—
152	C₂²	—	2	2	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	2	no	—
156	C₂²	C₂²×D₄×C₃₇	1	1	yes	characteristic
157	C₂²	C₂²×D₄×C₃₇	1	1	yes	characteristic
158	C₂²	C₂×#41×C₃₇	1	1	yes	characteristic
159	C₂²	2₋¹⁺⁴×C₃₇	1	4	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	4	no	—
162	C₄	—	4	4	no	—
163	C₄	—	4	2	no	—
164	C₄	—	4	4	no	—
165	C₄	#121×C₃₇	1	4	yes	—
166	C₂	—	4	2	no	—
167	C₂	C₂×#121×C₃₇	1	1	yes	characteristic
168	C₂	C₃₇×#536	1	2	yes	—
169	C₂	C₃₇×#536	1	2	yes	—
170	C₂	C₃₇×#541	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Q₈×C₃₇

C₂

yes

maximal

C₂×#117×C₃₇

C₂

yes

maximal, characteristic

C₃₇×#398

C₂

yes

maximal, complemented

C₃₇×#397

C₂

yes

maximal

C₃₇×#391

C₂

yes

maximal, complemented

C₃₇×#382

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₃₇

C₂²

yes

—

C₂³×C₄×C₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

—

C₂²⋊Q₈×C₃₇

—

#117×C₃₇

—

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₃₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#95×C₃₇

C₂²

yes

—

#95×C₃₇

C₂²

yes

—

C₂⁴×C₃₇

C₂³

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₂²⋊C₄×C₃₇

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₄²×C₃₇

D₄

yes

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂×D₄

yes

—

C₂³×C₃₇

C₂⁴

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂²×C₃₇

C₂×#41

yes

characteristic

C₂²×C₃₇

2₋¹⁺⁴

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#121

yes

—

#2621

C₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×#121

yes

characteristic

C₂×C₃₇

#536

yes

—

C₂×C₃₇

#536

yes

—

C₂×C₃₇

#541

yes

—

C₂×C₄⋊Q₈

C₂×C₃₇

yes

characteristic, complemented

C₂×C₂²⋊Q₈

C₂×C₃₇

yes

—

C₂×#117

C₂×C₃₇

yes

characteristic

#398

C₂×C₃₇

yes

complemented

#397

C₂×C₃₇

yes

—

#391

C₂×C₃₇

yes

complemented

#382

C₂×C₃₇

yes

characteristic, complemented

C₃₇

#2621

yes

characteristic, direct product factor, Sylow

C₂²×Q₈

C₂²×C₃₇

yes

—

C₂³×C₄

C₂²×C₃₇

yes

characteristic

C₄⋊Q₈

—

C₂²⋊Q₈

—

#117

—

C₂×C₄⋊C₄

C₂²×C₃₇

yes

—

100

C₂×C₄⋊C₄

C₂²×C₃₇

yes

—

101

C₂×C₄⋊C₄

C₂²×C₃₇

yes

—

102

C₂×C₄⋊C₄

C₂²×C₃₇

yes

—

103

C₂×C₂²⋊C₄

C₂²×C₃₇

yes

—

104

C₂×C₂²⋊C₄

C₂²×C₃₇

yes

—

105

C₂×C₄²

C₂²×C₃₇

yes

characteristic

106

C₂×C₄²

C₂²×C₃₇

yes

characteristic

107

#95

C₂²×C₃₇

yes

—

108

#95

C₂²×C₃₇

yes

—

109

C₂⁴

C₂³×C₃₇

yes

characteristic

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

D₄×C₃₇

yes

—

116

C₂²×C₄

C₂³×C₃₇

yes

—

117

C₂²×C₄

C₂³×C₃₇

yes

—

118

C₂²×C₄

C₂³×C₃₇

yes

—

119

C₂²×C₄

C₂³×C₃₇

yes

—

120

C₂²×C₄

C₂³×C₃₇

yes

characteristic

121

C₂²×C₄

C₂³×C₃₇

yes

characteristic

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

D₄×C₃₇

yes

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

C₂×D₄×C₃₇

yes

—

134

C₂³

C₂⁴×C₃₇

yes

characteristic

135

Q₈

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

C₂×D₄×C₃₇

yes

—

148

C₂×C₄

C₂×D₄×C₃₇

yes

—

149

C₂×C₄

C₂×D₄×C₃₇

yes

—

150

C₂×C₄

C₂×D₄×C₃₇

yes

—

151

C₂×C₄

#41×C₃₇

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

C₂²×D₄×C₃₇

yes

characteristic

157

C₂²

C₂²×D₄×C₃₇

yes

characteristic

158

C₂²

C₂×#41×C₃₇

yes

characteristic

159

C₂²

2₋¹⁺⁴×C₃₇

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#121×C₃₇

yes

—

166

C₂

—

167

C₂

C₂×#121×C₃₇

yes

characteristic

168

C₂

C₃₇×#536

yes

—

169

C₂

C₃₇×#536

yes

—

170

C₂

C₃₇×#541

yes

—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₇
prime radical	C₂⁴×C₃₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₃₇

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₃₇

prime radical

C₂⁴×C₃₇

C₂³

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₃₇ ◁ G

N	H ≅ G/N	type
1. C₂×C₄⋊Q₈×C₃₇	C₂	semidirect
4. C₃₇×#398	C₂	semidirect
6. C₃₇×#391	C₂	semidirect
7. C₃₇×#382	C₂	semidirect
80. #2621	C₃₇	direct (N × H)
86. C₂×C₄⋊Q₈	C₂×C₃₇	semidirect
89. #398	C₂×C₃₇	semidirect
91. #391	C₂×C₃₇	semidirect
92. #382	C₂×C₃₇	semidirect
93. C₃₇	#2621	direct (N × H)

H ≅ G/N

type

1. C₂×C₄⋊Q₈×C₃₇

C₂

semidirect

4. C₃₇×#398

C₂

semidirect

6. C₃₇×#391

C₂

semidirect

7. C₃₇×#382

C₂

semidirect

80. #2621

C₃₇

direct (N × H)

86. C₂×C₄⋊Q₈

C₂×C₃₇

semidirect

89. #398

C₂×C₃₇

semidirect

91. #391

C₂×C₃₇

semidirect

92. #382

C₂×C₃₇

semidirect

93. C₃₇

#2621

direct (N × H)

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

subgroup	index
C₂×C₄⋊Q₈×C₃₇	2
C₂×C₂²⋊Q₈×C₃₇	2
C₂×#117×C₃₇	2
C₃₇×#398	2
C₃₇×#397	2
C₃₇×#391	2
C₃₇×#382	2
#2621	37

subgroup

index

C₂×C₄⋊Q₈×C₃₇

C₂×C₂²⋊Q₈×C₃₇

C₂×#117×C₃₇

C₃₇×#398

C₃₇×#397

C₃₇×#391

C₃₇×#382

#2621

Subgroups — C₃₇×#2621 #17991439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C37×#2621 #17991439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#2621 #17991439