Group #18005243 — subgroups

Subgroups — #18005243

Order 4736.

386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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	#14558137	C₂	1	2	yes	maximal, complemented
2	#14557935	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557897	C₂	1	2	yes	maximal, complemented
4	#14557783	C₂	1	2	yes	maximal, complemented
5	#14557766	C₂	1	2	yes	maximal, complemented
6	#14557645	C₂	1	2	yes	maximal, complemented
7	D₈×Dic₃₇	C₂	1	2	yes	maximal, complemented
8	#14557487	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10596929	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
12	#11924528	C₂²	1	2	yes	—
13	#11924517	—	2	4	no	—
14	#11924508	—	2	4	no	—
15	#11924507	C₂²	1	4	yes	—
16	D₄×Dic₃₇	C₂²	1	4	yes	—
17	#11924493	C₂²	1	1	yes	characteristic
18	#11924470	C₂²	1	2	yes	—
19	#11924468	C₂²	1	2	yes	—
20	#11924457	C₂²	1	4	yes	—
21	#11924457	C₂²	1	4	yes	—
22	#11924447	C₂²	1	1	yes	characteristic
23	#11924446	C₂²	1	1	yes	characteristic
24	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
25	C₂×D₈×C₃₇	C₂²	1	2	yes	—
26	C₂×D₈×C₃₇	C₂²	1	2	yes	—
27	#127×C₃₇	C₂²	1	2	yes	—
28	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
29	C₂×D₄×C₃₇	—	2	4	no	—
30	C₂×D₄×C₃₇	C₂³	1	4	yes	—
31	C₂×C₂²⋊D₃₇	—	2	4	no	—
32	C₂²×Dic₃₇	—	2	4	no	—
33	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
34	D₈×C₃₇	—	2	4	no	—
35	D₈×C₃₇	—	2	4	no	—
36	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	C₄²×C₃₇	C₂³	1	1	yes	characteristic
39	C₂²⋊Dic₃₇	—	2	4	no	—
40	C₂²⋊Dic₃₇	—	2	4	no	—
41	#10596938	D₄	1	2	yes	—
42	C₄⋊Dic₃₇	C₂³	1	4	yes	—
43	#10596936	—	2	4	no	—
44	#10596936	D₄	1	2	yes	—
45	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
46	C₂×#64436	C₂³	1	1	yes	characteristic
47	C₂×#64436	C₂³	1	1	yes	characteristic
48	#10596929	—	2	2	no	—
49	C₈×D₃₇	—	2	2	no	—
50	C₂³×C₃₇	—	2	4	no	—
51	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
52	D₄×C₃₇	—	2	4	no	—
53	D₄×C₃₇	—	2	4	no	—
54	D₄×C₃₇	—	8	4	no	—
55	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
57	C₂²⋊D₃₇	—	8	4	no	—
58	C₂×Dic₃₇	—	2	4	no	—
59	C₂×Dic₃₇	—	8	4	no	—
60	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
61	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
62	C₄×D₃₇	—	2	1	no	—
63	C₄×D₃₇	—	2	1	no	—
64	C₈×C₃₇	#41	1	2	yes	—
65	C₈×C₃₇	#41	1	2	yes	—
66	#64436	—	2	1	no	—
67	#64436	—	2	1	no	—
68	C₂²×C₃₇	—	4	4	no	—
69	C₂²×C₃₇	—	4	4	no	—
70	C₂²×C₃₇	—	4	4	no	—
71	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
72	C₂×D₃₇	—	2	1	no	—
73	C₂×D₃₇	—	2	2	no	—
74	C₄×C₃₇	—	2	2	no	—
75	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
76	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
77	Dic₃₇	—	2	1	no	—
78	Dic₃₇	—	2	2	no	—
79	Dic₃₇	—	4	4	no	—
80	#3176	—	37	1	no	maximal, Sylow
81	C₂×C₃₇	—	8	4	no	—
82	C₂×C₃₇	#532	1	1	yes	characteristic
83	C₂×C₃₇	#576	1	2	yes	—
84	D₃₇	—	4	1	no	—
85	#540	—	37	2	no	—
86	#499	—	37	1	no	—
87	#493	D₃₇	1	1	yes	characteristic, complemented
88	#490	—	37	2	no	—
89	#469	—	37	2	no	—
90	#466	—	37	2	no	—
91	#442	—	37	2	no	—
92	C₄×D₈	—	37	2	no	—
93	#405	—	37	1	no	—
94	C₃₇	#3176	1	1	yes	characteristic, complemented, Sylow
95	C₂×D₈	C₂×D₃₇	1	2	yes	—
96	C₂×D₈	C₂×D₃₇	1	2	yes	—
97	C₂×M₁₆	—	37	1	no	—
98	C₂²×C₈	—	37	1	no	—
99	#127	C₂×D₃₇	1	2	yes	—
100	#125	—	37	2	no	—
101	#123	—	74	4	no	—
102	#121	—	37	4	no	—
103	#121	—	74	4	no	—
104	C₄×D₄	—	37	4	no	—
105	#117	—	37	1	no	—
106	#107	—	37	2	no	—
107	#106	—	37	2	no	—
108	#102	—	37	4	no	—
109	#102	—	37	4	no	—
110	#97	—	37	1	no	—
111	#97	—	37	1	no	—
112	C₄×C₈	—	37	1	no	—
113	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
114	C₂×D₄	—	2	4	no	—
115	C₂×D₄	—	74	4	no	—
116	C₂×D₄	C₂²×D₃₇	1	4	yes	—
117	C₂²×C₄	—	37	1	no	—
118	C₂²×C₄	—	74	4	no	—
119	D₈	—	2	4	no	—
120	D₈	—	2	4	no	—
121	M₁₆	—	74	2	no	—
122	C₂×C₈	—	37	1	no	—
123	C₂×C₈	—	37	1	no	—
124	C₂×C₈	—	74	2	no	—
125	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
126	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
127	C₄⋊C₄	—	37	2	no	—
128	C₄⋊C₄	—	37	4	no	—
129	C₄⋊C₄	—	74	4	no	—
130	C₂²⋊C₄	—	37	2	no	—
131	C₂²⋊C₄	—	74	4	no	—
132	C₂²⋊C₄	—	74	4	no	—
133	C₄²	—	37	1	no	—
134	C₄²	C₂²×D₃₇	1	1	yes	characteristic
135	C₂³	—	2	4	no	—
136	C₂³	—	37	1	no	—
137	D₄	—	2	4	no	—
138	D₄	—	2	4	no	—
139	D₄	—	8	4	no	—
140	D₄	—	296	4	no	—
141	C₂×C₄	—	37	2	no	—
142	C₂×C₄	—	37	1	no	—
143	C₂×C₄	—	296	4	no	—
144	C₂×C₄	—	74	1	no	—
145	C₂×C₄	—	74	1	no	—
146	C₂×C₄	—	74	4	no	—
147	C₂×C₄	D₄×D₃₇	1	2	yes	—
148	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
149	C₈	—	74	1	no	—
150	C₈	—	74	1	no	—
151	C₈	#10596964	1	2	yes	—
152	C₈	#10596964	1	2	yes	—
153	C₂²	—	4	4	no	—
154	C₂²	—	4	4	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	74	2	no	—
157	C₂²	—	74	1	no	—
158	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	74	2	no	—
161	C₄	—	74	1	no	—
162	C₄	—	148	4	no	—
163	C₄	C₂×#10596964	1	1	yes	characteristic
164	C₄	C₂×#10596964	1	1	yes	characteristic
165	C₂	—	8	4	no	—
166	C₂	—	148	1	no	—
167	C₂	#14558081	1	1	yes	characteristic
168	C₂	#14558334	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558137

C₂

yes

maximal, complemented

#14557935

C₂

yes

maximal, characteristic, complemented

#14557897

C₂

yes

maximal, complemented

#14557783

C₂

yes

maximal, complemented

#14557766

C₂

yes

maximal, complemented

#14557645

C₂

yes

maximal, complemented

D₈×Dic₃₇

C₂

yes

maximal, complemented

#14557487

C₂

yes

maximal, characteristic, complemented

C₃₇×#493

C₂

yes

maximal, characteristic, complemented

C₂×#10596929

C₂²

yes

characteristic

C₂×C₈×D₃₇

C₂²

yes

characteristic

#11924528

C₂²

yes

—

#11924517

—

#11924508

—

#11924507

C₂²

yes

—

D₄×Dic₃₇

C₂²

yes

—

#11924493

C₂²

yes

characteristic

#11924470

C₂²

yes

—

#11924468

C₂²

yes

—

#11924457

C₂²

yes

—

#11924457

C₂²

yes

—

#11924447

C₂²

yes

characteristic

#11924446

C₂²

yes

characteristic

C₈×Dic₃₇

C₂²

yes

characteristic

C₂×D₈×C₃₇

C₂²

yes

—

C₂×D₈×C₃₇

C₂²

yes

—

#127×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

C₂³

yes

—

C₂×C₂²⋊D₃₇

—

C₂²×Dic₃₇

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

D₈×C₃₇

—

D₈×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₄²×C₃₇

C₂³

yes

characteristic

C₂²⋊Dic₃₇

—

C₂²⋊Dic₃₇

—

#10596938

D₄

yes

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

#10596936

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂×#64436

C₂³

yes

characteristic

C₂×#64436

C₂³

yes

characteristic

#10596929

—

C₈×D₃₇

—

C₂³×C₃₇

—

C₂²×D₃₇

C₂×D₄

yes

characteristic

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₈×C₃₇

#41

yes

—

C₈×C₃₇

#41

yes

—

#64436

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×#41

yes

characteristic

C₄×C₃₇

C₂×#41

yes

characteristic

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

#3176

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#532

yes

characteristic

C₂×C₃₇

#576

yes

—

D₃₇

—

#540

—

#499

—

#493

D₃₇

yes

characteristic, complemented

—

—

—

—

C₄×D₈

—

#405

—

C₃₇

#3176

yes

characteristic, complemented, Sylow

C₂×D₈

C₂×D₃₇

yes

—

C₂×D₈

C₂×D₃₇

yes

—

C₂×M₁₆

—

C₂²×C₈

—

#127

C₂×D₃₇

yes

—

100

#125

—

101

#123

—

102

#121

—

103

#121

—

104

C₄×D₄

—

105

#117

—

106

#107

—

107

#106

—

108

#102

—

109

#102

—

110

#97

—

111

#97

—

112

C₄×C₈

—

113

C₄×C₈

C₂×D₃₇

yes

characteristic

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

C₂²×D₃₇

yes

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

D₈

—

120

D₈

—

121

M₁₆

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

C₂²×D₃₇

yes

characteristic

126

C₂×C₈

C₂²×D₃₇

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₄²

C₂²×D₃₇

yes

characteristic

135

C₂³

—

136

C₂³

—

137

D₄

—

138

D₄

—

139

D₄

—

140

D₄

—

296

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

296

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

D₄×D₃₇

yes

—

148

C₂×C₄

C₂³×D₃₇

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

#10596964

yes

—

152

C₈

#10596964

yes

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₂×D₄×D₃₇

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

148

—

163

C₄

C₂×#10596964

yes

characteristic

164

C₄

C₂×#10596964

yes

characteristic

165

C₂

—

166

C₂

—

148

—

167

C₂

#14558081

yes

characteristic

168

C₂

#14558334

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#493	C₂
socle	C₂²×C₃₇	C₂²×D₄
hypercenter	#493	D₃₇
hypocenter	C₃₇	#3176
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₃₇×#493

C₂

socle

C₂²×C₃₇

C₂²×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #14558137	C₂	semidirect
2. #14557935	C₂	semidirect
3. #14557897	C₂	semidirect
4. #14557783	C₂	semidirect
5. #14557766	C₂	semidirect
6. #14557645	C₂	semidirect
7. D₈×Dic₃₇	C₂	semidirect
8. #14557487	C₂	semidirect
9. C₃₇×#493	C₂	semidirect
87. #493	D₃₇	semidirect
94. C₃₇	#3176	semidirect

H ≅ G/N

type

1. #14558137

C₂

semidirect

2. #14557935

C₂

semidirect

3. #14557897

C₂

semidirect

4. #14557783

C₂

semidirect

5. #14557766

C₂

semidirect

6. #14557645

C₂

semidirect

7. D₈×Dic₃₇

C₂

semidirect

8. #14557487

C₂

semidirect

9. C₃₇×#493

C₂

semidirect

87. #493

D₃₇

semidirect

94. C₃₇

#3176

semidirect

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

subgroup	index
#14558137	2
#14557935	2
#14557897	2
#14557783	2
#14557766	2
#14557645	2
D₈×Dic₃₇	2
#14557487	2
C₃₇×#493	2
#3176	37

subgroup

index

D₈×Dic₃₇

C₃₇×#493

Subgroups — #18005243

Conjugacy classes of subgroups

Complements

Maximal subgroups