Group #17994778 — subgroups

Subgroups — #17994778

Order 4736.

324 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂²×#10596934	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924465	C₂	1	2	yes	maximal, complemented
3	C₄×C₂²⋊Dic₃₇	C₂	1	1	yes	maximal, characteristic
4	#14557287	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596934	—	2	2	no	—
7	C₂×#10596934	—	2	4	no	—
8	C₂²×#64436	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₃₇	C₄	1	2	yes	—
10	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
11	#11924465	—	2	4	no	—
12	#11924443	C₄	1	1	yes	characteristic
13	#11924443	C₄	1	1	yes	characteristic
14	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
17	C₂²⋊C₈×C₃₇	—	2	2	no	—
18	C₂⁴×C₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₃₇	—	2	2	no	—
20	C₂²×C₄×C₃₇	—	2	2	no	—
21	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₃₇	D₄	1	2	yes	—
23	C₂²×C₄×C₃₇	Q₈	1	2	yes	—
24	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
26	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₃₇	—	2	1	no	—
28	C₂×C₈×C₃₇	—	2	1	no	—
29	C₂×C₈×C₃₇	—	2	2	no	—
30	C₂×C₈×C₃₇	—	2	2	no	—
31	C₂²⋊Dic₃₇	—	2	4	no	—
32	C₄×Dic₃₇	—	2	2	no	—
33	C₄×Dic₃₇	—	2	2	no	—
34	#10596934	—	2	8	no	—
35	C₂×#64436	—	2	2	no	—
36	C₂×#64436	—	2	2	no	—
37	C₂×#64436	D₄	1	4	yes	—
38	C₂³×C₃₇	—	2	2	no	—
39	C₂³×C₃₇	—	2	2	no	—
40	C₂³×C₃₇	—	2	1	no	—
41	C₂³×C₃₇	—	2	1	no	—
42	C₂³×C₃₇	C₄⋊C₄	1	2	yes	—
43	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₇	—	2	2	no	—
45	C₂×C₄×C₃₇	—	2	2	no	—
46	C₂×C₄×C₃₇	—	2	4	no	—
47	C₂×C₄×C₃₇	—	2	2	no	—
48	C₂×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₃₇	#41	1	2	yes	—
53	C₂×Dic₃₇	—	2	2	no	—
54	C₂×Dic₃₇	—	2	2	no	—
55	C₂×Dic₃₇	—	2	2	no	—
56	C₂×Dic₃₇	—	2	2	no	—
57	C₂×Dic₃₇	—	2	2	no	—
58	C₂×Dic₃₇	M₁₆	1	4	yes	complemented
59	C₈×C₃₇	—	4	1	no	—
60	C₈×C₃₇	—	4	1	no	—
61	#64436	—	2	4	no	—
62	C₂²×C₃₇	—	2	2	no	—
63	C₂²×C₃₇	—	2	2	no	—
64	C₂²×C₃₇	—	2	2	no	—
65	C₂²×C₃₇	—	4	2	no	—
66	C₂²×C₃₇	—	4	2	no	—
67	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₃₇	C₄×D₄	1	2	yes	—
69	C₂²×C₃₇	C₂×M₁₆	1	2	yes	—
70	C₂²×C₃₇	#131	1	1	yes	characteristic
71	C₂²×C₃₇	#131	1	1	yes	characteristic
72	C₄×C₃₇	—	2	2	no	—
73	C₄×C₃₇	#120	1	1	yes	characteristic
74	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
75	C₄×C₃₇	#123	1	1	yes	characteristic
76	Dic₃₇	—	2	4	no	—
77	Dic₃₇	—	4	2	no	—
78	#1851	—	37	1	no	maximal, Sylow
79	C₂×C₃₇	—	4	2	no	—
80	C₂×C₃₇	#385	1	1	yes	characteristic
81	C₂×C₃₇	#423	1	1	yes	characteristic
82	C₂×C₃₇	#424	1	1	yes	characteristic
83	C₂×C₃₇	#435	1	2	yes	—
84	C₂×C₃₇	#436	1	2	yes	—
85	C₂²×M₁₆	—	37	1	no	—
86	C₂×C₄⋊C₈	—	37	2	no	—
87	C₂×C₂²⋊C₈	D₃₇	1	1	yes	characteristic
88	C₄×C₂²⋊C₄	—	37	1	no	—
89	#336	—	37	2	no	—
90	C₃₇	#1851	1	1	yes	characteristic, complemented, Sylow
91	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
92	C₂×M₁₆	—	74	4	no	—
93	C₂×M₁₆	—	74	2	no	—
94	C₂²×C₈	—	37	2	no	—
95	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
97	C₂×C₂²⋊C₄	—	37	2	no	—
98	C₂×C₄²	—	37	2	no	—
99	C₄⋊C₈	—	74	4	no	—
100	C₂²⋊C₈	—	2	2	no	—
101	#95	—	37	1	no	—
102	#95	—	37	1	no	—
103	C₂⁴	C₄×D₃₇	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	37	2	no	—
107	C₂²×C₄	—	37	2	no	—
108	C₂²×C₄	Dic₇₄	1	2	yes	—
109	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
112	M₁₆	—	74	8	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	37	4	no	—
118	C₂×C₈	—	74	2	no	—
119	C₂×C₈	—	74	2	no	—
120	C₂²⋊C₄	—	74	4	no	—
121	C₄²	—	74	2	no	—
122	C₄²	—	74	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	1	no	—
126	C₂³	—	2	1	no	—
127	C₂³	#10596936	1	2	yes	—
128	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	37	4	no	—
134	C₂×C₄	—	74	2	no	—
135	C₂×C₄	—	74	2	no	—
136	C₂×C₄	—	74	2	no	—
137	C₂×C₄	—	74	2	no	—
138	C₂×C₄	—	74	2	no	—
139	C₂×C₄	#10596936	1	2	yes	—
140	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
141	C₂×C₄	D₄×D₃₇	1	2	yes	—
142	C₂×C₄	#10596964	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
144	C₈	—	4	1	no	—
145	C₈	—	4	1	no	—
146	C₈	—	74	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	C₂×#10596936	1	1	yes	characteristic
153	C₂²	#11924496	1	2	yes	—
154	C₂²	M₁₆×D₃₇	1	2	yes	—
155	C₂²	#11924542	1	1	yes	characteristic
156	C₂²	#11924542	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	74	4	no	—
159	C₄	—	148	2	no	—
160	C₄	#11924504	1	1	yes	characteristic
161	C₄	#11924513	1	2	yes	—
162	C₄	#11924517	1	1	yes	characteristic
163	C₂	—	4	2	no	—
164	C₂	#14557405	1	1	yes	characteristic
165	C₂	#14557556	1	1	yes	characteristic
166	C₂	#14557559	1	1	yes	characteristic
167	C₂	#14557607	1	2	yes	—
168	C₂	#14557608	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10596934

C₂

yes

maximal, characteristic

C₂×#11924465

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₃₇

C₂

yes

maximal, characteristic

#14557287

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596934

—

C₂×#10596934

—

C₂²×#64436

C₂²

yes

—

C₂×C₂²⋊Dic₃₇

C₄

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

—

#11924465

—

#11924443

C₄

yes

characteristic

#11924443

C₄

yes

characteristic

C₂³×C₄×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂²⋊C₈×C₃₇

—

C₂⁴×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

Q₈

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂²⋊Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

—

—

—

D₄

yes

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₄⋊C₄

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

M₁₆

yes

complemented

C₈×C₃₇

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

C₄×D₄

yes

—

C₂²×C₃₇

C₂×M₁₆

yes

—

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₄×C₃₇

—

C₄×C₃₇

#120

yes

characteristic

C₄×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

#123

yes

characteristic

Dic₃₇

—

Dic₃₇

—

#1851

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#436

yes

—

C₂²×M₁₆

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

D₃₇

yes

characteristic

C₄×C₂²⋊C₄

—

#336

—

C₃₇

#1851

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₃₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

100

C₂²⋊C₈

—

101

#95

—

102

#95

—

103

C₂⁴

C₄×D₃₇

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

Dic₇₄

yes

—

109

C₂²×C₄

C₄×D₃₇

yes

characteristic

110

C₂²×C₄

C₂²⋊D₃₇

yes

—

111

C₂²×C₄

C₂²×D₃₇

yes

characteristic

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

#10596936

yes

—

128

C₂³

C₂×C₄×D₃₇

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#10596936

yes

—

140

C₂×C₄

C₂×Dic₇₄

yes

characteristic

141

C₂×C₄

D₄×D₃₇

yes

—

142

C₂×C₄

#10596964

yes

—

143

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×#10596936

yes

characteristic

153

C₂²

#11924496

yes

—

154

C₂²

M₁₆×D₃₇

yes

—

155

C₂²

#11924542

yes

characteristic

156

C₂²

#11924542

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

148

—

160

C₄

#11924504

yes

characteristic

161

C₄

#11924513

yes

—

162

C₄

#11924517

yes

characteristic

163

C₂

—

164

C₂

#14557405

yes

characteristic

165

C₂

#14557556

yes

characteristic

166

C₂

#14557559

yes

characteristic

167

C₂

#14557607

yes

—

168

C₂

#14557608

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃₇
commutator, socle	C₂³×C₃₇	C₂²×C₄
Fitting	C₂×C₂²⋊C₈×C₃₇	C₂
prime radical	C₂⁴×C₃₇	C₂×C₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×C₂²⋊C₈	D₃₇
hypocenter	C₃₇	#1851
LCS	G ▷ C₂³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃₇

commutator, socle

C₂³×C₃₇

C₂²×C₄

Fitting

C₂×C₂²⋊C₈×C₃₇

C₂

prime radical

C₂⁴×C₃₇

C₂×C₄

cosocle

C₂²×C₄×C₃₇

C₂³

hypercenter

C₂×C₂²⋊C₈

D₃₇

C₃₇

G ▷ C₂³×C₃₇ ▷ C₃₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×#11924465	C₂	semidirect
4. #14557287	C₂	semidirect
58. C₂×Dic₃₇	M₁₆	semidirect
90. C₃₇	#1851	semidirect

H ≅ G/N

type

2. C₂×#11924465

C₂

semidirect

4. #14557287

C₂

semidirect

58. C₂×Dic₃₇

M₁₆

semidirect

90. C₃₇

#1851

semidirect

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

subgroup	index
C₂²×#10596934	2
C₂×#11924465	2
C₄×C₂²⋊Dic₃₇	2
#14557287	2
C₂×C₂²⋊C₈×C₃₇	2
#1851	37

subgroup

index

C₄×C₂²⋊Dic₃₇

C₂×C₂²⋊C₈×C₃₇

Subgroups — #17994778

Conjugacy classes of subgroups

Complements

Maximal subgroups