Group #17995141 — subgroups

Subgroups — #17995141

Order 4736.

340 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 85 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	#14558300	C₂	1	1	yes	maximal, characteristic, complemented
2	#14558097	C₂	1	1	yes	maximal, characteristic
3	C₂×#11924452	C₂	1	2	yes	maximal, complemented
4	#14557304	C₂	1	2	yes	maximal, complemented
5	C₃₇×#413	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₄₈	C₂²	1	1	yes	characteristic
7	#11924543	—	2	4	no	—
8	C₂×#10596934	—	2	2	no	—
9	C₂×#10596934	C₂²	1	2	yes	—
10	#11924532	—	2	1	no	—
11	#11924525	—	2	1	no	—
12	#11924510	—	2	2	no	—
13	#11924504	C₄	1	4	yes	—
14	#11924495	C₂²	1	1	yes	characteristic
15	#11924452	—	2	4	no	—
16	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
18	#101×C₃₇	—	2	1	no	—
19	#100×C₃₇	—	2	1	no	—
20	C₂³×D₃₇	C₂×C₄	1	2	yes	—
21	#41×C₃₇	—	2	2	no	—
22	#41×C₃₇	—	2	2	no	—
23	C₂×Q₈×C₃₇	D₄	1	1	yes	characteristic
24	C₂×D₄×C₃₇	C₂×C₄	1	2	yes	—
25	C₂×D₄×C₃₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₃₇	D₄	1	2	yes	—
27	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₃₇	—	2	4	no	—
29	C₂×D₁₄₈	—	2	1	no	—
30	C₂×D₁₄₈	—	2	1	no	—
31	C₂×D₁₄₈	—	2	2	no	—
32	C₂×D₁₄₈	—	2	2	no	—
33	M₁₆×C₃₇	—	2	2	no	—
34	M₁₆×C₃₇	—	2	2	no	—
35	C₂×C₈×C₃₇	—	2	2	no	—
36	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
37	#10596938	—	2	4	no	—
38	C₄⋊Dic₃₇	—	2	1	no	—
39	C₄×Dic₃₇	—	2	1	no	—
40	#10596934	—	2	4	no	—
41	#10596934	—	2	2	no	—
42	#10596934	D₄	1	4	yes	complemented
43	C₂×#64436	—	2	4	no	—
44	C₂×#64436	—	2	2	no	—
45	C₂³×C₃₇	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×D₃₇	—	2	2	no	—
48	C₂²×D₃₇	—	2	4	no	—
49	C₂²×D₃₇	—	8	2	no	—
50	Q₈×C₃₇	—	2	2	no	—
51	D₄×C₃₇	—	2	4	no	—
52	D₄×C₃₇	—	2	2	no	—
53	C₂×C₄×C₃₇	—	2	2	no	—
54	C₂×C₄×C₃₇	—	2	2	no	—
55	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₃₇	#41	1	2	yes	—
60	C₂²⋊D₃₇	—	8	4	no	—
61	C₂×Dic₃₇	—	2	2	no	—
62	D₁₄₈	—	4	1	no	—
63	D₁₄₈	—	4	2	no	—
64	D₁₄₈	—	4	1	no	—
65	C₈×C₃₇	—	4	2	no	—
66	#64436	—	2	4	no	—
67	C₂²×C₃₇	—	2	2	no	—
68	C₂²×C₃₇	—	4	1	no	—
69	C₂²×C₃₇	—	4	2	no	—
70	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₃₇	C₄×D₄	1	2	yes	—
72	C₂×D₃₇	—	4	4	no	—
73	C₂×D₃₇	—	4	2	no	—
74	C₂×D₃₇	—	8	4	no	—
75	C₄×C₃₇	—	2	2	no	—
76	C₄×C₃₇	#120	1	1	yes	characteristic
77	C₄×C₃₇	#121	1	2	yes	—
78	C₄×C₃₇	#123	1	1	yes	characteristic
79	Dic₃₇	—	4	2	no	—
80	#1914	—	37	1	no	maximal, Sylow
81	C₂×C₃₇	—	2	1	no	—
82	C₂×C₃₇	—	2	2	no	—
83	C₂×C₃₇	—	4	2	no	—
84	C₂×C₃₇	#386	1	1	yes	characteristic
85	D₃₇	—	8	2	no	—
86	#568	—	37	1	no	—
87	#535	—	37	1	no	—
88	#413	D₃₇	1	1	yes	characteristic, complemented
89	C₂×#100	—	37	2	no	—
90	#344	—	37	2	no	—
91	C₃₇	#1914	1	1	yes	characteristic, complemented, Sylow
92	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
93	C₂²×D₄	—	37	1	no	—
94	#131	—	74	4	no	—
95	C₂×M₁₆	—	37	2	no	—
96	C₂×M₁₆	—	74	2	no	—
97	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
98	#127	—	74	1	no	—
99	#124	—	74	1	no	—
100	#121	—	74	2	no	—
101	#120	—	37	4	no	—
102	#117	—	37	1	no	—
103	#101	—	2	1	no	—
104	#100	—	2	1	no	—
105	#100	—	74	4	no	—
106	C₂⁴	—	37	2	no	—
107	#41	—	2	2	no	—
108	#41	—	2	2	no	—
109	C₂×Q₈	C₂²⋊D₃₇	1	1	yes	characteristic
110	C₂×D₄	—	74	4	no	—
111	C₂×D₄	—	74	1	no	—
112	C₂×D₄	—	74	2	no	—
113	C₂×D₄	—	74	2	no	—
114	C₂×D₄	—	74	1	no	—
115	C₂×D₄	C₄×D₃₇	1	2	yes	—
116	C₂×D₄	C₂²⋊D₃₇	1	1	yes	characteristic
117	C₂²×C₄	D₁₄₈	1	2	yes	—
118	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
119	M₁₆	—	2	2	no	—
120	M₁₆	—	2	2	no	—
121	M₁₆	—	37	4	no	—
122	M₁₆	—	74	4	no	—
123	M₁₆	—	74	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	74	4	no	—
126	C₂×C₈	—	74	2	no	—
127	C₄⋊C₄	—	74	1	no	—
128	C₂²⋊C₄	—	37	2	no	—
129	C₂²⋊C₄	—	74	4	no	—
130	C₄²	—	74	1	no	—
131	C₂³	—	296	2	no	—
132	C₂³	—	74	4	no	—
133	C₂³	—	74	2	no	—
134	C₂³	#10596938	1	2	yes	—
135	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
136	Q₈	—	2	2	no	—
137	D₄	—	2	4	no	—
138	D₄	—	2	2	no	—
139	D₄	—	296	4	no	—
140	D₄	—	148	1	no	—
141	D₄	—	148	2	no	—
142	D₄	—	148	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	74	2	no	—
146	C₂×C₄	#10596938	1	2	yes	—
147	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
148	C₂×C₄	D₄×D₃₇	1	2	yes	—
149	C₂×C₄	#10596964	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
151	C₈	—	4	2	no	—
152	C₈	—	74	4	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	296	4	no	—
157	C₂²	—	148	4	no	—
158	C₂²	—	148	2	no	—
159	C₂²	C₂×#10596938	1	1	yes	characteristic
160	C₂²	#11924496	1	2	yes	—
161	C₄	—	2	2	no	—
162	C₄	—	148	2	no	—
163	C₄	#11924506	1	1	yes	characteristic
164	C₄	#11924508	1	2	yes	—
165	C₄	#11924520	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	—	4	2	no	—
169	C₂	—	296	2	no	—
170	C₂	#14557410	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558300

C₂

yes

maximal, characteristic, complemented

#14558097

C₂

yes

maximal, characteristic

C₂×#11924452

C₂

yes

maximal, complemented

#14557304

C₂

yes

maximal, complemented

C₃₇×#413

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₄₈

C₂²

yes

—

—

C₂²

yes

—

—

—

—

C₄

yes

—

#11924495

C₂²

yes

characteristic

#11924452

—

C₂×#41×C₃₇

C₂²

yes

characteristic

C₂×M₁₆×C₃₇

C₂²

yes

—

#101×C₃₇

—

#100×C₃₇

—

C₂³×D₃₇

C₂×C₄

yes

—

#41×C₃₇

—

#41×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

characteristic

C₂×D₄×C₃₇

C₂×C₄

yes

—

C₂×D₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₂²⋊D₃₇

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

M₁₆×C₃₇

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₂²⋊Dic₃₇

C₂×C₄

yes

—

#10596938

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

#10596934

—

#10596934

—

#10596934

D₄

yes

complemented

C₂×#64436

—

C₂×#64436

—

C₂³×C₃₇

C₂²⋊C₄

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

C₄×D₄

yes

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#120

yes

characteristic

C₄×C₃₇

#121

yes

—

C₄×C₃₇

#123

yes

characteristic

Dic₃₇

—

#1914

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#386

yes

characteristic

D₃₇

—

#568

—

#535

—

#413

D₃₇

yes

characteristic, complemented

C₂×#100

—

#344

—

C₃₇

#1914

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₃₇

yes

characteristic

C₂²×D₄

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

#127

—

#124

—

100

#121

—

101

#120

—

102

#117

—

103

#101

—

104

#100

—

105

#100

—

106

C₂⁴

—

107

#41

—

108

#41

—

109

C₂×Q₈

C₂²⋊D₃₇

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

C₄×D₃₇

yes

—

116

C₂×D₄

C₂²⋊D₃₇

yes

characteristic

117

C₂²×C₄

D₁₄₈

yes

—

118

C₂²×C₄

C₂²×D₃₇

yes

characteristic

119

M₁₆

—

120

M₁₆

—

121

M₁₆

—

122

M₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₂²⋊C₄

—

130

C₄²

—

131

C₂³

—

296

—

132

C₂³

—

133

C₂³

—

134

C₂³

#10596938

yes

—

135

C₂³

C₂×C₄×D₃₇

yes

characteristic

136

Q₈

—

137

D₄

—

138

D₄

—

139

D₄

—

296

—

140

D₄

—

148

—

141

D₄

—

148

—

142

D₄

—

148

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#10596938

yes

—

147

C₂×C₄

C₂×D₁₄₈

yes

characteristic

148

C₂×C₄

D₄×D₃₇

yes

—

149

C₂×C₄

#10596964

yes

—

150

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

151

C₈

—

152

C₈

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

296

—

157

C₂²

—

148

—

158

C₂²

—

148

—

159

C₂²

C₂×#10596938

yes

characteristic

160

C₂²

#11924496

yes

—

161

C₄

—

162

C₄

—

148

—

163

C₄

#11924506

yes

characteristic

164

C₄

#11924508

yes

—

165

C₄

#11924520

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

296

—

170

C₂

#14557410

yes

characteristic

	subgroup	G/H
center	C₂	#14557410
commutator	C₂³×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#413	C₂
socle	C₂×C₃₇	#386
prime radical	#14558097	C₂
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#413	D₃₇
hypocenter	C₃₇	#1914
LCS	G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₂

#14557410

commutator

C₂³×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#413

C₂

C₂×C₃₇

C₂

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
1. #14558300	C₂	semidirect
3. C₂×#11924452	C₂	semidirect
4. #14557304	C₂	semidirect
5. C₃₇×#413	C₂	semidirect
42. #10596934	D₄	semidirect
88. #413	D₃₇	semidirect
91. C₃₇	#1914	semidirect

H ≅ G/N

type

1. #14558300

C₂

semidirect

3. C₂×#11924452

C₂

semidirect

4. #14557304

C₂

semidirect

5. C₃₇×#413

C₂

semidirect

42. #10596934

D₄

semidirect

88. #413

D₃₇

semidirect

91. C₃₇

#1914

semidirect

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

subgroup	index
#14558300	2
#14558097	2
C₂×#11924452	2
#14557304	2
C₃₇×#413	2
#1914	37

subgroup

index

C₃₇×#413

Subgroups — #17995141

Conjugacy classes of subgroups

Complements

Maximal subgroups