Group #20252411 — subgroups

Subgroups — #20252411

Order 6016.

500 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 217 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×D₁₈₈	C₂	1	1	yes	maximal, characteristic
2	C₈×D₁₈₈	C₂	1	4	yes	maximal, complemented
3	#15689188	C₂	1	4	yes	maximal, complemented
4	#15689176	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊C₈×D₄₇	C₂	1	4	yes	maximal, complemented
6	C₄₇×#431	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₈₈	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
9	C₂×C₈×D₄₇	C₂²	1	8	yes	complemented
10	C₄×D₁₈₈	C₄	1	2	yes	—
11	C₄×D₁₈₈	C₄	1	2	yes	—
12	C₄×D₁₈₈	C₂²	1	2	yes	—
13	C₄×D₁₈₈	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
15	C₂×(C₂×D₄₇)⋊C₄	C₄	1	2	yes	—
16	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	complemented
17	C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
18	(C₂×D₄₇)⋊C₈	C₂²	1	8	yes	complemented
19	C₄×C₄₇⋊C₈	C₂²	1	2	yes	complemented
20	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₇	C₂²	1	2	yes	complemented
22	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
23	C₄×C₈×C₄₇	C₂²	1	2	yes	complemented
24	C₂³×D₄₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×D₁₈₈	C₈	1	8	yes	complemented
30	C₂×D₁₈₈	C₂×C₄	1	2	yes	—
31	C₂×D₁₈₈	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₄₇	—	4	4	no	—
33	C₂×C₄×D₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄×D₄₇	C₂³	1	4	yes	—
35	C₂×C₈×C₄₇	C₂³	1	4	yes	—
36	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	(C₂×D₄₇)⋊C₄	C₂×C₄	1	4	yes	—
41	(C₂×D₄₇)⋊C₄	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
43	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
45	C₈×D₄₇	—	4	8	no	—
46	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×D₄₇	—	4	2	no	—
48	C₂²×D₄₇	—	4	2	no	—
49	C₂²×D₄₇	C₂×C₈	1	8	yes	complemented
50	C₂²×D₄₇	C₂²×C₄	1	4	yes	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	4	1	no	—
53	C₂×C₄×C₄₇	C₂×C₈	1	4	yes	complemented
54	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₄₇	—	4	2	no	—
61	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
62	D₁₈₈	C₂×C₈	1	8	yes	complemented
63	D₁₈₈	C₂×C₈	1	8	yes	complemented
64	C₄×D₄₇	—	2	8	no	—
65	C₄×D₄₇	—	4	4	no	—
66	C₈×C₄₇	—	2	4	no	—
67	C₄₇⋊C₈	—	2	4	no	—
68	C₂²×C₄₇	—	2	1	no	—
69	C₂²×C₄₇	—	2	1	no	—
70	C₂²×C₄₇	C₂²×C₈	1	2	yes	—
71	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
72	C₂×D₄₇	—	2	4	no	—
73	C₂×D₄₇	—	2	4	no	—
74	C₂×D₄₇	—	4	8	no	—
75	C₂×D₄₇	C₂²×C₈	1	8	yes	—
76	C₄×C₄₇	—	2	1	no	—
77	C₄×C₄₇	—	2	1	no	—
78	C₄×C₄₇	—	2	1	no	—
79	C₄×C₄₇	C₂²×C₈	1	2	yes	—
80	C₄×C₄₇	C₂²×C₈	1	2	yes	—
81	C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
82	Dic₄₇	—	2	4	no	—
83	#2972	—	47	1	no	maximal, Sylow
84	C₂×C₄₇	—	2	2	no	—
85	C₂×C₄₇	#518	1	1	yes	characteristic
86	C₂×C₄₇	C₂³×C₈	1	1	yes	characteristic
87	C₂×C₄₇	#568	1	1	yes	characteristic
88	D₄₇	—	2	8	no	—
89	C₂×C₄×D₄	—	47	1	no	—
90	C₈×D₄	—	47	4	no	—
91	C₈×D₄	—	47	4	no	—
92	#431	—	47	1	no	—
93	#431	D₄₇	1	1	yes	characteristic, complemented
94	C₂×C₂²⋊C₈	—	47	4	no	—
95	C₄₇	#2972	1	1	yes	characteristic, complemented, Sylow
96	C₂²×D₄	—	47	1	no	—
97	C₂³×C₄	—	47	2	no	—
98	C₂²×C₈	—	47	8	no	—
99	C₄×D₄	—	47	2	no	—
100	C₄×D₄	—	47	2	no	—
101	C₄×D₄	—	47	2	no	—
102	C₄×D₄	—	47	2	no	—
103	C₂×C₄⋊C₄	—	47	1	no	—
104	C₂×C₂²⋊C₄	—	47	2	no	—
105	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
106	C₄⋊C₈	—	47	2	no	—
107	C₄⋊C₈	C₂×D₄₇	1	2	yes	complemented
108	C₂²⋊C₈	—	47	2	no	—
109	C₂²⋊C₈	—	47	8	no	—
110	C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
111	C₄×C₈	—	47	2	no	—
112	C₄×C₈	C₂×D₄₇	1	2	yes	complemented
113	C₂⁴	—	47	2	no	—
114	C₂×D₄	—	47	8	no	—
115	C₂×D₄	—	47	2	no	—
116	C₂×D₄	—	47	2	no	—
117	C₂²×C₄	—	47	2	no	—
118	C₂²×C₄	—	47	4	no	—
119	C₂²×C₄	—	47	4	no	—
120	C₂²×C₄	—	188	4	no	—
121	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
122	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
124	C₂×C₈	—	47	4	no	—
125	C₂×C₈	—	188	8	no	—
126	C₂×C₈	C₂²×D₄₇	1	4	yes	—
127	C₄⋊C₄	—	47	2	no	—
128	C₄⋊C₄	—	47	2	no	—
129	C₂²⋊C₄	—	47	4	no	—
130	C₂²⋊C₄	—	47	4	no	—
131	C₄²	C₄×D₄₇	1	1	yes	characteristic
132	C₄²	C₄×D₄₇	1	1	yes	characteristic
133	C₄²	C₂²×D₄₇	1	1	yes	characteristic
134	C₄²	C₂²×D₄₇	1	1	yes	characteristic
135	C₂³	—	47	4	no	—
136	C₂³	—	47	8	no	—
137	C₂³	—	188	2	no	—
138	C₂³	—	188	2	no	—
139	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
140	D₄	—	47	8	no	—
141	D₄	—	47	8	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	47	4	no	—
145	C₂×C₄	—	94	8	no	—
146	C₂×C₄	—	188	4	no	—
147	C₂×C₄	—	188	2	no	—
148	C₂×C₄	C₈×D₄₇	1	4	yes	complemented
149	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
154	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₈	—	94	4	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	47	8	no	—
160	C₂²	—	94	4	no	—
161	C₂²	—	94	4	no	—
162	C₂²	—	188	8	no	—
163	C₂²	C₂×C₈×D₄₇	1	2	yes	—
164	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	94	4	no	—
169	C₄	C₂×C₈×D₄₇	1	2	yes	—
170	C₄	C₂×C₈×D₄₇	1	2	yes	—
171	C₄	#13094070	1	2	yes	—
172	C₂	—	2	2	no	—
173	C₂	—	94	8	no	—
174	C₂	#15689597	1	1	yes	characteristic
175	C₂	C₂²×C₈×D₄₇	1	1	yes	characteristic
176	C₂	#15689889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₁₈₈

C₂

yes

maximal, characteristic

C₈×D₁₈₈

C₂

yes

maximal, complemented

#15689188

C₂

yes

maximal, complemented

#15689176

C₂

yes

maximal, characteristic, complemented

C₂²⋊C₈×D₄₇

C₂

yes

maximal, complemented

C₄₇×#431

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₄

yes

characteristic

C₂²×C₄×D₄₇

C₂²

yes

—

C₂×C₈×D₄₇

C₂²

yes

complemented

C₄×D₁₈₈

C₄

yes

—

C₄×D₁₈₈

C₄

yes

—

C₄×D₁₈₈

C₂²

yes

—

C₄×D₁₈₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₄

yes

characteristic

C₂×(C₂×D₄₇)⋊C₄

C₄

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

complemented

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

(C₂×D₄₇)⋊C₈

C₂²

yes

complemented

C₄×C₄₇⋊C₈

C₂²

yes

complemented

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

complemented

C₂²⋊C₈×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

complemented

C₂³×D₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂×D₁₈₈

C₈

yes

complemented

C₂×D₁₈₈

C₂×C₄

yes

—

C₂×D₁₈₈

C₂×C₄

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

C₂³

yes

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

(C₂×D₄₇)⋊C₄

C₂×C₄

yes

—

(C₂×D₄₇)⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₈×D₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂×C₈

yes

complemented

C₂²×D₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₈

yes

complemented

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

D₁₈₈

C₂×C₈

yes

complemented

D₁₈₈

C₂×C₈

yes

complemented

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×C₈

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₂²×C₈

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²×C₈

yes

—

C₄×C₄₇

C₂²×C₈

yes

—

C₄×C₄₇

H(5,𝔽₂)

yes

—

Dic₄₇

—

#2972

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#518

yes

characteristic

C₂×C₄₇

C₂³×C₈

yes

characteristic

C₂×C₄₇

#568

yes

characteristic

D₄₇

—

C₂×C₄×D₄

—

C₈×D₄

—

C₈×D₄

—

#431

—

#431

D₄₇

yes

characteristic, complemented

C₂×C₂²⋊C₈

—

C₄₇

#2972

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

C₂²×C₈

—

C₄×D₄

—

100

C₄×D₄

—

101

C₄×D₄

—

102

C₄×D₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₂²⋊C₄

—

105

C₂×C₄²

C₂×D₄₇

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

C₂×D₄₇

yes

complemented

108

C₂²⋊C₈

—

109

C₂²⋊C₈

—

110

C₂²⋊C₈

C₂×D₄₇

yes

—

111

C₄×C₈

—

112

C₄×C₈

C₂×D₄₇

yes

complemented

113

C₂⁴

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

188

—

121

C₂²×C₄

C₄×D₄₇

yes

characteristic

122

C₂²×C₄

C₄×D₄₇

yes

characteristic

123

C₂²×C₄

C₂²×D₄₇

yes

characteristic

124

C₂×C₈

—

125

C₂×C₈

—

188

—

126

C₂×C₈

C₂²×D₄₇

yes

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₂²⋊C₄

—

130

C₂²⋊C₄

—

131

C₄²

C₄×D₄₇

yes

characteristic

132

C₄²

C₄×D₄₇

yes

characteristic

133

C₄²

C₂²×D₄₇

yes

characteristic

134

C₄²

C₂²×D₄₇

yes

characteristic

135

C₂³

—

136

C₂³

—

137

C₂³

—

188

—

138

C₂³

—

188

—

139

C₂³

C₂×C₄×D₄₇

yes

characteristic

140

D₄

—

141

D₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

188

—

147

C₂×C₄

—

188

—

148

C₂×C₄

C₈×D₄₇

yes

complemented

149

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

150

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

151

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

152

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

153

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

154

C₂×C₄

C₂³×D₄₇

yes

characteristic

155

C₈

—

156

C₈

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

188

—

163

C₂²

C₂×C₈×D₄₇

yes

—

164

C₂²

C₂²×C₄×D₄₇

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

C₂×C₈×D₄₇

yes

—

170

C₄

C₂×C₈×D₄₇

yes

—

171

C₄

#13094070

yes

—

172

C₂

—

173

C₂

—

174

C₂

#15689597

yes

characteristic

175

C₂

C₂²×C₈×D₄₇

yes

characteristic

176

C₂

#15689889

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator	C₂×C₄₇	C₂³×C₈
Fitting	C₄₇×#431	C₂
socle	C₂²×C₄₇	C₂³×C₄
hypercenter	#431	D₄₇
hypocenter	C₄₇	#2972
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₄₇

commutator

C₂×C₄₇

C₂³×C₈

Fitting

C₄₇×#431

C₂

socle

C₂²×C₄₇

C₂³×C₄

D₄₇

C₄₇

G ▷ C₂×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. C₈×D₁₈₈	C₂	semidirect
3. #15689188	C₂	semidirect
4. #15689176	C₂	semidirect
5. C₂²⋊C₈×D₄₇	C₂	semidirect
6. C₄₇×#431	C₂	semidirect
9. C₂×C₈×D₄₇	C₂²	semidirect
16. C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
18. (C₂×D₄₇)⋊C₈	C₂²	semidirect
19. C₄×C₄₇⋊C₈	C₂²	semidirect
21. C₄⋊C₈×C₄₇	C₂²	semidirect
23. C₄×C₈×C₄₇	C₂²	semidirect
29. C₂×D₁₈₈	C₈	semidirect
49. C₂²×D₄₇	C₂×C₈	semidirect
53. C₂×C₄×C₄₇	C₂×C₈	semidirect
62. D₁₈₈	C₂×C₈	semidirect
63. D₁₈₈	C₂×C₈	semidirect
93. #431	D₄₇	semidirect
95. C₄₇	#2972	semidirect
107. C₄⋊C₈	C₂×D₄₇	semidirect
112. C₄×C₈	C₂×D₄₇	semidirect
148. C₂×C₄	C₈×D₄₇	semidirect

H ≅ G/N

type

2. C₈×D₁₈₈

C₂

semidirect

3. #15689188

C₂

semidirect

4. #15689176

C₂

semidirect

5. C₂²⋊C₈×D₄₇

C₂

semidirect

6. C₄₇×#431

C₂

semidirect

9. C₂×C₈×D₄₇

C₂²

semidirect

16. C₄⋊(C₄₇⋊C₈)

C₂²

semidirect

18. (C₂×D₄₇)⋊C₈

C₂²

semidirect

19. C₄×C₄₇⋊C₈

C₂²

semidirect

21. C₄⋊C₈×C₄₇

C₂²

semidirect

23. C₄×C₈×C₄₇

C₂²

semidirect

29. C₂×D₁₈₈

C₈

semidirect

49. C₂²×D₄₇

C₂×C₈

semidirect

53. C₂×C₄×C₄₇

C₂×C₈

semidirect

62. D₁₈₈

C₂×C₈

semidirect

63. D₁₈₈

C₂×C₈

semidirect

93. #431

D₄₇

semidirect

95. C₄₇

#2972

semidirect

107. C₄⋊C₈

C₂×D₄₇

semidirect

112. C₄×C₈

C₂×D₄₇

semidirect

148. C₂×C₄

C₈×D₄₇

semidirect

Subgroups — #20252411

Conjugacy classes of subgroups

Complements