Group #20243125 — subgroups

Subgroups — #20243125

Order 6016.

388 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 137 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×M₁₆×D₄₇	C₂	1	2	yes	maximal
2	C₂×#13093982	C₂	1	1	yes	maximal, characteristic
3	#15688892	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688891	C₂	1	2	yes	maximal, complemented
5	C₄₇×#343	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
7	C₂×#10625329	C₂²	1	2	yes	—
8	C₂×#10625324	—	2	2	no	—
9	M₁₆×D₄₇	—	2	4	no	—
10	M₁₆×D₄₇	C₄	1	8	yes	complemented
11	C₂×C₈×D₄₇	—	2	2	no	—
12	#13093982	—	2	2	no	—
13	#13093982	—	2	2	no	—
14	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂×(C₂×D₄₇)⋊C₄	C₄	1	2	yes	—
16	C₂×C₄×Dic₄₇	C₄	1	2	yes	—
17	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂³×D₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
23	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₄₇	C₂×C₄	1	4	yes	—
25	C₂×C₄×D₄₇	D₄	1	2	yes	—
26	C₂×C₄×D₄₇	D₄	1	2	yes	—
27	C₂×C₄×D₄₇	D₄	1	2	yes	—
28	C₂×C₄×D₄₇	Q₈	1	2	yes	—
29	M₁₆×C₄₇	—	2	2	no	—
30	M₁₆×C₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₈×C₄₇	—	2	2	no	—
32	C₄⋊C₄×C₄₇	—	2	2	no	—
33	(C₂×D₄₇)⋊C₄	—	2	4	no	—
34	C₄⋊Dic₄₇	—	2	2	no	—
35	C₄×Dic₄₇	—	2	2	no	—
36	C₄×Dic₄₇	—	2	2	no	—
37	#10625329	—	2	2	no	—
38	#10625329	C₂×C₄	1	4	yes	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	#10625324	—	2	8	no	—
41	C₈×D₄₇	—	2	8	no	—
42	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×D₄₇	—	4	1	no	—
44	C₂²×D₄₇	—	4	1	no	—
45	C₂²×D₄₇	C₄²	1	2	yes	—
46	C₂²×D₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×D₄₇	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	4	2	no	—
50	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×Dic₄₇	—	2	2	no	—
55	C₂×Dic₄₇	—	4	2	no	—
56	C₂×Dic₄₇	C₄²	1	2	yes	—
57	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×Dic₄₇	C₄⋊C₄	1	2	yes	—
60	C₄×D₄₇	C₂²⋊C₄	1	4	yes	—
61	C₄×D₄₇	C₂²⋊C₄	1	4	yes	—
62	C₄×D₄₇	C₄⋊C₄	1	4	yes	—
63	C₄×D₄₇	C₄⋊C₄	1	4	yes	—
64	C₈×C₄₇	—	2	4	no	—
65	C₄₇⋊C₈	—	2	4	no	—
66	C₂²×C₄₇	—	2	1	no	—
67	C₂²×C₄₇	—	2	1	no	—
68	C₂²×C₄₇	C₂×C₄²	1	1	yes	characteristic
69	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×D₄₇	—	2	2	no	—
72	C₂×D₄₇	—	2	2	no	—
73	C₂×D₄₇	—	4	2	no	—
74	C₂×D₄₇	—	4	2	no	—
75	C₂×D₄₇	#95	1	4	yes	—
76	C₄×C₄₇	—	4	2	no	—
77	C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
78	C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
79	Dic₄₇	—	4	2	no	—
80	Dic₄₇	#95	1	2	yes	—
81	Dic₄₇	#95	1	2	yes	—
82	#1742	—	47	1	no	maximal, Sylow
83	C₂×C₄₇	—	2	1	no	—
84	C₂×C₄₇	—	2	1	no	—
85	C₂×C₄₇	C₂×#95	1	1	yes	characteristic
86	C₂×C₄₇	#413	1	1	yes	characteristic
87	C₂×C₄₇	#413	1	1	yes	characteristic
88	D₄₇	—	2	4	no	—
89	C₂²×M₁₆	—	47	2	no	—
90	C₂×#117	—	47	1	no	—
91	#343	—	47	2	no	—
92	#343	—	47	1	no	—
93	#343	D₄₇	1	1	yes	characteristic, complemented
94	C₄₇	#1742	1	1	yes	characteristic, complemented, Sylow
95	C₂³×C₄	—	47	1	no	—
96	C₂×M₁₆	—	47	2	no	—
97	C₂×M₁₆	—	47	8	no	—
98	C₂×M₁₆	—	94	2	no	—
99	C₂×M₁₆	—	94	4	no	—
100	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
101	C₂²×C₈	—	94	2	no	—
102	#117	—	94	2	no	—
103	#117	—	94	2	no	—
104	C₂×C₄⋊C₄	—	47	1	no	—
105	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
106	C₂×C₂²⋊C₄	—	47	2	no	—
107	C₂×C₄²	—	47	2	no	—
108	C₂⁴	—	47	1	no	—
109	C₂²×C₄	—	47	1	no	—
110	C₂²×C₄	—	47	4	no	—
111	C₂²×C₄	—	47	2	no	—
112	C₂²×C₄	—	47	2	no	—
113	C₂²×C₄	—	47	2	no	—
114	C₂²×C₄	—	47	2	no	—
115	C₂²×C₄	—	47	2	no	—
116	C₂²×C₄	C₄×D₄₇	1	2	yes	—
117	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
118	M₁₆	—	2	2	no	—
119	M₁₆	—	47	4	no	—
120	M₁₆	—	94	8	no	—
121	M₁₆	—	94	2	no	—
122	M₁₆	C₄×D₄₇	1	4	yes	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	94	8	no	—
125	C₂×C₈	—	94	2	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	94	2	no	—
128	C₂²⋊C₄	—	94	4	no	—
129	C₄²	—	94	2	no	—
130	C₄²	—	94	2	no	—
131	C₂³	—	47	2	no	—
132	C₂³	—	47	2	no	—
133	C₂³	—	47	2	no	—
134	C₂³	—	188	1	no	—
135	C₂³	—	188	1	no	—
136	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	47	2	no	—
140	C₂×C₄	—	47	1	no	—
141	C₂×C₄	—	47	1	no	—
142	C₂×C₄	—	47	2	no	—
143	C₂×C₄	—	47	4	no	—
144	C₂×C₄	—	47	4	no	—
145	C₂×C₄	—	47	4	no	—
146	C₂×C₄	—	47	4	no	—
147	C₂×C₄	—	94	2	no	—
148	C₂×C₄	—	188	2	no	—
149	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
150	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
151	C₂×C₄	D₄×D₄₇	1	2	yes	—
152	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
153	C₈	—	2	4	no	—
154	C₈	—	94	4	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	47	4	no	—
158	C₂²	—	94	2	no	—
159	C₂²	—	94	2	no	—
160	C₂²	—	188	2	no	—
161	C₂²	—	188	2	no	—
162	C₂²	C₄²×D₄₇	1	1	yes	characteristic
163	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
164	C₂²	C₄⋊C₄×D₄₇	1	1	yes	characteristic
165	C₄	—	4	2	no	—
166	C₄	—	47	2	no	—
167	C₄	—	47	2	no	—
168	C₄	—	188	2	no	—
169	C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
170	C₄	C₄⋊C₄×D₄₇	1	2	yes	—
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	94	4	no	—
174	C₂	#95×D₄₇	1	1	yes	characteristic
175	C₂	#15689104	1	1	yes	characteristic
176	C₂	#15689106	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×D₄₇

C₂

yes

maximal

C₂×#13093982

C₂

yes

maximal, characteristic

#15688892

C₂

yes

maximal, characteristic, complemented

#15688891

C₂

yes

maximal, complemented

C₄₇×#343

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₄₇

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

—

C₂×#10625324

—

M₁₆×D₄₇

—

M₁₆×D₄₇

C₄

yes

complemented

C₂×C₈×D₄₇

—

#13093982

—

#13093982

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×(C₂×D₄₇)⋊C₄

C₄

yes

—

C₂×C₄×Dic₄₇

C₄

yes

—

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

D₄

yes

—

C₂×C₄×D₄₇

D₄

yes

—

C₂×C₄×D₄₇

D₄

yes

—

C₂×C₄×D₄₇

Q₈

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

—

#10625324

—

C₈×D₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₄²

yes

—

C₂²×D₄₇

C₂²⋊C₄

yes

—

C₂²×D₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₄²

yes

—

C₂×Dic₄₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₇

C₄⋊C₄

yes

—

C₄×D₄₇

C₂²⋊C₄

yes

—

C₄×D₄₇

C₂²⋊C₄

yes

—

C₄×D₄₇

C₄⋊C₄

yes

—

C₄×D₄₇

C₄⋊C₄

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄²

yes

characteristic

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

#95

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

Dic₄₇

—

Dic₄₇

#95

yes

—

Dic₄₇

#95

yes

—

#1742

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#95

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

C₂²×M₁₆

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄₇

#1742

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂×M₁₆

C₂×D₄₇

yes

—

101

C₂²×C₈

—

102

#117

—

103

#117

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

106

C₂×C₂²⋊C₄

—

107

C₂×C₄²

—

108

C₂⁴

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₄×D₄₇

yes

—

117

C₂²×C₄

C₂²×D₄₇

yes

characteristic

118

M₁₆

—

119

M₁₆

—

120

M₁₆

—

121

M₁₆

—

122

M₁₆

C₄×D₄₇

yes

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

—

188

—

135

C₂³

—

188

—

136

C₂³

C₂×C₄×D₄₇

yes

characteristic

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

—

148

C₂×C₄

—

188

—

149

C₂×C₄

C₂×C₄×D₄₇

yes

—

150

C₂×C₄

D₄×D₄₇

yes

characteristic

151

C₂×C₄

D₄×D₄₇

yes

—

152

C₂×C₄

Q₈×D₄₇

yes

characteristic

153

C₈

—

154

C₈

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

188

—

161

C₂²

—

188

—

162

C₂²

C₄²×D₄₇

yes

characteristic

163

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

164

C₂²

C₄⋊C₄×D₄₇

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

188

—

169

C₄

C₂²⋊C₄×D₄₇

yes

—

170

C₄

C₄⋊C₄×D₄₇

yes

—

171

C₂

—

172

C₂

—

173

C₂

—

174

C₂

#95×D₄₇

yes

characteristic

175

C₂

#15689104

yes

characteristic

176

C₂

#15689106

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₄₇
commutator	C₂²×C₄₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#343	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#343	D₄₇
hypocenter	C₄₇	#1742
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₄₇

commutator

C₂²×C₄₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#343

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
3. #15688892	C₂	semidirect
4. #15688891	C₂	semidirect
5. C₄₇×#343	C₂	semidirect
10. M₁₆×D₄₇	C₄	semidirect
93. #343	D₄₇	semidirect
94. C₄₇	#1742	semidirect

H ≅ G/N

type

3. #15688892

C₂

semidirect

4. #15688891

C₂

semidirect

5. C₄₇×#343

C₂

semidirect

10. M₁₆×D₄₇

C₄

semidirect

93. #343

D₄₇

semidirect

94. C₄₇

#1742

semidirect

Subgroups — #20243125

Conjugacy classes of subgroups

Complements