Group #20248776 — subgroups

Subgroups — #20248776

Order 6016.

468 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 173 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×#13094016	C₂	1	1	yes	maximal, characteristic
2	#15689054	C₂	1	2	yes	maximal
3	#15689042	C₂	1	2	yes	maximal
4	#15689021	C₂	1	2	yes	maximal
5	#15689021	C₂	1	2	yes	maximal
6	#15689018	C₂	1	4	yes	maximal
7	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	#13094016	—	2	4	no	—
10	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
14	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
15	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
16	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
17	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
18	C₄₇⋊#95	C₂²	1	2	yes	—
19	C₄₇⋊#95	C₂²	1	4	yes	—
20	C₄₇⋊#95	C₂²	1	4	yes	—
21	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₄₇	—	2	4	no	—
23	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×Q₈×C₄₇	—	2	2	no	—
28	C₂×Q₈×C₄₇	—	2	4	no	—
29	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
30	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²×Dic₄₇	C₂³	1	2	yes	—
35	C₂²×Dic₄₇	C₂³	1	2	yes	—
36	C₂²×Dic₄₇	C₂³	1	4	yes	—
37	C₄⋊C₄×C₄₇	—	2	4	no	—
38	C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₄²×C₄₇	—	2	4	no	—
40	C₄²×C₄₇	D₄	1	4	yes	—
41	C₄⋊Dic₄₇	—	2	2	no	—
42	C₄⋊Dic₄₇	—	2	2	no	—
43	C₄⋊Dic₄₇	Q₈	1	8	yes	complemented
44	Dic₄₇⋊C₄	—	2	4	no	—
45	Dic₄₇⋊C₄	—	2	4	no	—
46	Dic₄₇⋊C₄	—	2	8	no	—
47	Dic₄₇⋊C₄	—	4	4	no	—
48	C₄×Dic₄₇	—	4	2	no	—
49	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
50	Q₈×C₄₇	—	2	8	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	—	2	1	no	—
55	C₂×C₄×C₄₇	—	2	1	no	—
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
63	C₂×C₄×C₄₇	#41	1	4	yes	—
64	C₂×Dic₄₇	—	2	4	no	—
65	C₂×Dic₄₇	—	2	4	no	—
66	C₂×Dic₄₇	—	2	2	no	—
67	C₂×Dic₄₇	—	2	2	no	—
68	C₂×Dic₄₇	—	2	4	no	—
69	C₂×Dic₄₇	—	2	2	no	—
70	C₂×Dic₄₇	—	2	2	no	—
71	C₂×Dic₄₇	C₂×Q₈	1	8	yes	—
72	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₄₇	C₂²×Q₈	1	2	yes	—
74	C₂²×C₄₇	C₂×#41	1	2	yes	—
75	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
76	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
77	C₄×C₄₇	—	2	2	no	—
78	C₄×C₄₇	—	2	4	no	—
79	C₄×C₄₇	—	2	2	no	—
80	C₄×C₄₇	—	4	2	no	—
81	C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
82	Dic₄₇	—	2	8	no	—
83	Dic₄₇	—	4	2	no	—
84	Dic₄₇	—	4	2	no	—
85	#2584	—	47	1	no	maximal, Sylow
86	C₂×C₄₇	#525	1	1	yes	characteristic
87	C₂×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄₇	#537	1	1	yes	characteristic
89	C₂×C₄₇	#554	1	2	yes	—
90	C₂×C₄₇	#557	1	2	yes	—
91	C₂×#125	—	47	1	no	—
92	C₂×C₄×Q₈	D₄₇	1	1	yes	characteristic
93	#398	—	47	2	no	—
94	#395	—	47	2	no	—
95	#389	—	47	4	no	—
96	#389	—	47	2	no	—
97	#389	—	47	2	no	—
98	C₄×C₄⋊C₄	—	47	1	no	—
99	C₄₇	#2584	1	1	yes	characteristic, complemented, Sylow
100	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
101	#125	—	94	4	no	—
102	C₄×Q₈	—	2	4	no	—
103	C₂×C₄⋊C₄	—	47	2	no	—
104	C₂×C₄⋊C₄	—	47	2	no	—
105	C₂×C₄⋊C₄	—	47	2	no	—
106	C₂×C₄⋊C₄	—	47	1	no	—
107	C₂×C₄⋊C₄	—	47	1	no	—
108	C₂×C₄⋊C₄	—	47	4	no	—
109	C₂×C₄⋊C₄	—	47	4	no	—
110	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
111	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
112	C₂×C₄²	—	47	2	no	—
113	C₂×C₄²	C₂×D₄₇	1	2	yes	—
114	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
115	#95	—	47	2	no	—
116	#95	—	47	4	no	—
117	#95	—	47	4	no	—
118	C₂×Q₈	—	2	4	no	—
119	C₂×Q₈	—	2	2	no	—
120	C₂²×C₄	—	47	4	no	—
121	C₂²×C₄	—	47	2	no	—
122	C₂²×C₄	—	47	2	no	—
123	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
124	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
125	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	47	8	no	—
131	C₄⋊C₄	—	94	8	no	—
132	C₄⋊C₄	—	94	4	no	—
133	C₄⋊C₄	—	94	4	no	—
134	C₄⋊C₄	—	94	2	no	—
135	C₄⋊C₄	—	94	2	no	—
136	C₄⋊C₄	—	188	4	no	—
137	C₄²	—	2	4	no	—
138	C₄²	—	188	2	no	—
139	C₄²	C₂²⋊D₄₇	1	4	yes	—
140	C₂³	C₂³×D₄₇	1	1	yes	characteristic
141	Q₈	—	2	8	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	47	8	no	—
151	C₂×C₄	—	94	4	no	—
152	C₂×C₄	—	94	4	no	—
153	C₂×C₄	—	94	2	no	—
154	C₂×C₄	—	94	2	no	—
155	C₂×C₄	—	94	4	no	—
156	C₂×C₄	—	94	2	no	—
157	C₂×C₄	—	94	2	no	—
158	C₂×C₄	#10625350	1	4	yes	—
159	C₂×C₄	Q₈×D₄₇	1	4	yes	—
160	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
161	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
163	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
164	C₂²	C₂×Q₈×D₄₇	1	2	yes	—
165	C₂²	C₂×#10625350	1	2	yes	—
166	C₂²	#13094070	1	1	yes	characteristic
167	C₂²	#13094071	1	1	yes	characteristic
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	2	no	—
171	C₄	—	4	2	no	—
172	C₄	—	94	8	no	—
173	C₄	—	188	2	no	—
174	C₄	—	188	2	no	—
175	C₄	#13094022	1	4	yes	—
176	C₂	#15689634	1	1	yes	characteristic
177	C₂	C₂×#13094022	1	1	yes	characteristic
178	C₂	#15689701	1	1	yes	characteristic
179	C₂	#15689833	1	2	yes	—
180	C₂	#15689851	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094016

C₂

yes

maximal, characteristic

#15689054

C₂

yes

maximal

#15689042

C₂

yes

maximal

#15689021

C₂

yes

maximal

#15689021

C₂

yes

maximal

#15689018

C₂

yes

maximal

C₄×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₄₇

C₂

yes

maximal, characteristic

#13094016

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

Q₈

yes

complemented

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×Q₈

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

—

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

characteristic

C₂²×C₄₇

2₋¹⁺⁴

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄⋊Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂×C₄⋊Q₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#557

yes

—

C₂×#125

—

C₂×C₄×Q₈

D₄₇

yes

—

—

—

—

—

C₄×C₄⋊C₄

—

C₄₇

#2584

yes

characteristic, complemented, Sylow

100

C₂²×Q₈

C₂×D₄₇

yes

characteristic

101

#125

—

102

C₄×Q₈

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

111

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

112

C₂×C₄²

—

113

C₂×C₄²

C₂×D₄₇

yes

—

114

C₂×C₄²

C₂×D₄₇

yes

characteristic

115

#95

—

116

#95

—

117

#95

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂²×D₄₇

yes

—

124

C₂²×C₄

C₂²×D₄₇

yes

—

125

C₂²×C₄

C₂²×D₄₇

yes

characteristic

126

C₂²×C₄

C₂²×D₄₇

yes

characteristic

127

C₂²×C₄

C₂²×D₄₇

yes

characteristic

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

188

—

137

C₄²

—

138

C₄²

—

188

—

139

C₄²

C₂²⋊D₄₇

yes

—

140

C₂³

C₂³×D₄₇

yes

characteristic

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#10625350

yes

—

159

C₂×C₄

Q₈×D₄₇

yes

—

160

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

161

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

162

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

163

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

164

C₂²

C₂×Q₈×D₄₇

yes

—

165

C₂²

C₂×#10625350

yes

—

166

C₂²

#13094070

yes

characteristic

167

C₂²

#13094071

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

188

—

174

C₄

—

188

—

175

C₄

#13094022

yes

—

176

C₂

#15689634

yes

characteristic

177

C₂

C₂×#13094022

yes

characteristic

178

C₂

#15689701

yes

characteristic

179

C₂

#15689833

yes

—

180

C₂

#15689851

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×C₄×Q₈×C₄₇	C₂
hypercenter	C₂×C₄×Q₈	D₄₇
hypocenter	C₄₇	#2584
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×C₄×Q₈×C₄₇

C₂

hypercenter

C₂×C₄×Q₈

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
43. C₄⋊Dic₄₇	Q₈	semidirect
99. C₄₇	#2584	semidirect

H ≅ G/N

type

43. C₄⋊Dic₄₇

Q₈

semidirect

99. C₄₇

#2584

semidirect

Subgroups — #20248776

Conjugacy classes of subgroups

Complements