Group #20246301 — subgroups

Subgroups — Q₈×Dic₄₇⋊C₄ #20246301

Order 6016.

568 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 287 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×Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#15689021	C₂	1	3	yes	maximal
4	#15689019	C₂	1	3	yes	maximal
5	#15688992	C₂	1	3	yes	maximal
6	C₄×Dic₄₇⋊C₄	C₂	1	3	yes	maximal
7	C₂×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₄₇	—	2	4	no	—
9	Q₈×Dic₄₇	C₄	1	8	yes	complemented
10	C₂×C₄⋊Dic₄₇	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	3	yes	—
12	C₂×Dic₄₇⋊C₄	C₂²	1	3	yes	—
13	C₂×Dic₄₇⋊C₄	C₂²	1	6	yes	—
14	C₂×Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₄₇	C₂²	1	3	yes	—
16	C₂×C₄×Dic₄₇	C₂²	1	3	yes	—
17	C₄₇⋊#95	C₂²	1	3	yes	—
18	C₄₇⋊#95	C₂²	1	3	yes	—
19	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₄₇	—	2	4	no	—
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
22	C₂×C₄²×C₄₇	C₂²	1	3	yes	—
23	C₂×Q₈×C₄₇	C₂×C₄	1	4	yes	—
24	C₂×Q₈×C₄₇	D₄	1	4	yes	—
25	C₂×Q₈×C₄₇	Q₈	1	4	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
29	C₂²×Dic₄₇	C₂³	1	3	yes	—
30	C₂²×Dic₄₇	C₂³	1	3	yes	—
31	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
32	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₄₇	—	2	6	no	—
34	C₄²×C₄₇	—	2	6	no	—
35	C₄⋊Dic₄₇	—	2	6	no	—
36	C₄⋊Dic₄₇	C₂×C₄	1	12	yes	—
37	Dic₄₇⋊C₄	—	2	6	no	—
38	Dic₄₇⋊C₄	—	4	6	no	—
39	Dic₄₇⋊C₄	Q₈	1	4	yes	direct product factor
40	C₄×Dic₄₇	—	2	6	no	—
41	C₄×Dic₄₇	C₂×C₄	1	12	yes	—
42	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
43	Q₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	—	2	1	no	—
46	C₂×C₄×C₄₇	—	2	3	no	—
47	C₂×C₄×C₄₇	—	2	3	no	—
48	C₂×C₄×C₄₇	—	2	3	no	—
49	C₂×C₄×C₄₇	C₂²×C₄	1	6	yes	—
50	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
51	C₂×C₄×C₄₇	C₂×Q₈	1	6	yes	—
52	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
53	C₂×Dic₄₇	—	2	1	no	—
54	C₂×Dic₄₇	—	2	3	no	—
55	C₂×Dic₄₇	—	2	3	no	—
56	C₂×Dic₄₇	—	2	1	no	—
57	C₂×Dic₄₇	—	2	3	no	—
58	C₂×Dic₄₇	—	2	3	no	—
59	C₂×Dic₄₇	—	2	3	no	—
60	C₂×Dic₄₇	C₂²×C₄	1	6	yes	—
61	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
62	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
63	C₂×Dic₄₇	C₂×Q₈	1	2	yes	—
64	C₂×Dic₄₇	#41	1	2	yes	—
65	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
66	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
67	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
68	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
69	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
70	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
71	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
72	C₄×C₄₇	—	2	2	no	—
73	C₄×C₄₇	—	4	3	no	—
74	C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
75	Dic₄₇	—	2	6	no	—
76	Dic₄₇	—	2	2	no	—
77	Dic₄₇	—	4	3	no	—
78	Dic₄₇	C₄×Q₈	1	4	yes	complemented
79	Q₈×C₄⋊C₄	—	47	1	no	maximal, Sylow
80	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
82	C₂×C₄₇	#520	1	1	yes	characteristic
83	C₂×C₄₇	D₄×Q₈	1	1	yes	characteristic
84	C₂×C₄₇	#550	1	1	yes	characteristic
85	C₂×C₄₇	#557	1	1	yes	characteristic
86	C₂×C₄₇	Q₈²	1	1	yes	characteristic
87	C₂×C₄×Q₈	—	47	1	no	—
88	C₂×C₄×Q₈	—	47	1	no	—
89	C₂×C₄×Q₈	D₄₇	1	1	yes	characteristic
90	#389	—	47	3	no	—
91	#389	—	47	3	no	—
92	#384	—	47	3	no	—
93	C₄×C₄⋊C₄	—	47	3	no	—
94	C₄₇	Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
96	C₄×Q₈	—	2	4	no	—
97	C₄×Q₈	—	47	8	no	—
98	C₄×Q₈	—	94	4	no	—
99	C₂×C₄⋊C₄	—	47	3	no	—
100	C₂×C₄⋊C₄	—	47	1	no	—
101	C₂×C₄⋊C₄	—	47	3	no	—
102	C₂×C₄⋊C₄	—	47	3	no	—
103	C₂×C₄⋊C₄	—	47	6	no	—
104	C₂×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
105	C₂×C₄²	—	47	3	no	—
106	C₂×C₄²	—	47	3	no	—
107	C₂×C₄²	C₂×D₄₇	1	3	yes	—
108	#95	—	47	3	no	—
109	#95	—	47	3	no	—
110	C₂×Q₈	Dic₉₄	1	4	yes	—
111	C₂×Q₈	C₄×D₄₇	1	4	yes	—
112	C₂×Q₈	C₂²⋊D₄₇	1	4	yes	—
113	C₂²×C₄	—	47	1	no	—
114	C₂²×C₄	—	47	1	no	—
115	C₂²×C₄	—	47	3	no	—
116	C₂²×C₄	—	47	3	no	—
117	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
118	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
119	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
120	C₄⋊C₄	—	2	6	no	—
121	C₄⋊C₄	—	47	12	no	—
122	C₄⋊C₄	—	47	4	no	—
123	C₄⋊C₄	—	94	6	no	—
124	C₄⋊C₄	—	94	6	no	—
125	C₄⋊C₄	—	188	6	no	—
126	C₄²	—	2	6	no	—
127	C₄²	—	47	12	no	—
128	C₄²	—	94	6	no	—
129	C₂³	C₂³×D₄₇	1	1	yes	characteristic
130	Q₈	Dic₄₇⋊C₄	1	16	yes	direct product factor
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	2	3	no	—
135	C₂×C₄	—	2	3	no	—
136	C₂×C₄	—	47	2	no	—
137	C₂×C₄	—	47	2	no	—
138	C₂×C₄	—	47	2	no	—
139	C₂×C₄	—	47	6	no	—
140	C₂×C₄	—	47	2	no	—
141	C₂×C₄	—	94	1	no	—
142	C₂×C₄	—	94	3	no	—
143	C₂×C₄	—	94	1	no	—
144	C₂×C₄	—	94	3	no	—
145	C₂×C₄	—	94	3	no	—
146	C₂×C₄	—	94	3	no	—
147	C₂×C₄	—	94	3	no	—
148	C₂×C₄	C₂×Dic₉₄	1	6	yes	—
149	C₂×C₄	C₂×C₄×D₄₇	1	6	yes	—
150	C₂×C₄	Q₈×D₄₇	1	2	yes	—
151	C₂×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
152	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
153	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
154	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
155	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
156	C₂²	#41×D₄₇	1	1	yes	characteristic
157	C₂²	#13094070	1	1	yes	characteristic
158	C₂²	#13094072	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	3	no	—
161	C₄	—	47	4	no	—
162	C₄	—	94	2	no	—
163	C₄	—	94	6	no	—
164	C₄	—	188	3	no	—
165	C₄	C₂×Dic₄₇⋊C₄	1	12	yes	—
166	C₂	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
167	C₂	C₄×Q₈×D₄₇	1	1	yes	characteristic
168	C₂	#15689605	1	1	yes	characteristic
169	C₂	Q₈×C₂²⋊D₄₇	1	1	yes	characteristic
170	C₂	#15689792	1	1	yes	characteristic
171	C₂	#15689849	1	1	yes	characteristic
172	C₂	Q₈×Dic₉₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₄₇

C₂

yes

maximal, characteristic

C₂×Q₈×Dic₄₇

C₂

yes

maximal, characteristic

#15689021

C₂

yes

maximal

#15689019

C₂

yes

maximal

#15688992

C₂

yes

maximal

C₄×Dic₄₇⋊C₄

C₂

yes

maximal

C₂×C₄×Q₈×C₄₇

C₂

yes

maximal, characteristic

Q₈×Dic₄₇

—

Q₈×Dic₄₇

C₄

yes

complemented

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

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×C₄²×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

Q₈

yes

direct product factor

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

Q₈×C₄₇

C₄⋊C₄

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

#41

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

characteristic

C₂²×C₄₇

2₋¹⁺⁴

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₄×Q₈

yes

complemented

Q₈×C₄⋊C₄

—

maximal, Sylow

C₂×C₄₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₄₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₄₇

#520

yes

characteristic

C₂×C₄₇

D₄×Q₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

Q₈²

yes

characteristic

C₂×C₄×Q₈

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₄₇

yes

—

—

—

C₄×C₄⋊C₄

—

C₄₇

Q₈×C₄⋊C₄

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

105

C₂×C₄²

—

106

C₂×C₄²

—

107

C₂×C₄²

C₂×D₄₇

yes

—

108

#95

—

109

#95

—

110

C₂×Q₈

Dic₉₄

yes

—

111

C₂×Q₈

C₄×D₄₇

yes

—

112

C₂×Q₈

C₂²⋊D₄₇

yes

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂²×D₄₇

yes

—

118

C₂²×C₄

C₂²×D₄₇

yes

characteristic

119

C₂²×C₄

C₂²×D₄₇

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

188

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₂³

C₂³×D₄₇

yes

characteristic

130

Q₈

Dic₄₇⋊C₄

yes

direct product factor

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

—

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

C₂×Dic₉₄

yes

—

149

C₂×C₄

C₂×C₄×D₄₇

yes

—

150

C₂×C₄

Q₈×D₄₇

yes

—

151

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

152

C₂²

C₂²×C₄×D₄₇

yes

characteristic

153

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

154

C₂²

C₂×Q₈×D₄₇

yes

characteristic

155

C₂²

C₂²×Dic₉₄

yes

characteristic

156

C₂²

#41×D₄₇

yes

characteristic

157

C₂²

#13094070

yes

characteristic

158

C₂²

#13094072

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

188

—

165

C₄

C₂×Dic₄₇⋊C₄

yes

—

166

C₂

C₂²×Dic₄₇⋊C₄

yes

characteristic

167

C₂

C₄×Q₈×D₄₇

yes

characteristic

168

C₂

#15689605

yes

characteristic

169

C₂

Q₈×C₂²⋊D₄₇

yes

characteristic

170

C₂

#15689792

yes

characteristic

171

C₂

#15689849

yes

characteristic

172

C₂

Q₈×Dic₉₄

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₄₇
hypocenter	C₄₇	Q₈×C₄⋊C₄
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂⁴

hypercenter

C₂×C₄×Q₈

D₄₇

hypocenter

C₄₇

Q₈×C₄⋊C₄

LCS

G ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
9. Q₈×Dic₄₇	C₄	semidirect
39. Dic₄₇⋊C₄	Q₈	direct (N × H)
43. Q₈×C₄₇	C₄⋊C₄	semidirect
78. Dic₄₇	C₄×Q₈	semidirect
94. C₄₇	Q₈×C₄⋊C₄	semidirect
130. Q₈	Dic₄₇⋊C₄	direct (N × H)

H ≅ G/N

type

9. Q₈×Dic₄₇

C₄

semidirect

39. Dic₄₇⋊C₄

Q₈

direct (N × H)

43. Q₈×C₄₇

C₄⋊C₄

semidirect

78. Dic₄₇

C₄×Q₈

semidirect

94. C₄₇

Q₈×C₄⋊C₄

semidirect

130. Q₈

Dic₄₇⋊C₄

direct (N × H)

Subgroups — Q₈×Dic₄₇⋊C₄ #20246301

Conjugacy classes of subgroups

Complements

Subgroups — Q8×Dic47⋊C4 #20246301

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×Dic₄₇⋊C₄ #20246301