Group #20241765 — subgroups

Subgroups — C₂×Q₈⋊(C₄₇⋊C₈) #20241765

Order 6016.

280 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 141 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⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₄₇⋊C₈)	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₈	—	2	1	no	—
6	C₂²×C₄₇⋊C₈	—	2	1	no	—
7	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₄×C₄₇⋊C₈	—	2	1	no	—
10	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
11	C₂²×Q₈×C₄₇	C₄	1	1	yes	characteristic
12	C₄×Q₈×C₄₇	—	2	1	no	—
13	C₄×Q₈×C₄₇	—	2	1	no	—
14	C₄×Q₈×C₄₇	C₄	1	2	yes	—
15	C₄×Q₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
17	C₂×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
18	C₂×C₄²×C₄₇	—	2	1	no	—
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₄₇	—	2	1	no	—
21	C₂×Q₈×C₄₇	—	2	2	no	—
22	C₂×Q₈×C₄₇	C₈	1	4	yes	complemented
23	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₄₇	—	2	1	no	—
25	C₂²×C₄×C₄₇	—	2	1	no	—
26	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₄⋊C₄×C₄₇	—	2	1	no	—
31	C₄⋊C₄×C₄₇	—	2	2	no	—
32	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₄²×C₄₇	—	2	2	no	—
34	C₄²×C₄₇	—	2	2	no	—
35	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₄₇	D₄	1	1	yes	characteristic
37	C₄²×C₄₇	D₄	1	1	yes	characteristic
38	C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄₇⋊C₈	—	2	4	no	—
40	C₂×C₄₇⋊C₈	—	2	2	no	—
41	C₂×C₄₇⋊C₈	—	2	2	no	—
42	C₂×C₄₇⋊C₈	—	4	2	no	—
43	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
44	Q₈×C₄₇	—	2	4	no	—
45	Q₈×C₄₇	—	2	2	no	—
46	Q₈×C₄₇	C₂×C₈	1	4	yes	complemented
47	C₂×C₄×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	4	2	no	—
52	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×C₈	1	2	yes	—
56	C₂×C₄×C₄₇	M₁₆	1	2	yes	—
57	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
58	C₂×C₄×C₄₇	Q₁₆	1	2	yes	—
59	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₄₇⋊C₈	—	2	4	no	—
63	C₄₇⋊C₈	—	4	2	no	—
64	C₂²×C₄₇	C₂²⋊C₈	1	2	yes	—
65	C₂²×C₄₇	#103	1	2	yes	—
66	C₂²×C₄₇	#104	1	2	yes	—
67	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₄₇	—	2	4	no	—
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	2	1	no	—
71	C₄×C₄₇	—	4	2	no	—
72	C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
73	C₄×C₄₇	#103	1	2	yes	—
74	C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
75	C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
76	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×Q₈⋊C₈	—	47	1	no	maximal, Sylow
79	C₂×C₄₇	Q₈⋊C₈	1	4	yes	complemented
80	C₂×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄₇	C₂×#103	1	1	yes	characteristic
82	C₂×C₄₇	C₂×#104	1	1	yes	characteristic
83	C₂×C₄×Q₈	D₄₇	1	1	yes	characteristic
84	C₂×C₄⋊C₈	—	47	1	no	—
85	C₂×C₄×C₈	—	47	1	no	—
86	Q₈⋊C₈	—	47	4	no	—
87	C₄₇	C₂×Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
88	C₂²×Q₈	Dic₄₇	1	1	yes	characteristic
89	C₂²×C₈	—	94	1	no	—
90	C₂²×C₈	—	94	1	no	—
91	C₄×Q₈	—	2	1	no	—
92	C₄×Q₈	—	2	1	no	—
93	C₄×Q₈	Dic₄₇	1	2	yes	—
94	C₄×Q₈	C₂×D₄₇	1	2	yes	—
95	C₂×C₄⋊C₄	—	2	1	no	—
96	C₂×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
97	C₂×C₄²	—	2	1	no	—
98	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
99	C₄⋊C₈	—	47	2	no	—
100	C₄⋊C₈	—	94	1	no	—
101	C₄×C₈	—	47	2	no	—
102	C₄×C₈	—	94	1	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	C₄₇⋊C₈	1	4	yes	complemented
106	C₂×Q₈	C₂×Dic₄₇	1	2	yes	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
111	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
112	C₂×C₈	—	94	4	no	—
113	C₂×C₈	—	94	2	no	—
114	C₂×C₈	—	94	2	no	—
115	C₂×C₈	—	188	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
120	C₄²	—	2	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
123	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
124	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
125	C₄²	C₂²×D₄₇	1	1	yes	characteristic
126	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	Q₈	—	2	2	no	—
129	Q₈	C₂×C₄₇⋊C₈	1	4	yes	complemented
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
136	C₂×C₄	#10625329	1	2	yes	—
137	C₂×C₄	#10625336	1	2	yes	—
138	C₂×C₄	#10625337	1	2	yes	—
139	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
140	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
141	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
142	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
145	C₈	—	94	4	no	—
146	C₈	—	188	2	no	—
147	C₂²	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
148	C₂²	#13093948	1	2	yes	—
149	C₂²	#13093951	1	2	yes	—
150	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
151	C₄	—	2	4	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	4	2	no	—
155	C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
156	C₄	#13093948	1	2	yes	—
157	C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
158	C₄	C₂×#10625329	1	1	yes	characteristic
159	C₄	C₂×#10625336	1	1	yes	characteristic
160	C₄	C₂×#10625337	1	1	yes	characteristic
161	C₂	Q₈⋊(C₄₇⋊C₈)	1	4	yes	direct product factor
162	C₂	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
163	C₂	C₂×#13093948	1	1	yes	characteristic
164	C₂	C₂×#13093951	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic

Q₈⋊(C₄₇⋊C₈)

C₂

yes

maximal, direct product factor

C₂×C₄×Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

C₂²

yes

—

C₂²×Q₈×C₄₇

C₄

yes

characteristic

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

C₄

yes

—

C₄×Q₈×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₄

yes

characteristic

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₈

yes

complemented

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

C₂×C₈

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

M₁₆

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

Q₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂²⋊C₈

yes

—

C₂²×C₄₇

#103

yes

—

C₂²×C₄₇

#104

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊C₈

yes

—

C₄×C₄₇

#103

yes

—

C₄×C₄₇

C₂²×C₈

yes

characteristic

C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×Q₈⋊C₈

—

maximal, Sylow

C₂×C₄₇

Q₈⋊C₈

yes

complemented

C₂×C₄₇

C₂×C₂²⋊C₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄×Q₈

D₄₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

Q₈⋊C₈

—

C₄₇

C₂×Q₈⋊C₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₄₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

Dic₄₇

yes

—

C₄×Q₈

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₄₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

C₄₇⋊C₈

yes

complemented

106

C₂×Q₈

C₂×Dic₄₇

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

110

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

111

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

188

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

C₂×Dic₄₇

yes

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂×Dic₄₇

yes

characteristic

123

C₄²

C₂²⋊D₄₇

yes

characteristic

124

C₄²

C₂²⋊D₄₇

yes

characteristic

125

C₄²

C₂²×D₄₇

yes

characteristic

126

C₂³

C₂²⋊Dic₄₇

yes

characteristic

127

Q₈

—

128

Q₈

—

129

Q₈

C₂×C₄₇⋊C₈

yes

complemented

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄₇⋊C₈

yes

—

136

C₂×C₄

#10625329

yes

—

137

C₂×C₄

#10625336

yes

—

138

C₂×C₄

#10625337

yes

—

139

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

140

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

141

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

142

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

145

C₈

—

146

C₈

—

188

—

147

C₂²

C₂²⋊(C₄₇⋊C₈)

yes

—

148

C₂²

#13093948

yes

—

149

C₂²

#13093951

yes

—

150

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

156

C₄

#13093948

yes

—

157

C₄

C₂²×C₄₇⋊C₈

yes

characteristic

158

C₄

C₂×#10625329

yes

characteristic

159

C₄

C₂×#10625336

yes

characteristic

160

C₄

C₂×#10625337

yes

characteristic

161

C₂

Q₈⋊(C₄₇⋊C₈)

yes

direct product factor

162

C₂

C₂×C₂²⋊(C₄₇⋊C₈)

yes

characteristic

163

C₂

C₂×#13093948

yes

characteristic

164

C₂

C₂×#13093951

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂²×C₈
Frattini	C₄²	C₂²×D₄₇
Fitting	C₂×C₄×Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²⋊C₄
hypercenter	C₂×C₄×Q₈	D₄₇
hypocenter	C₄₇	C₂×Q₈⋊C₈
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²×C₄

C₂²⋊D₄₇

commutator

C₄×C₄₇

C₂²×C₈

Frattini

C₄²

C₂²×D₄₇

Fitting

C₂×C₄×Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂²⋊C₄

hypercenter

C₂×C₄×Q₈

D₄₇

hypocenter

C₄₇

C₂×Q₈⋊C₈

LCS

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
3. Q₈⋊(C₄₇⋊C₈)	C₂	direct (N × H)
22. C₂×Q₈×C₄₇	C₈	semidirect
46. Q₈×C₄₇	C₂×C₈	semidirect
79. C₂×C₄₇	Q₈⋊C₈	semidirect
87. C₄₇	C₂×Q₈⋊C₈	semidirect
105. C₂×Q₈	C₄₇⋊C₈	semidirect
129. Q₈	C₂×C₄₇⋊C₈	semidirect
161. C₂	Q₈⋊(C₄₇⋊C₈)	direct (N × H)

H ≅ G/N

type

3. Q₈⋊(C₄₇⋊C₈)

C₂

direct (N × H)

22. C₂×Q₈×C₄₇

C₈

semidirect

46. Q₈×C₄₇

C₂×C₈

semidirect

79. C₂×C₄₇

Q₈⋊C₈

semidirect

87. C₄₇

C₂×Q₈⋊C₈

semidirect

105. C₂×Q₈

C₄₇⋊C₈

semidirect

129. Q₈

C₂×C₄₇⋊C₈

semidirect

161. C₂

Q₈⋊(C₄₇⋊C₈)

direct (N × H)

Subgroups — C₂×Q₈⋊(C₄₇⋊C₈) #20241765

Conjugacy classes of subgroups

Complements

Subgroups — C2×Q8⋊(C47⋊C8) #20241765

Conjugacy classes of subgroups

Complements

Subgroups — C₂×Q₈⋊(C₄₇⋊C₈) #20241765