Group #19132972 — subgroups

Subgroups — C₂×Q₈⋊(C₄₃⋊C₈) #19132972

Order 5504.

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⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₄₃⋊C₈)	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₃⋊C₈	—	2	1	no	—
6	C₂²×C₄₃⋊C₈	—	2	1	no	—
7	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
8	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
9	C₄×C₄₃⋊C₈	—	2	1	no	—
10	C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
11	C₂²×Q₈×C₄₃	C₄	1	1	yes	characteristic
12	C₄×Q₈×C₄₃	—	2	1	no	—
13	C₄×Q₈×C₄₃	—	2	1	no	—
14	C₄×Q₈×C₄₃	C₄	1	2	yes	—
15	C₄×Q₈×C₄₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
17	C₂×C₄⋊C₄×C₄₃	C₄	1	1	yes	characteristic
18	C₂×C₄²×C₄₃	—	2	1	no	—
19	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₄₃	—	2	1	no	—
21	C₂×Q₈×C₄₃	—	2	2	no	—
22	C₂×Q₈×C₄₃	C₈	1	4	yes	complemented
23	C₂×Q₈×C₄₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₄₃	—	2	1	no	—
25	C₂²×C₄×C₄₃	—	2	1	no	—
26	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₄⋊C₄×C₄₃	—	2	1	no	—
31	C₄⋊C₄×C₄₃	—	2	2	no	—
32	C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
33	C₄²×C₄₃	—	2	2	no	—
34	C₄²×C₄₃	—	2	2	no	—
35	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₄₃	D₄	1	1	yes	characteristic
37	C₄²×C₄₃	D₄	1	1	yes	characteristic
38	C₄²×C₄₃	C₂³	1	1	yes	characteristic
39	C₂×C₄₃⋊C₈	—	2	4	no	—
40	C₂×C₄₃⋊C₈	—	2	2	no	—
41	C₂×C₄₃⋊C₈	—	2	2	no	—
42	C₂×C₄₃⋊C₈	—	4	2	no	—
43	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
44	Q₈×C₄₃	—	2	4	no	—
45	Q₈×C₄₃	—	2	2	no	—
46	Q₈×C₄₃	C₂×C₈	1	4	yes	complemented
47	C₂×C₄×C₄₃	—	2	4	no	—
48	C₂×C₄×C₄₃	—	2	2	no	—
49	C₂×C₄×C₄₃	—	2	2	no	—
50	C₂×C₄×C₄₃	—	2	2	no	—
51	C₂×C₄×C₄₃	—	4	2	no	—
52	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₃	C₂×C₈	1	2	yes	—
56	C₂×C₄×C₄₃	M₁₆	1	2	yes	—
57	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
58	C₂×C₄×C₄₃	Q₁₆	1	2	yes	—
59	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₄₃⋊C₈	—	2	4	no	—
63	C₄₃⋊C₈	—	4	2	no	—
64	C₂²×C₄₃	C₂²⋊C₈	1	2	yes	—
65	C₂²×C₄₃	#103	1	2	yes	—
66	C₂²×C₄₃	#104	1	2	yes	—
67	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₄₃	—	2	4	no	—
69	C₄×C₄₃	—	2	1	no	—
70	C₄×C₄₃	—	2	1	no	—
71	C₄×C₄₃	—	4	2	no	—
72	C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
73	C₄×C₄₃	#103	1	2	yes	—
74	C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
75	C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
76	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×Q₈⋊C₈	—	43	1	no	maximal, Sylow
79	C₂×C₄₃	Q₈⋊C₈	1	4	yes	complemented
80	C₂×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄₃	C₂×#103	1	1	yes	characteristic
82	C₂×C₄₃	C₂×#104	1	1	yes	characteristic
83	C₂×C₄×Q₈	D₄₃	1	1	yes	characteristic
84	C₂×C₄⋊C₈	—	43	1	no	—
85	C₂×C₄×C₈	—	43	1	no	—
86	Q₈⋊C₈	—	43	4	no	—
87	C₄₃	C₂×Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
88	C₂²×Q₈	Dic₄₃	1	1	yes	characteristic
89	C₂²×C₈	—	86	1	no	—
90	C₂²×C₈	—	86	1	no	—
91	C₄×Q₈	—	2	1	no	—
92	C₄×Q₈	—	2	1	no	—
93	C₄×Q₈	Dic₄₃	1	2	yes	—
94	C₄×Q₈	C₂×D₄₃	1	2	yes	—
95	C₂×C₄⋊C₄	—	2	1	no	—
96	C₂×C₄⋊C₄	Dic₄₃	1	1	yes	characteristic
97	C₂×C₄²	—	2	1	no	—
98	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
99	C₄⋊C₈	—	43	2	no	—
100	C₄⋊C₈	—	86	1	no	—
101	C₄×C₈	—	43	2	no	—
102	C₄×C₈	—	86	1	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	C₄₃⋊C₈	1	4	yes	complemented
106	C₂×Q₈	C₂×Dic₄₃	1	2	yes	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
111	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
112	C₂×C₈	—	86	4	no	—
113	C₂×C₈	—	86	2	no	—
114	C₂×C₈	—	86	2	no	—
115	C₂×C₈	—	172	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
120	C₄²	—	2	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₂×Dic₄₃	1	1	yes	characteristic
123	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
124	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
125	C₄²	C₂²×D₄₃	1	1	yes	characteristic
126	C₂³	C₂²⋊Dic₄₃	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	Q₈	—	2	2	no	—
129	Q₈	C₂×C₄₃⋊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×C₄₃⋊C₈	1	2	yes	—
136	C₂×C₄	#10621877	1	2	yes	—
137	C₂×C₄	#10621884	1	2	yes	—
138	C₂×C₄	#10621885	1	2	yes	—
139	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
140	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
141	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
142	C₂×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
145	C₈	—	86	4	no	—
146	C₈	—	172	2	no	—
147	C₂²	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
148	C₂²	#13063249	1	2	yes	—
149	C₂²	#13063252	1	2	yes	—
150	C₂²	C₂×C₂²⋊Dic₄₃	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⋊(C₄₃⋊C₈)	1	2	yes	—
156	C₄	#13063249	1	2	yes	—
157	C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
158	C₄	C₂×#10621877	1	1	yes	characteristic
159	C₄	C₂×#10621884	1	1	yes	characteristic
160	C₄	C₂×#10621885	1	1	yes	characteristic
161	C₂	Q₈⋊(C₄₃⋊C₈)	1	4	yes	direct product factor
162	C₂	C₂×C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
163	C₂	C₂×#13063249	1	1	yes	characteristic
164	C₂	C₂×#13063252	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₄×C₄₃⋊C₈

C₂

yes

maximal, characteristic

Q₈⋊(C₄₃⋊C₈)

C₂

yes

maximal, direct product factor

C₂×C₄×Q₈×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

—

C₂²×C₄₃⋊C₈

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

C₂²

yes

—

C₂²×Q₈×C₄₃

C₄

yes

characteristic

C₄×Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

C₄×Q₈×C₄₃

C₄

yes

—

C₄×Q₈×C₄₃

C₂²

yes

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₄

yes

characteristic

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

C₈

yes

complemented

C₂×Q₈×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

C₂×C₈

yes

complemented

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

M₁₆

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

Q₁₆

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂²⋊C₈

yes

—

C₂²×C₄₃

#103

yes

—

C₂²×C₄₃

#104

yes

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊C₈

yes

—

C₄×C₄₃

#103

yes

—

C₄×C₄₃

C₂²×C₈

yes

characteristic

C₄×C₄₃

C₂×M₁₆

yes

characteristic

C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₄×C₄₃

C₂×Q₁₆

yes

characteristic

C₂×Q₈⋊C₈

—

maximal, Sylow

C₂×C₄₃

Q₈⋊C₈

yes

complemented

C₂×C₄₃

C₂×C₂²⋊C₈

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄×Q₈

D₄₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

Q₈⋊C₈

—

C₄₃

C₂×Q₈⋊C₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₄₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

Dic₄₃

yes

—

C₄×Q₈

C₂×D₄₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₄₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

C₄₃⋊C₈

yes

complemented

106

C₂×Q₈

C₂×Dic₄₃

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

110

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

111

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

172

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

C₂×Dic₄₃

yes

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂×Dic₄₃

yes

characteristic

123

C₄²

C₂²⋊D₄₃

yes

characteristic

124

C₄²

C₂²⋊D₄₃

yes

characteristic

125

C₄²

C₂²×D₄₃

yes

characteristic

126

C₂³

C₂²⋊Dic₄₃

yes

characteristic

127

Q₈

—

128

Q₈

—

129

Q₈

C₂×C₄₃⋊C₈

yes

complemented

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄₃⋊C₈

yes

—

136

C₂×C₄

#10621877

yes

—

137

C₂×C₄

#10621884

yes

—

138

C₂×C₄

#10621885

yes

—

139

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

140

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

141

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

142

C₂×C₄

C₂²×Dic₄₃

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

145

C₈

—

146

C₈

—

172

—

147

C₂²

C₂²⋊(C₄₃⋊C₈)

yes

—

148

C₂²

#13063249

yes

—

149

C₂²

#13063252

yes

—

150

C₂²

C₂×C₂²⋊Dic₄₃

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

C₂²⋊(C₄₃⋊C₈)

yes

—

156

C₄

#13063249

yes

—

157

C₄

C₂²×C₄₃⋊C₈

yes

characteristic

158

C₄

C₂×#10621877

yes

characteristic

159

C₄

C₂×#10621884

yes

characteristic

160

C₄

C₂×#10621885

yes

characteristic

161

C₂

Q₈⋊(C₄₃⋊C₈)

yes

direct product factor

162

C₂

C₂×C₂²⋊(C₄₃⋊C₈)

yes

characteristic

163

C₂

C₂×#13063249

yes

characteristic

164

C₂

C₂×#13063252

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₄₃
commutator	C₄×C₄₃	C₂²×C₈
Frattini	C₄²	C₂²×D₄₃
Fitting	C₂×C₄×Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²⋊C₄
hypercenter	C₂×C₄×Q₈	D₄₃
hypocenter	C₄₃	C₂×Q₈⋊C₈
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²×C₄

C₂²⋊D₄₃

commutator

C₄×C₄₃

C₂²×C₈

Frattini

C₄²

C₂²×D₄₃

Fitting

C₂×C₄×Q₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂²⋊C₄

hypercenter

C₂×C₄×Q₈

D₄₃

hypocenter

C₄₃

C₂×Q₈⋊C₈

LCS

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
3. Q₈⋊(C₄₃⋊C₈)	C₂	direct (N × H)
22. C₂×Q₈×C₄₃	C₈	semidirect
46. Q₈×C₄₃	C₂×C₈	semidirect
79. C₂×C₄₃	Q₈⋊C₈	semidirect
87. C₄₃	C₂×Q₈⋊C₈	semidirect
105. C₂×Q₈	C₄₃⋊C₈	semidirect
129. Q₈	C₂×C₄₃⋊C₈	semidirect
161. C₂	Q₈⋊(C₄₃⋊C₈)	direct (N × H)

H ≅ G/N

type

3. Q₈⋊(C₄₃⋊C₈)

C₂

direct (N × H)

22. C₂×Q₈×C₄₃

C₈

semidirect

46. Q₈×C₄₃

C₂×C₈

semidirect

79. C₂×C₄₃

Q₈⋊C₈

semidirect

87. C₄₃

C₂×Q₈⋊C₈

semidirect

105. C₂×Q₈

C₄₃⋊C₈

semidirect

129. Q₈

C₂×C₄₃⋊C₈

semidirect

161. C₂

Q₈⋊(C₄₃⋊C₈)

direct (N × H)

Subgroups — C₂×Q₈⋊(C₄₃⋊C₈) #19132972

Conjugacy classes of subgroups

Complements

Subgroups — C2×Q8⋊(C43⋊C8) #19132972

Conjugacy classes of subgroups

Complements

Subgroups — C₂×Q₈⋊(C₄₃⋊C₈) #19132972