Group #19135390 — subgroups

Subgroups — #19135390

Order 5504.

396 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 145 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, complemented
2	C₂×C₂²⋊(C₄₃⋊C₈)	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
7	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
8	C₂²⋊(C₄₃⋊C₈)	—	2	8	no	—
9	C₄×C₄₃⋊C₈	—	2	2	no	—
10	C₂²×D₄×C₄₃	C₄	1	1	yes	characteristic
11	C₂³×C₄×C₄₃	C₂²	1	2	yes	—
12	C₄×D₄×C₄₃	—	2	2	no	—
13	C₄×D₄×C₄₃	—	2	2	no	—
14	C₂×C₄⋊C₄×C₄₃	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₄₃	C₄	1	2	yes	—
16	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄₃	C₂×C₄	1	2	yes	—
18	C₂×D₄×C₄₃	—	2	2	no	—
19	C₂×D₄×C₄₃	—	2	2	no	—
20	C₂×D₄×C₄₃	C₈	1	4	yes	complemented
21	C₂²×C₄×C₄₃	—	2	2	no	—
22	C₂²×C₄×C₄₃	—	2	2	no	—
23	C₂²×C₄×C₄₃	—	2	4	no	—
24	C₂²×C₄×C₄₃	—	2	4	no	—
25	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₂²⋊C₄×C₄₃	—	2	4	no	—
31	C₄²×C₄₃	D₄	1	2	yes	—
32	C₄²×C₄₃	D₄	1	2	yes	—
33	C₂×C₄₃⋊C₈	—	2	2	no	—
34	C₂×C₄₃⋊C₈	—	2	2	no	—
35	C₂×C₄₃⋊C₈	—	2	4	no	—
36	C₂×C₄₃⋊C₈	—	2	2	no	—
37	C₂×C₄₃⋊C₈	D₄	1	4	yes	complemented
38	C₂³×C₄₃	—	2	2	no	—
39	C₂³×C₄₃	—	2	4	no	—
40	C₂³×C₄₃	—	2	2	no	—
41	C₂³×C₄₃	—	2	4	no	—
42	C₂³×C₄₃	C₂×C₈	1	4	yes	—
43	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
44	D₄×C₄₃	—	2	8	no	—
45	C₂×C₄×C₄₃	—	2	2	no	—
46	C₂×C₄×C₄₃	—	2	2	no	—
47	C₂×C₄×C₄₃	—	2	4	no	—
48	C₂×C₄×C₄₃	—	4	4	no	—
49	C₂×C₄×C₄₃	—	4	2	no	—
50	C₂×C₄×C₄₃	—	4	2	no	—
51	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₄₃	C₂×C₈	1	2	yes	—
54	C₂×C₄×C₄₃	M₁₆	1	2	yes	—
55	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₃	#41	1	2	yes	—
59	C₄₃⋊C₈	—	2	4	no	—
60	C₄₃⋊C₈	—	4	2	no	—
61	C₂²×C₄₃	—	2	4	no	—
62	C₂²×C₄₃	—	2	8	no	—
63	C₂²×C₄₃	—	4	2	no	—
64	C₂²×C₄₃	—	4	2	no	—
65	C₂²×C₄₃	—	4	4	no	—
66	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄₃	C₄×D₄	1	2	yes	—
68	C₂²×C₄₃	C₂²×C₈	1	1	yes	characteristic
69	C₂²×C₄₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂²×C₄₃	#131	1	2	yes	—
71	C₄×C₄₃	—	2	2	no	—
72	C₄×C₄₃	—	4	2	no	—
73	C₄×C₄₃	C₂²⋊C₈	1	4	yes	complemented
74	C₄×C₄₃	#121	1	2	yes	—
75	C₄×C₄₃	#124	1	1	yes	characteristic
76	C₄×C₄₃	#127	1	1	yes	characteristic
77	#1954	—	43	1	no	maximal, Sylow
78	C₂×C₄₃	—	4	4	no	—
79	C₂×C₄₃	#390	1	1	yes	characteristic
80	C₂×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄₃	#408	1	1	yes	characteristic
82	C₂×C₄₃	C₈×D₄	1	2	yes	—
83	C₂×C₄₃	#436	1	2	yes	—
84	C₂×C₄×D₄	D₄₃	1	1	yes	characteristic
85	C₂×C₄⋊C₈	—	43	1	no	—
86	C₂×C₂²⋊C₈	—	43	4	no	—
87	C₂×C₄×C₈	—	43	1	no	—
88	C₄₃	#1954	1	1	yes	characteristic, complemented, Sylow
89	C₂²×D₄	Dic₄₃	1	1	yes	characteristic
90	C₂³×C₄	C₂×D₄₃	1	2	yes	—
91	C₂²×C₈	—	43	2	no	—
92	C₂²×C₈	—	43	2	no	—
93	C₄×D₄	—	2	2	no	—
94	C₄×D₄	—	2	2	no	—
95	C₂×C₄⋊C₄	Dic₄₃	1	1	yes	characteristic
96	C₂×C₂²⋊C₄	Dic₄₃	1	2	yes	—
97	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
98	C₄⋊C₈	—	86	2	no	—
99	C₂²⋊C₈	—	86	8	no	—
100	C₄×C₈	—	86	2	no	—
101	C₂⁴	C₂×Dic₄₃	1	2	yes	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	C₄₃⋊C₈	1	4	yes	complemented
105	C₂²×C₄	—	2	4	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	C₂×Dic₄₃	1	2	yes	—
110	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
111	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
113	C₂×C₈	—	43	4	no	—
114	C₂×C₈	—	86	2	no	—
115	C₂×C₈	—	86	2	no	—
116	C₂×C₈	—	86	4	no	—
117	C₂×C₈	—	86	2	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₄²	C₂²⋊D₄₃	1	2	yes	—
121	C₄²	C₂²⋊D₄₃	1	2	yes	—
122	C₂³	—	2	2	no	—
123	C₂³	—	2	4	no	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	4	no	—
126	C₂³	C₂×C₄₃⋊C₈	1	4	yes	—
127	C₂³	C₂²×Dic₄₃	1	1	yes	characteristic
128	D₄	—	2	8	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	4	4	no	—
133	C₂×C₄	—	4	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	C₂²⋊Dic₄₃	1	2	yes	—
138	C₂×C₄	C₂²⋊Dic₄₃	1	2	yes	—
139	C₂×C₄	D₄×D₄₃	1	2	yes	—
140	C₂×C₄	#10621900	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
142	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
143	C₈	—	86	4	no	—
144	C₈	—	172	2	no	—
145	C₂²	—	2	4	no	—
146	C₂²	—	2	8	no	—
147	C₂²	—	4	2	no	—
148	C₂²	—	4	2	no	—
149	C₂²	—	4	4	no	—
150	C₂²	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
151	C₂²	D₄×Dic₄₃	1	2	yes	—
152	C₂²	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
153	C₂²	C₂×#10621877	1	1	yes	characteristic
154	C₂²	#13063332	1	2	yes	—
155	C₄	—	2	2	no	—
156	C₄	—	4	2	no	—
157	C₄	C₂²⋊(C₄₃⋊C₈)	1	4	yes	complemented
158	C₄	#13063296	1	2	yes	—
159	C₄	#13063312	1	1	yes	characteristic
160	C₄	#13063321	1	1	yes	characteristic
161	C₂	—	4	4	no	—
162	C₂	#14582802	1	1	yes	characteristic
163	C₂	C₂×C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
164	C₂	#14582866	1	1	yes	characteristic
165	C₂	D₄×C₄₃⋊C₈	1	2	yes	—
166	C₂	#14582977	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊(C₄₃⋊C₈)

C₂

yes

maximal, complemented

C₂×C₄×C₄₃⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₄×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

—

C₂²×C₄₃⋊C₈

C₂²

yes

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₂²×D₄×C₄₃

C₄

yes

characteristic

C₂³×C₄×C₄₃

C₂²

yes

—

C₄×D₄×C₄₃

—

C₄×D₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₄₃

C₄

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂⁴×C₄₃

C₂×C₄

yes

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

C₈

yes

complemented

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

D₄

yes

—

C₄²×C₄₃

D₄

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₄

yes

complemented

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂×C₈

yes

—

C₂³×C₄₃

C₂²×C₄

yes

characteristic

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

M₁₆

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

#41

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

C₄×D₄

yes

—

C₂²×C₄₃

C₂²×C₈

yes

characteristic

C₂²×C₄₃

C₂×M₁₆

yes

characteristic

C₂²×C₄₃

#131

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊C₈

yes

complemented

C₄×C₄₃

#121

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₂×C₄₃

—

C₂×C₄₃

#390

yes

characteristic

C₂×C₄₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₄₃

#408

yes

characteristic

C₂×C₄₃

C₈×D₄

yes

—

C₂×C₄₃

#436

yes

—

C₂×C₄×D₄

D₄₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

—

C₄₃

#1954

yes

characteristic, complemented, Sylow

C₂²×D₄

Dic₄₃

yes

characteristic

C₂³×C₄

C₂×D₄₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

Dic₄₃

yes

characteristic

C₂×C₂²⋊C₄

Dic₄₃

yes

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

100

C₄×C₈

—

101

C₂⁴

C₂×Dic₄₃

yes

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

C₄₃⋊C₈

yes

complemented

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×Dic₄₃

yes

—

110

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

111

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

112

C₂²×C₄

C₂²×D₄₃

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

C₂²⋊D₄₃

yes

—

121

C₄²

C₂²⋊D₄₃

yes

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

C₂×C₄₃⋊C₈

yes

—

127

C₂³

C₂²×Dic₄₃

yes

characteristic

128

D₄

—

129

C₂×C₄

—

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

C₂²⋊Dic₄₃

yes

—

138

C₂×C₄

C₂²⋊Dic₄₃

yes

—

139

C₂×C₄

D₄×D₄₃

yes

—

140

C₂×C₄

#10621900

yes

—

141

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

142

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

143

C₈

—

144

C₈

—

172

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂×C₂²⋊Dic₄₃

yes

characteristic

151

C₂²

D₄×Dic₄₃

yes

—

152

C₂²

C₂²×C₄₃⋊C₈

yes

characteristic

153

C₂²

C₂×#10621877

yes

characteristic

154

C₂²

#13063332

yes

—

155

C₄

—

156

C₄

—

157

C₄

C₂²⋊(C₄₃⋊C₈)

yes

complemented

158

C₄

#13063296

yes

—

159

C₄

#13063312

yes

characteristic

160

C₄

#13063321

yes

characteristic

161

C₂

—

162

C₂

#14582802

yes

characteristic

163

C₂

C₂×C₂²⋊(C₄₃⋊C₈)

yes

characteristic

164

C₂

#14582866

yes

characteristic

165

C₂

D₄×C₄₃⋊C₈

yes

—

166

C₂

#14582977

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂²×C₈
Fitting	C₂×C₄×D₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×C₄
hypercenter	C₂×C₄×D₄	D₄₃
hypocenter	C₄₃	#1954
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₃

commutator

C₂²×C₄₃

C₂²×C₈

Fitting

C₂×C₄×D₄×C₄₃

C₂

socle

C₂³×C₄₃

C₂²×C₄

hypercenter

C₂×C₄×D₄

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. C₂×C₄⋊(C₄₃⋊C₈)	C₂	semidirect
2. C₂×C₂²⋊(C₄₃⋊C₈)	C₂	semidirect
3. C₂×C₄×C₄₃⋊C₈	C₂	semidirect
20. C₂×D₄×C₄₃	C₈	semidirect
37. C₂×C₄₃⋊C₈	D₄	semidirect
73. C₄×C₄₃	C₂²⋊C₈	semidirect
88. C₄₃	#1954	semidirect
104. C₂×D₄	C₄₃⋊C₈	semidirect
157. C₄	C₂²⋊(C₄₃⋊C₈)	semidirect

H ≅ G/N

type

1. C₂×C₄⋊(C₄₃⋊C₈)

C₂

semidirect

2. C₂×C₂²⋊(C₄₃⋊C₈)

C₂

semidirect

3. C₂×C₄×C₄₃⋊C₈

C₂

semidirect

20. C₂×D₄×C₄₃

C₈

semidirect

37. C₂×C₄₃⋊C₈

D₄

semidirect

73. C₄×C₄₃

C₂²⋊C₈

semidirect

88. C₄₃

#1954

semidirect

104. C₂×D₄

C₄₃⋊C₈

semidirect

157. C₄

C₂²⋊(C₄₃⋊C₈)

semidirect

Subgroups — #19135390

Conjugacy classes of subgroups

Complements