Group #19132909 — subgroups

Subgroups — #19132909

Order 5504.

264 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 109 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	#14582952	C₂	1	2	yes	maximal, complemented
2	#14582625	C₂	1	2	yes	maximal, complemented
3	#14582623	C₂	1	2	yes	maximal
4	C₂×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
6	C₂²⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
7	C₄×C₄₃⋊C₈	—	2	2	no	—
8	C₂³×C₄×C₄₃	C₄	1	1	yes	characteristic
9	#117×C₄₃	—	2	2	no	—
10	#117×C₄₃	—	2	2	no	—
11	C₂×C₄⋊C₄×C₄₃	C₄	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₄₃	C₄	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₄₃	C₄	1	1	yes	characteristic
16	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₃	—	2	1	no	—
19	C₂²×C₄×C₄₃	—	2	1	no	—
20	C₂²×C₄×C₄₃	—	2	2	no	—
21	C₂²×C₄×C₄₃	C₈	1	4	yes	complemented
22	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₄₃	—	2	2	no	—
29	C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₂²⋊C₄×C₄₃	—	2	2	no	—
31	C₂²⋊C₄×C₄₃	—	2	2	no	—
32	C₄²×C₄₃	—	2	1	no	—
33	C₄²×C₄₃	—	2	1	no	—
34	C₄²×C₄₃	D₄	1	2	yes	—
35	C₄²×C₄₃	D₄	1	2	yes	—
36	C₂×C₄₃⋊C₈	—	2	4	no	—
37	C₂³×C₄₃	—	2	1	no	—
38	C₂³×C₄₃	—	2	1	no	—
39	C₂³×C₄₃	—	4	1	no	—
40	C₂³×C₄₃	—	4	1	no	—
41	C₂³×C₄₃	C₂×C₈	1	2	yes	—
42	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₃	—	2	1	no	—
44	C₂×C₄×C₄₃	—	2	1	no	—
45	C₂×C₄×C₄₃	—	2	4	no	—
46	C₂×C₄×C₄₃	—	2	4	no	—
47	C₂×C₄×C₄₃	—	2	1	no	—
48	C₂×C₄×C₄₃	—	2	1	no	—
49	C₂×C₄×C₄₃	—	4	1	no	—
50	C₂×C₄×C₄₃	—	4	1	no	—
51	C₂×C₄×C₄₃	—	4	1	no	—
52	C₂×C₄×C₄₃	—	4	1	no	—
53	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₂×C₈	1	2	yes	—
57	C₂×C₄×C₄₃	C₂×C₈	1	2	yes	—
58	C₂×C₄×C₄₃	M₁₆	1	2	yes	—
59	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
60	C₄₃⋊C₈	—	4	4	no	—
61	C₂²×C₄₃	—	2	1	no	—
62	C₂²×C₄₃	—	2	1	no	—
63	C₂²×C₄₃	—	2	2	no	—
64	C₂²×C₄₃	—	4	2	no	—
65	C₂²×C₄₃	—	4	2	no	—
66	C₂²×C₄₃	—	4	1	no	—
67	C₂²×C₄₃	—	4	1	no	—
68	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄₃	C₂²×C₈	1	1	yes	characteristic
70	C₂²×C₄₃	C₂×M₁₆	1	1	yes	characteristic
71	C₄×C₄₃	—	2	2	no	—
72	C₄×C₄₃	—	2	2	no	—
73	C₄×C₄₃	—	2	2	no	—
74	C₄×C₄₃	—	4	1	no	—
75	C₄×C₄₃	—	4	1	no	—
76	C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
77	C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
78	#1458	—	43	1	no	maximal, Sylow
79	C₂×C₄₃	—	2	1	no	—
80	C₂×C₄₃	—	2	1	no	—
81	C₂×C₄₃	—	4	2	no	—
82	C₂×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄₃	#410	1	1	yes	characteristic
84	C₂×C₄₃	#413	1	1	yes	characteristic
85	C₂×#117	D₄₃	1	1	yes	characteristic
86	#431	—	43	2	no	—
87	#324	—	43	2	no	—
88	#323	—	43	2	no	—
89	C₄₃	#1458	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₄	Dic₄₃	1	1	yes	characteristic
91	#117	—	2	2	no	—
92	#117	—	2	2	no	—
93	C₂×C₄⋊C₄	Dic₄₃	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
95	C₂×C₂²⋊C₄	Dic₄₃	1	1	yes	characteristic
96	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
97	C₂×C₄²	Dic₄₃	1	1	yes	characteristic
98	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
99	C₄⋊C₈	—	86	2	no	—
100	C₂²⋊C₈	—	43	4	no	—
101	C₄×C₈	—	86	2	no	—
102	C₂⁴	C₂×Dic₄₃	1	1	yes	characteristic
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	C₄₃⋊C₈	1	4	yes	complemented
107	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
108	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
109	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
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	—	86	4	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	C₂²⋊D₄₃	1	2	yes	—
121	C₄²	C₂²⋊D₄₃	1	2	yes	—
122	C₂³	—	2	1	no	—
123	C₂³	—	2	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	—	4	1	no	—
126	C₂³	C₂×C₄₃⋊C₈	1	2	yes	—
127	C₂³	C₂²×Dic₄₃	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
139	C₂×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
140	C₂×C₄	#10621877	1	2	yes	—
141	C₂×C₄	C₂²⋊Dic₄₃	1	2	yes	—
142	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
145	C₈	—	172	4	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
154	C₂²	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
155	C₂²	C₂×#10621877	1	1	yes	characteristic
156	C₄	—	2	2	no	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	1	no	—
160	C₄	—	4	1	no	—
161	C₄	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
162	C₄	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	4	2	no	—
166	C₂	C₂×C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
167	C₂	#14582873	1	1	yes	characteristic
168	C₂	#14582883	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14582952

C₂

yes

maximal, complemented

#14582625

C₂

yes

maximal, complemented

#14582623

C₂

yes

maximal

C₂×#117×C₄₃

C₂

yes

maximal, characteristic

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

C₂²

yes

—

C₄×C₄₃⋊C₈

—

C₂³×C₄×C₄₃

C₄

yes

characteristic

#117×C₄₃

—

#117×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₄

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₃

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₄

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂⁴×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₈

yes

complemented

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

D₄

yes

—

C₄²×C₄₃

D₄

yes

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂×C₈

yes

—

C₂³×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×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

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

M₁₆

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

C₂²×C₈

yes

characteristic

C₂²×C₄₃

C₂×M₁₆

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊C₈

yes

—

C₄×C₄₃

C₂²⋊C₈

yes

—

#1458

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

C₂×C₂²⋊C₈

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

yes

—

—

—

C₄₃

#1458

yes

characteristic, complemented, Sylow

C₂³×C₄

Dic₄₃

yes

characteristic

#117

—

#117

—

C₂×C₄⋊C₄

Dic₄₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₂×C₂²⋊C₄

Dic₄₃

yes

characteristic

C₂×C₂²⋊C₄

C₂×D₄₃

yes

characteristic

C₂×C₄²

Dic₄₃

yes

characteristic

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

100

C₂²⋊C₈

—

101

C₄×C₈

—

102

C₂⁴

C₂×Dic₄₃

yes

characteristic

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₄₃⋊C₈

yes

complemented

107

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

108

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

109

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

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

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

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

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂×C₄₃⋊C₈

yes

—

139

C₂×C₄

C₂×C₄₃⋊C₈

yes

—

140

C₂×C₄

#10621877

yes

—

141

C₂×C₄

C₂²⋊Dic₄₃

yes

—

142

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

143

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

145

C₈

—

172

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

C₂×C₂²⋊Dic₄₃

yes

characteristic

154

C₂²

C₂²×C₄₃⋊C₈

yes

characteristic

155

C₂²

C₂×#10621877

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

C₂²⋊(C₄₃⋊C₈)

yes

—

162

C₄

C₂²⋊(C₄₃⋊C₈)

yes

—

163

C₂

—

164

C₂

—

165

C₂

—

166

C₂

C₂×C₂²⋊(C₄₃⋊C₈)

yes

characteristic

167

C₂

#14582873

yes

characteristic

168

C₂

#14582883

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₄₃
commutator	C₂²×C₄₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×#117×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	C₂×#117	D₄₃
hypocenter	C₄₃	#1458
LCS	G ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#117

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₄₃

commutator

C₂²×C₄₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×#117×C₄₃

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#117

N	H ≅ G/N	type
1. #14582952	C₂	semidirect
2. #14582625	C₂	semidirect
21. C₂²×C₄×C₄₃	C₈	semidirect
89. C₄₃	#1458	semidirect
106. C₂²×C₄	C₄₃⋊C₈	semidirect

H ≅ G/N

type

1. #14582952

C₂

semidirect

2. #14582625

C₂

semidirect

21. C₂²×C₄×C₄₃

C₈

semidirect

89. C₄₃

#1458

semidirect

106. C₂²×C₄

C₄₃⋊C₈

semidirect

Subgroups — #19132909

Conjugacy classes of subgroups

Complements