Group #25027771 — subgroups

Subgroups — #25027771

Order 11008.

2286 conjugacy classes of subgroups; only the 555 normal subgroups are shown.

normal subgroups only

This group has 2286 conjugacy classes of subgroups — too many to draw. Only its 555 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×#14582767	C₂	1	2	yes	maximal, complemented
2	#19132822	C₂	1	6	yes	maximal, complemented
3	#19132818	C₂	1	6	yes	maximal
4	C₂³×C₄²×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
6	#14582767	—	2	8	no	—
7	C₂×C₄₃⋊#95	C₂²	1	12	yes	—
8	C₂×C₄₃⋊#95	C₂²	1	12	yes	—
9	C₂⁴×C₄×C₄₃	C₂²	1	3	yes	—
10	C₂²×C₄²×C₄₃	—	2	8	no	—
11	C₂²×C₄²×C₄₃	C₄	1	8	yes	complemented
12	C₂²×C₄²×C₄₃	C₂²	1	3	yes	—
13	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
14	C₂³×Dic₄₃	C₂³	1	8	yes	—
15	C₂×C₂²⋊Dic₄₃	—	2	24	no	—
16	C₄₃⋊#95	—	2	24	no	—
17	C₄₃⋊#95	—	2	24	no	—
18	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
19	C₂³×C₄×C₄₃	—	2	12	no	—
20	C₂³×C₄×C₄₃	—	2	12	no	—
21	C₂³×C₄×C₄₃	—	2	12	no	—
22	C₂³×C₄×C₄₃	C₂×C₄	1	12	yes	—
23	C₂³×C₄×C₄₃	C₂³	1	3	yes	—
24	C₂³×C₄×C₄₃	C₂³	1	3	yes	—
25	C₂×C₄²×C₄₃	—	2	12	no	—
26	C₂×C₄²×C₄₃	—	2	32	no	—
27	C₂×C₄²×C₄₃	C₂×C₄	1	12	yes	—
28	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₄₃	D₄	1	8	yes	—
30	C₂⁴×C₄₃	—	2	2	no	—
31	C₂⁴×C₄₃	—	2	6	no	—
32	C₂⁴×C₄₃	C₂²×C₄	1	2	yes	—
33	C₂⁴×C₄₃	#41	1	12	yes	—
34	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	—	2	96	no	—
36	C₂²×C₄×C₄₃	—	2	24	no	—
37	C₂²×C₄×C₄₃	—	2	24	no	—
38	C₂²×C₄×C₄₃	—	2	24	no	—
39	C₂²×C₄×C₄₃	—	2	6	no	—
40	C₂²×C₄×C₄₃	—	2	6	no	—
41	C₂²×C₄×C₄₃	—	2	6	no	—
42	C₂²×C₄×C₄₃	—	2	6	no	—
43	C₂²×C₄×C₄₃	C₂²×C₄	1	6	yes	—
44	C₂²×C₄×C₄₃	C₂²×C₄	1	6	yes	—
45	C₂²×C₄×C₄₃	C₂×D₄	1	12	yes	—
46	C₂²×C₄×C₄₃	#41	1	12	yes	—
47	C₂²×Dic₄₃	—	2	8	no	—
48	C₂²×Dic₄₃	—	2	24	no	—
49	C₂²×Dic₄₃	—	2	24	no	—
50	C₄²×C₄₃	—	2	24	no	—
51	C₄²×C₄₃	C₂²⋊C₄	1	16	yes	complemented
52	C₂²⋊Dic₄₃	—	4	16	no	—
53	C₂³×C₄₃	—	2	2	no	—
54	C₂³×C₄₃	—	2	6	no	—
55	C₂³×C₄₃	—	2	24	no	—
56	C₂³×C₄₃	—	2	24	no	—
57	C₂³×C₄₃	#117	1	12	yes	—
58	C₂³×C₄₃	#126	1	16	yes	—
59	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
60	C₂³×C₄₃	C₂²×D₄	1	2	yes	—
61	C₂³×C₄₃	C₂×#41	1	6	yes	—
62	C₂³×C₄₃	C₂×#41	1	6	yes	—
63	C₂×C₄×C₄₃	—	2	96	no	—
64	C₂×C₄×C₄₃	—	2	6	no	—
65	C₂×C₄×C₄₃	—	2	6	no	—
66	C₂×C₄×C₄₃	—	2	6	no	—
67	C₂×C₄×C₄₃	—	2	6	no	—
68	C₂×C₄×C₄₃	—	2	24	no	—
69	C₂×C₄×C₄₃	—	2	24	no	—
70	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	12	yes	—
71	C₂×C₄×C₄₃	#117	1	12	yes	—
72	C₂×Dic₄₃	—	4	24	no	—
73	C₂×Dic₄₃	—	4	8	no	—
74	C₂×Dic₄₃	—	4	24	no	—
75	#14683	—	43	1	no	maximal, Sylow
76	C₂²×C₄₃	—	2	32	no	—
77	C₂²×C₄₃	—	2	12	no	—
78	C₂²×C₄₃	—	2	12	no	—
79	C₂²×C₄₃	#383	1	8	yes	complemented
80	C₂²×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄₃	C₂×#117	1	3	yes	—
82	C₂²×C₄₃	C₂×#117	1	3	yes	—
83	C₂²×C₄₃	#525	1	12	yes	—
84	C₂²×C₄₃	C₂×#126	1	4	yes	—
85	C₂²×C₄₃	#529	1	12	yes	—
86	C₄×C₄₃	—	2	12	no	—
87	C₄×C₄₃	—	2	12	no	—
88	Dic₄₃	—	8	8	no	—
89	C₂³×C₄²	D₄₃	1	1	yes	characteristic
90	C₂×#383	—	43	2	no	—
91	#1434	—	43	6	no	—
92	C₂²⋊#95	—	43	6	no	—
93	C₂×C₄₃	—	2	8	no	—
94	C₂×C₄₃	#2280	1	3	yes	—
95	C₂×C₄₃	C₂×#383	1	1	yes	characteristic
96	C₂×C₄₃	#2283	1	3	yes	—
97	C₂×C₄₃	#2852	1	8	yes	—
98	C₂⁴×C₄	C₂×D₄₃	1	3	yes	—
99	C₂²×C₂²⋊C₄	—	43	4	no	—
100	C₂²×C₄²	—	2	8	no	—
101	C₂²×C₄²	Dic₄₃	1	8	yes	complemented
102	C₂²×C₄²	C₂×D₄₃	1	3	yes	—
103	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
104	#383	—	86	8	no	—
105	C₂×#95	—	43	12	no	—
106	C₂×#95	—	43	12	no	—
107	C₄₃	#14683	1	1	yes	characteristic, complemented, Sylow
108	C₂⁵	C₂²×D₄₃	1	1	yes	characteristic
109	C₂³×C₄	—	2	12	no	—
110	C₂³×C₄	—	2	12	no	—
111	C₂³×C₄	—	2	12	no	—
112	C₂³×C₄	—	43	8	no	—
113	C₂³×C₄	C₂×Dic₄₃	1	12	yes	—
114	C₂³×C₄	C₂²×D₄₃	1	3	yes	—
115	C₂³×C₄	C₂²×D₄₃	1	3	yes	—
116	C₂×C₂²⋊C₄	—	86	24	no	—
117	C₂×C₄²	—	2	12	no	—
118	C₂×C₄²	—	2	32	no	—
119	C₂×C₄²	C₂×Dic₄₃	1	12	yes	—
120	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
121	C₂×C₄²	C₂²⋊D₄₃	1	8	yes	—
122	#95	—	86	24	no	—
123	#95	—	86	24	no	—
124	C₂⁴	—	2	2	no	—
125	C₂⁴	—	2	6	no	—
126	C₂⁴	#10621898	1	12	yes	—
127	C₂⁴	C₂²×Dic₄₃	1	2	yes	—
128	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
129	C₂²×C₄	—	2	96	no	—
130	C₂²×C₄	—	2	24	no	—
131	C₂²×C₄	—	2	6	no	—
132	C₂²×C₄	—	2	6	no	—
133	C₂²×C₄	—	2	6	no	—
134	C₂²×C₄	—	2	6	no	—
135	C₂²×C₄	—	2	24	no	—
136	C₂²×C₄	—	2	24	no	—
137	C₂²×C₄	—	86	8	no	—
138	C₂²×C₄	—	86	24	no	—
139	C₂²×C₄	—	86	24	no	—
140	C₂²×C₄	#10621898	1	12	yes	—
141	C₂²×C₄	C₂²×Dic₄₃	1	6	yes	—
142	C₂²×C₄	C₂²×Dic₄₃	1	6	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₄₃	1	12	yes	—
144	C₂²⋊C₄	—	172	16	no	—
145	C₄²	—	2	24	no	—
146	C₄²	C₂²⋊Dic₄₃	1	16	yes	complemented
147	C₂³	—	2	2	no	—
148	C₂³	—	2	6	no	—
149	C₂³	—	2	24	no	—
150	C₂³	—	2	24	no	—
151	C₂³	#13063284	1	12	yes	—
152	C₂³	#13063319	1	16	yes	—
153	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
154	C₂³	C₂²×C₂²⋊D₄₃	1	2	yes	—
155	C₂³	C₂×#10621898	1	6	yes	—
156	C₂³	C₂×#10621898	1	6	yes	—
157	C₂×C₄	—	2	96	no	—
158	C₂×C₄	—	2	6	no	—
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	2	6	no	—
161	C₂×C₄	—	2	6	no	—
162	C₂×C₄	—	2	24	no	—
163	C₂×C₄	—	2	24	no	—
164	C₂×C₄	—	172	24	no	—
165	C₂×C₄	—	172	8	no	—
166	C₂×C₄	—	172	24	no	—
167	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	12	yes	—
168	C₂×C₄	#13063284	1	12	yes	—
169	C₂²	—	2	12	no	—
170	C₂²	—	2	12	no	—
171	C₂²	—	2	32	no	—
172	C₂²	#14582767	1	8	yes	complemented
173	C₂²	C₂²×C₂²⋊Dic₄₃	1	1	yes	characteristic
174	C₂²	C₂×#13063284	1	3	yes	—
175	C₂²	C₂×#13063284	1	3	yes	—
176	C₂²	#14583410	1	12	yes	—
177	C₂²	C₂×#13063319	1	4	yes	—
178	C₂²	#14583433	1	12	yes	—
179	C₄	—	2	12	no	—
180	C₄	—	2	12	no	—
181	C₄	—	344	8	no	—
182	C₂	—	2	8	no	—
183	C₂	#19136978	1	3	yes	—
184	C₂	C₂×#14582767	1	1	yes	characteristic
185	C₂	#19136993	1	3	yes	—
186	C₂	#19142910	1	8	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582767

C₂

yes

maximal, complemented

#19132822

C₂

yes

maximal, complemented

#19132818

C₂

yes

maximal

C₂³×C₄²×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₂²⋊Dic₄₃

C₂²

yes

—

#14582767

—

C₂×C₄₃⋊#95

C₂²

yes

—

C₂×C₄₃⋊#95

C₂²

yes

—

C₂⁴×C₄×C₄₃

C₂²

yes

—

C₂²×C₄²×C₄₃

—

C₂²×C₄²×C₄₃

C₄

yes

complemented

C₂²×C₄²×C₄₃

C₂²

yes

—

C₂²×C₄²×C₄₃

C₂²

yes

characteristic

C₂³×Dic₄₃

C₂³

yes

—

C₂×C₂²⋊Dic₄₃

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₂⁵×C₄₃

C₂³

yes

characteristic

C₂³×C₄×C₄₃

—

C₂³×C₄×C₄₃

—

C₂³×C₄×C₄₃

—

C₂³×C₄×C₄₃

C₂×C₄

yes

—

C₂³×C₄×C₄₃

C₂³

yes

—

C₂³×C₄×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂⁴×C₄₃

—

C₂⁴×C₄₃

—

C₂⁴×C₄₃

C₂²×C₄

yes

—

C₂⁴×C₄₃

#41

yes

—

C₂⁴×C₄₃

C₂⁴

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

#41

yes

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂²⋊C₄

yes

complemented

C₂²⋊Dic₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

#117

yes

—

C₂³×C₄₃

#126

yes

—

C₂³×C₄₃

C₂³×C₄

yes

characteristic

C₂³×C₄₃

C₂²×D₄

yes

—

C₂³×C₄₃

C₂×#41

yes

—

C₂³×C₄₃

C₂×#41

yes

—

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×C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

#117

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

#14683

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#383

yes

complemented

C₂²×C₄₃

C₂²×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

C₂×#117

yes

—

C₂²×C₄₃

C₂×#117

yes

—

C₂²×C₄₃

#525

yes

—

C₂²×C₄₃

C₂×#126

yes

—

C₂²×C₄₃

#529

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

Dic₄₃

—

C₂³×C₄²

D₄₃

yes

characteristic

C₂×#383

—

#1434

—

C₂²⋊#95

—

C₂×C₄₃

—

C₂×C₄₃

#2280

yes

—

C₂×C₄₃

C₂×#383

yes

characteristic

C₂×C₄₃

#2283

yes

—

C₂×C₄₃

#2852

yes

—

C₂⁴×C₄

C₂×D₄₃

yes

—

C₂²×C₂²⋊C₄

—

100

C₂²×C₄²

—

101

C₂²×C₄²

Dic₄₃

yes

complemented

102

C₂²×C₄²

C₂×D₄₃

yes

—

103

C₂²×C₄²

C₂×D₄₃

yes

characteristic

104

#383

—

105

C₂×#95

—

106

C₂×#95

—

107

C₄₃

#14683

yes

characteristic, complemented, Sylow

108

C₂⁵

C₂²×D₄₃

yes

characteristic

109

C₂³×C₄

—

110

C₂³×C₄

—

111

C₂³×C₄

—

112

C₂³×C₄

—

113

C₂³×C₄

C₂×Dic₄₃

yes

—

114

C₂³×C₄

C₂²×D₄₃

yes

—

115

C₂³×C₄

C₂²×D₄₃

yes

—

116

C₂×C₂²⋊C₄

—

117

C₂×C₄²

—

118

C₂×C₄²

—

119

C₂×C₄²

C₂×Dic₄₃

yes

—

120

C₂×C₄²

C₂×Dic₄₃

yes

—

121

C₂×C₄²

C₂²⋊D₄₃

yes

—

122

#95

—

123

#95

—

124

C₂⁴

—

125

C₂⁴

—

126

C₂⁴

#10621898

yes

—

127

C₂⁴

C₂²×Dic₄₃

yes

—

128

C₂⁴

C₂³×D₄₃

yes

characteristic

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

#10621898

yes

—

141

C₂²×C₄

C₂²×Dic₄₃

yes

—

142

C₂²×C₄

C₂²×Dic₄₃

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

—

144

C₂²⋊C₄

—

172

—

145

C₄²

—

146

C₄²

C₂²⋊Dic₄₃

yes

complemented

147

C₂³

—

148

C₂³

—

149

C₂³

—

150

C₂³

—

151

C₂³

#13063284

yes

—

152

C₂³

#13063319

yes

—

153

C₂³

C₂³×Dic₄₃

yes

characteristic

154

C₂³

C₂²×C₂²⋊D₄₃

yes

—

155

C₂³

C₂×#10621898

yes

—

156

C₂³

C₂×#10621898

yes

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

172

—

165

C₂×C₄

—

172

—

166

C₂×C₄

—

172

—

167

C₂×C₄

C₂×C₂²⋊Dic₄₃

yes

—

168

C₂×C₄

#13063284

yes

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

#14582767

yes

complemented

173

C₂²

C₂²×C₂²⋊Dic₄₃

yes

characteristic

174

C₂²

C₂×#13063284

yes

—

175

C₂²

C₂×#13063284

yes

—

176

C₂²

#14583410

yes

—

177

C₂²

C₂×#13063319

yes

—

178

C₂²

#14583433

yes

—

179

C₄

—

180

C₄

—

181

C₄

—

344

—

182

C₂

—

183

C₂

#19136978

yes

—

184

C₂

C₂×#14582767

yes

characteristic

185

C₂

#19136993

yes

—

186

C₂

#19142910

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₃
commutator	C₂³×C₄₃	C₂³×C₄
Fitting	C₂³×C₄²×C₄₃	C₂
socle, cosocle	C₂⁴×C₄₃	C₂⁴
prime radical	C₂⁵×C₄₃	C₂³
hypercenter	C₂³×C₄²	D₄₃
hypocenter	C₄₃	#14683
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄²

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₄₃

commutator

C₂³×C₄₃

C₂³×C₄

Fitting

C₂³×C₄²×C₄₃

C₂

socle, cosocle

C₂⁴×C₄₃

C₂⁴

prime radical

C₂⁵×C₄₃

C₂³

hypercenter

C₂³×C₄²

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂⁴ ◁ C₂³×C₄²

N	H ≅ G/N	type
1. C₂×#14582767	C₂	semidirect
2. #19132822	C₂	semidirect
11. C₂²×C₄²×C₄₃	C₄	semidirect
51. C₄²×C₄₃	C₂²⋊C₄	semidirect
79. C₂²×C₄₃	#383	semidirect
101. C₂²×C₄²	Dic₄₃	semidirect
107. C₄₃	#14683	semidirect
146. C₄²	C₂²⋊Dic₄₃	semidirect
172. C₂²	#14582767	semidirect

H ≅ G/N

type

1. C₂×#14582767

C₂

semidirect

2. #19132822

C₂

semidirect

11. C₂²×C₄²×C₄₃

C₄

semidirect

51. C₄²×C₄₃

C₂²⋊C₄

semidirect

79. C₂²×C₄₃

#383

semidirect

101. C₂²×C₄²

Dic₄₃

semidirect

107. C₄₃

#14683

semidirect

146. C₄²

C₂²⋊Dic₄₃

semidirect

172. C₂²

#14582767

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#14582767	2
#19132822	2
#19132818	2
C₂³×C₄²×C₄₃	2
#14683	43

subgroup

index

C₂³×C₄²×C₄₃

Subgroups — #25027771

Conjugacy classes of subgroups

Complements

Maximal subgroups