Group #25015120 — subgroups

Subgroups — #25015120

Order 11008.

386 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 159 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×C₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#19132672	C₂	1	4	yes	maximal
4	C₂×C₄₃×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₈₆	C₄	1	1	yes	characteristic
6	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
7	C₈×Dic₈₆	—	2	2	no	—
8	C₈×Dic₈₆	—	2	2	no	—
9	C₂×C₄⋊(C₄₃⋊C₈)	C₄	1	1	yes	characteristic
10	C₂×Dic₄₃⋊C₈	C₄	1	2	yes	—
11	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
12	C₄⋊(C₄₃⋊C₁₆)	—	2	2	no	—
13	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#346	—	2	2	no	—
16	C₂²×Dic₈₆	C₈	1	1	yes	characteristic
17	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
18	C₄×Dic₈₆	—	2	2	no	—
19	C₄×Dic₈₆	—	2	2	no	—
20	C₂×C₄⋊Dic₄₃	C₈	1	1	yes	characteristic
21	C₂×Dic₄₃⋊C₄	C₈	1	2	yes	—
22	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
23	C₂×C₄₃⋊C₁₆	—	2	2	no	—
24	C₂×C₄₃⋊C₁₆	—	2	4	no	—
25	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
26	Dic₄₃⋊C₈	—	2	4	no	—
27	C₈×Dic₄₃	—	2	2	no	—
28	C₈×Dic₄₃	—	2	2	no	—
29	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₄₃	—	2	2	no	—
33	C₂×C₁₆×C₄₃	—	2	2	no	—
34	C₂×C₁₆×C₄₃	Q₈	1	4	yes	complemented
35	C₄×C₈×C₄₃	D₄	1	2	yes	—
36	C₄×C₈×C₄₃	D₄	1	2	yes	—
37	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₄₃	C₂×C₈	1	2	yes	—
41	C₂²×Dic₄₃	C₂×C₈	1	2	yes	—
42	C₂×Dic₈₆	—	2	2	no	—
43	C₂×Dic₈₆	—	2	4	no	—
44	C₂×C₈×C₄₃	—	2	2	no	—
45	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₄₃	#41	1	2	yes	—
50	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
51	C₄²×C₄₃	M₁₆	1	2	yes	—
52	C₄⋊Dic₄₃	—	2	2	no	—
53	Dic₄₃⋊C₄	—	2	4	no	—
54	C₄×Dic₄₃	—	2	2	no	—
55	C₄×Dic₄₃	—	2	2	no	—
56	C₂×C₄₃⋊C₈	—	2	2	no	—
57	C₂×C₄₃⋊C₈	—	2	4	no	—
58	C₁₆×C₄₃	—	2	4	no	—
59	C₄₃⋊C₁₆	—	4	2	no	—
60	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₄₃	—	2	2	no	—
62	C₂×C₄×C₄₃	—	2	2	no	—
63	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
65	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₄₃	#131	1	2	yes	—
69	C₂×Dic₄₃	—	2	2	no	—
70	C₂×Dic₄₃	—	2	2	no	—
71	C₂×Dic₄₃	—	2	2	no	—
72	C₂×Dic₄₃	—	2	4	no	—
73	C₂×Dic₄₃	M₃₂	1	4	yes	—
74	Dic₈₆	—	2	8	no	—
75	C₈×C₄₃	—	2	2	no	—
76	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₄₃	#124	1	1	yes	characteristic
78	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₄₃⋊C₈	—	4	2	no	—
80	#13264	—	43	1	no	maximal, Sylow
81	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
82	C₂²×C₄₃	C₈×Q₈	1	2	yes	—
83	C₂²×C₄₃	C₂×M₃₂	1	2	yes	—
84	C₂²×C₄₃	#504	1	2	yes	—
85	C₄×C₄₃	—	2	2	no	—
86	C₄×C₄₃	—	2	2	no	—
87	C₄×C₄₃	#391	1	1	yes	characteristic
88	C₄×C₄₃	#408	1	1	yes	characteristic
89	C₄×C₄₃	#446	1	2	yes	—
90	Dic₄₃	—	2	4	no	—
91	Dic₄₃	—	4	2	no	—
92	C₂×C₈×Q₈	—	43	1	no	—
93	C₂×C₄⋊C₁₆	—	43	1	no	—
94	C₂×#346	D₄₃	1	1	yes	characteristic
95	#1374	—	43	4	no	—
96	C₂×C₄₃	#1984	1	1	yes	characteristic
97	C₂×C₄₃	#2112	1	1	yes	characteristic
98	C₂×C₄₃	#2113	1	1	yes	characteristic
99	C₂×C₄₃	#2183	1	4	yes	—
100	C₂×C₄×Q₈	—	43	1	no	—
101	C₂²×C₁₆	—	43	2	no	—
102	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
103	C₈×Q₈	—	86	2	no	—
104	C₈×Q₈	—	86	2	no	—
105	C₂×C₄⋊C₈	—	43	2	no	—
106	C₂×C₄⋊C₈	—	43	1	no	—
107	C₂×C₄×C₈	—	43	2	no	—
108	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
109	C₄⋊C₁₆	—	86	2	no	—
110	#346	—	2	2	no	—
111	C₄₃	#13264	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	43	1	no	—
113	C₂²×C₈	—	43	2	no	—
114	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
116	C₄×Q₈	—	86	2	no	—
117	C₄×Q₈	—	86	2	no	—
118	C₂×C₄⋊C₄	—	43	2	no	—
119	C₂×C₄⋊C₄	—	43	1	no	—
120	C₂×C₄²	—	43	2	no	—
121	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	86	2	no	—
125	C₂×C₁₆	—	86	4	no	—
126	C₂×C₁₆	Dic₈₆	1	4	yes	complemented
127	C₄⋊C₈	—	86	4	no	—
128	C₄⋊C₈	—	86	2	no	—
129	C₄×C₈	—	86	2	no	—
130	C₄×C₈	—	86	2	no	—
131	C₄×C₈	D₁₇₂	1	2	yes	—
132	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
133	C₂×Q₈	—	86	4	no	—
134	C₂×Q₈	—	86	2	no	—
135	C₂²×C₄	—	43	2	no	—
136	C₂²×C₄	—	43	2	no	—
137	C₂²×C₄	C₈×D₄₃	1	1	yes	characteristic
138	C₂²×C₄	C₈×D₄₃	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	86	2	no	—
142	C₂×C₈	—	86	4	no	—
143	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
144	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
145	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
146	C₂×C₈	#10621898	1	2	yes	—
147	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
148	C₄⋊C₄	—	86	4	no	—
149	C₄⋊C₄	—	86	2	no	—
150	C₄²	—	86	2	no	—
151	C₄²	—	86	2	no	—
152	C₄²	#10621872	1	2	yes	—
153	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
154	C₁₆	—	2	4	no	—
155	C₁₆	—	172	2	no	—
156	C₂³	C₂×C₈×D₄₃	1	1	yes	characteristic
157	Q₈	—	86	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	43	4	no	—
161	C₂×C₄	—	86	2	no	—
162	C₂×C₄	—	86	2	no	—
163	C₂×C₄	—	86	4	no	—
164	C₂×C₄	—	86	2	no	—
165	C₂×C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
166	C₂×C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
167	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₈₆	1	2	yes	—
169	C₂×C₄	C₂×#10621872	1	1	yes	characteristic
170	C₂×C₄	#13063331	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	172	2	no	—
173	C₈	#13063304	1	2	yes	—
174	C₈	#13063313	1	1	yes	characteristic
175	C₈	#13063325	1	1	yes	characteristic
176	C₂²	C₂×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
177	C₂²	C₈×Dic₈₆	1	2	yes	—
178	C₂²	M₃₂×D₄₃	1	2	yes	—
179	C₂²	#14583317	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	86	4	no	—
183	C₄	—	172	2	no	—
184	C₄	#14582805	1	1	yes	characteristic
185	C₄	#14582865	1	1	yes	characteristic
186	C₄	#14583029	1	2	yes	—
187	C₂	#19135531	1	1	yes	characteristic
188	C₂	#19136205	1	1	yes	characteristic
189	C₂	#19136210	1	1	yes	characteristic
190	C₂	#19136501	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₈₆

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₃⋊C₁₆)

C₂

yes

maximal, characteristic

#19132672

C₂

yes

maximal

C₂×C₄₃×#346

C₂

yes

maximal, characteristic

C₂×C₄×Dic₈₆

C₄

yes

characteristic

C₂²×C₄₃⋊C₁₆

C₂²

yes

—

C₈×Dic₈₆

—

C₈×Dic₈₆

—

C₂×C₄⋊(C₄₃⋊C₈)

C₄

yes

characteristic

C₂×Dic₄₃⋊C₈

C₄

yes

—

C₂×C₈×Dic₄₃

C₂²

yes

—

C₄⋊(C₄₃⋊C₁₆)

—

C₂²×C₁₆×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#346

—

C₂²×Dic₈₆

C₈

yes

characteristic

C₂²×C₄₃⋊C₈

C₂×C₄

yes

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

C₂×C₄⋊Dic₄₃

C₈

yes

characteristic

C₂×Dic₄₃⋊C₄

C₈

yes

—

C₂×C₄×Dic₄₃

C₂×C₄

yes

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

—

C₄⋊(C₄₃⋊C₈)

—

Dic₄₃⋊C₈

—

C₈×Dic₄₃

—

C₈×Dic₄₃

—

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

Q₈

yes

complemented

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×Dic₄₃

C₂×C₈

yes

—

C₂²×Dic₄₃

C₂×C₈

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×Q₈

yes

—

C₂×C₈×C₄₃

#41

yes

—

C₄²×C₄₃

C₂²⋊C₄

yes

—

C₄²×C₄₃

M₁₆

yes

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₁₆×C₄₃

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₈

yes

—

C₂×C₄×C₄₃

C₂²⋊C₈

yes

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×Q₈

yes

—

C₂×C₄×C₄₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₃

#131

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

M₃₂

yes

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂²⋊Q₈

yes

—

C₈×C₄₃

#124

yes

characteristic

C₈×C₄₃

C₄⋊Q₈

yes

characteristic

C₄₃⋊C₈

—

#13264

—

maximal, Sylow

C₂²×C₄₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₄₃

C₈×Q₈

yes

—

C₂²×C₄₃

C₂×M₃₂

yes

—

C₂²×C₄₃

#504

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

Dic₄₃

—

Dic₄₃

—

C₂×C₈×Q₈

—

C₂×C₄⋊C₁₆

—

C₂×#346

D₄₃

yes

characteristic

#1374

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

100

C₂×C₄×Q₈

—

101

C₂²×C₁₆

—

102

C₂²×C₁₆

C₂×D₄₃

yes

—

103

C₈×Q₈

—

104

C₈×Q₈

—

105

C₂×C₄⋊C₈

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

109

C₄⋊C₁₆

—

110

#346

—

111

C₄₃

#13264

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₄×D₄₃

yes

characteristic

115

C₂²×C₈

C₂²×D₄₃

yes

characteristic

116

C₄×Q₈

—

117

C₄×Q₈

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄²

—

121

C₂×C₄²

C₄×D₄₃

yes

characteristic

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

Dic₈₆

yes

complemented

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

D₁₇₂

yes

—

132

C₄×C₈

C₂²⋊D₄₃

yes

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₈×D₄₃

yes

characteristic

138

C₂²×C₄

C₈×D₄₃

yes

characteristic

139

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

144

C₂×C₈

C₂×Dic₈₆

yes

—

145

C₂×C₈

C₂×D₁₇₂

yes

characteristic

146

C₂×C₈

#10621898

yes

—

147

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄²

—

151

C₄²

—

152

C₄²

#10621872

yes

—

153

C₄²

(C₂×D₄₃)⋊C₄

yes

—

154

C₁₆

—

155

C₁₆

—

172

—

156

C₂³

C₂×C₈×D₄₃

yes

characteristic

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

(C₂×D₄₃)⋊C₈

yes

—

166

C₂×C₄

(C₂×D₄₃)⋊C₈

yes

—

167

C₂×C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

168

C₂×C₄

C₄×Dic₈₆

yes

—

169

C₂×C₄

C₂×#10621872

yes

characteristic

170

C₂×C₄

#13063331

yes

—

171

C₈

—

172

C₈

—

172

—

173

C₈

#13063304

yes

—

174

C₈

#13063313

yes

characteristic

175

C₈

#13063325

yes

characteristic

176

C₂²

C₂×(C₂×D₄₃)⋊C₈

yes

characteristic

177

C₂²

C₈×Dic₈₆

yes

—

178

C₂²

M₃₂×D₄₃

yes

—

179

C₂²

#14583317

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

172

—

184

C₄

#14582805

yes

characteristic

185

C₄

#14582865

yes

characteristic

186

C₄

#14583029

yes

—

187

C₂

#19135531

yes

characteristic

188

C₂

#19136205

yes

characteristic

189

C₂

#19136210

yes

characteristic

190

C₂

#19136501

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₃
commutator, socle, prime radical	C₂³×C₄₃	C₂²×C₈
Fitting	C₂×C₄₃×#346	C₂
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#346	D₄₃
hypocenter	C₄₃	#13264
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₄₃

commutator, socle, prime radical

C₂³×C₄₃

C₂²×C₈

Fitting

C₂×C₄₃×#346

C₂

cosocle

C₂²×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂²×C₈ ◁ C₂×#346

N	H ≅ G/N	type
34. C₂×C₁₆×C₄₃	Q₈	semidirect
111. C₄₃	#13264	semidirect
126. C₂×C₁₆	Dic₈₆	semidirect

H ≅ G/N

type

34. C₂×C₁₆×C₄₃

Q₈

semidirect

111. C₄₃

#13264

semidirect

126. C₂×C₁₆

Dic₈₆

semidirect

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

subgroup	index
C₂×C₈×Dic₈₆	2
C₂×C₄⋊(C₄₃⋊C₁₆)	2
#19132672	2
C₂×C₄₃×#346	2
#13264	43

subgroup

index

C₂×C₈×Dic₈₆

C₂×C₄⋊(C₄₃⋊C₁₆)

#19132672

C₂×C₄₃×#346

#13264

Subgroups — #25015120

Conjugacy classes of subgroups

Complements

Maximal subgroups