Group #26028604 — subgroups

Subgroups — #26028604

Order 11392.

644 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 237 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×#15668012	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15667991	C₂	1	2	yes	maximal
3	#19151504	C₂	1	2	yes	maximal, complemented
4	#19151496	C₂	1	4	yes	maximal, complemented
5	#19151468	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151458	C₂	1	4	yes	maximal
7	C₂²×C₄²×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
9	#15668012	—	2	4	no	—
10	#15667991	—	2	8	no	—
11	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
14	C₂×#13083553	C₂²	1	4	yes	—
15	C₂×C₂²⋊Dic₈₉	C₂²	1	4	yes	—
16	#15667919	C₂²	1	8	yes	—
17	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₈₉	—	2	4	no	—
20	C₂×C₄²×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₇₈	—	2	4	no	—
24	C₂×Dic₁₇₈	—	2	4	no	—
25	C₂×Dic₁₇₈	—	2	4	no	—
26	C₂²×Dic₈₉	C₂³	1	4	yes	—
27	C₂²×Dic₈₉	C₂³	1	4	yes	—
28	C₄⋊Dic₈₉	—	2	4	no	—
29	C₄⋊Dic₈₉	—	2	8	no	—
30	C₄⋊Dic₈₉	—	2	8	no	—
31	#13083553	—	2	8	no	—
32	C₂²⋊Dic₈₉	—	2	8	no	—
33	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₈₉	—	2	2	no	—
35	C₂²×C₄×C₈₉	—	2	4	no	—
36	C₂²×C₄×C₈₉	—	2	2	no	—
37	C₂²×C₄×C₈₉	—	2	4	no	—
38	C₂²×C₄×C₈₉	D₄	1	4	yes	—
39	C₂²×C₄×C₈₉	Q₈	1	8	yes	complemented
40	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
44	C₄²×C₈₉	—	2	2	no	—
45	C₄²×C₈₉	—	2	4	no	—
46	C₄²×C₈₉	D₄	1	4	yes	—
47	C₂³×C₈₉	—	2	1	no	—
48	C₂³×C₈₉	—	2	1	no	—
49	C₂³×C₈₉	—	2	2	no	—
50	C₂³×C₈₉	C₂×D₄	1	2	yes	—
51	C₂³×C₈₉	C₂×Q₈	1	4	yes	—
52	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
53	C₂×C₄×C₈₉	—	2	1	no	—
54	C₂×C₄×C₈₉	—	2	2	no	—
55	C₂×C₄×C₈₉	—	2	1	no	—
56	C₂×C₄×C₈₉	—	2	2	no	—
57	C₂×C₄×C₈₉	—	2	8	no	—
58	C₂×C₄×C₈₉	—	2	16	no	—
59	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
61	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
64	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
65	C₂×C₄×C₈₉	#41	1	2	yes	—
66	C₂×C₄×C₈₉	#41	1	4	yes	—
67	C₂×Dic₈₉	—	2	4	no	—
68	C₂×Dic₈₉	—	2	4	no	—
69	C₂×Dic₈₉	—	2	4	no	—
70	C₂×Dic₈₉	—	2	8	no	—
71	C₂×Dic₈₉	—	2	4	no	—
72	Dic₁₇₈	—	4	8	no	—
73	C₂²×C₈₉	—	2	2	no	—
74	C₂²×C₈₉	—	2	4	no	—
75	C₂²×C₈₉	—	2	2	no	—
76	C₂²×C₈₉	—	2	4	no	—
77	C₂²×C₈₉	C₄⋊Q₈	1	4	yes	complemented
78	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
79	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₈₉	C₂²×Q₈	1	2	yes	—
81	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
82	C₂²×C₈₉	C₂×#41	1	2	yes	—
83	C₄×C₈₉	—	2	2	no	—
84	C₄×C₈₉	—	2	4	no	—
85	C₄×C₈₉	#121	1	4	yes	—
86	C₄×C₈₉	C₂²⋊Q₈	1	8	yes	complemented
87	Dic₈₉	—	4	4	no	—
88	Dic₈₉	—	4	4	no	—
89	C₂×C₈₉	—	2	4	no	—
90	C₂×C₈₉	C₂×#121	1	1	yes	characteristic
91	C₂×C₈₉	C₂×C₂²⋊Q₈	1	2	yes	—
92	C₂×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₈₉	#532	1	1	yes	characteristic
94	C₂×C₈₉	#533	1	2	yes	—
95	#2857	—	89	1	no	maximal, Sylow
96	C₈₉	#2857	1	1	yes	characteristic, complemented, Sylow
97	C₂×C₄⋊Q₈	—	89	1	no	—
98	C₂×C₂²⋊Q₈	—	89	2	no	—
99	C₂²×C₄²	D₈₉	1	1	yes	characteristic
100	#391	—	89	2	no	—
101	#389	—	89	4	no	—
102	#384	—	89	1	no	—
103	#380	—	89	4	no	—
104	C₂²×Q₈	—	89	2	no	—
105	C₂³×C₄	C₂×D₈₉	1	2	yes	—
106	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
107	C₄⋊Q₈	—	178	4	no	—
108	C₂²⋊Q₈	—	178	8	no	—
109	C₂×C₄⋊C₄	—	89	2	no	—
110	C₂×C₄⋊C₄	—	89	4	no	—
111	C₂×C₄⋊C₄	—	89	4	no	—
112	C₂×C₄⋊C₄	—	89	4	no	—
113	C₂×C₂²⋊C₄	—	89	4	no	—
114	C₂×C₄²	—	2	4	no	—
115	C₂×C₄²	C₂×D₈₉	1	2	yes	—
116	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
117	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
118	#95	—	89	8	no	—
119	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
120	C₂×Q₈	—	178	4	no	—
121	C₂×Q₈	—	178	4	no	—
122	C₂×Q₈	—	178	4	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	2	4	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	89	4	no	—
128	C₂²×C₄	—	89	4	no	—
129	C₂²×C₄	Dic₁₇₈	1	8	yes	complemented
130	C₂²×C₄	D₃₅₆	1	4	yes	—
131	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
132	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
133	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
135	C₄⋊C₄	—	178	8	no	—
136	C₄⋊C₄	—	178	8	no	—
137	C₄⋊C₄	—	178	4	no	—
138	C₄⋊C₄	—	178	8	no	—
139	C₂²⋊C₄	—	178	8	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	4	no	—
142	C₄²	C₂²⋊D₈₉	1	4	yes	—
143	C₂³	—	2	1	no	—
144	C₂³	—	2	2	no	—
145	C₂³	—	2	1	no	—
146	C₂³	C₂×D₃₅₆	1	2	yes	—
147	C₂³	C₂×Dic₁₇₈	1	4	yes	—
148	C₂³	C₂³×D₈₉	1	1	yes	characteristic
149	Q₈	—	356	8	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	8	no	—
155	C₂×C₄	—	2	16	no	—
156	C₂×C₄	—	178	4	no	—
157	C₂×C₄	—	178	4	no	—
158	C₂×C₄	—	178	4	no	—
159	C₂×C₄	—	178	8	no	—
160	C₂×C₄	—	178	4	no	—
161	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊D₈₉	1	4	yes	—
163	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
164	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
165	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	—
166	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	—
167	C₂×C₄	#13083574	1	2	yes	—
168	C₂×C₄	#13083574	1	4	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	4	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	4	no	—
173	C₂²	#15668012	1	4	yes	complemented
174	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
175	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
176	C₂²	C₂²×Dic₁₇₈	1	2	yes	—
177	C₂²	C₂×#13083574	1	2	yes	—
178	C₂²	C₂×#13083574	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	4	no	—
181	C₄	—	356	4	no	—
182	C₄	—	356	4	no	—
183	C₄	#15667986	1	4	yes	—
184	C₄	#15667991	1	8	yes	complemented
185	C₂	—	2	4	no	—
186	C₂	C₂×#15667986	1	1	yes	characteristic
187	C₂	C₂×#15667991	1	2	yes	—
188	C₂	C₂×#15668012	1	1	yes	characteristic
189	C₂	#19152148	1	1	yes	characteristic
190	C₂	#19152155	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668012

C₂

yes

maximal, characteristic, complemented

C₂×#15667991

C₂

yes

maximal

#19151504

C₂

yes

maximal, complemented

#19151496

C₂

yes

maximal, complemented

#19151468

C₂

yes

maximal, characteristic, complemented

#19151458

C₂

yes

maximal

C₂²×C₄²×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

#15668012

—

#15667991

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

C₂³×C₄×C₈₉

C₂²

yes

characteristic

C₂³×C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂⁴×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

Q₈

yes

complemented

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

C₂×Q₈

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄⋊Q₈

yes

complemented

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

—

C₂²×C₈₉

C₂×#41

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#121

yes

—

C₄×C₈₉

C₂²⋊Q₈

yes

complemented

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#121

yes

characteristic

C₂×C₈₉

C₂×C₂²⋊Q₈

yes

—

C₂×C₈₉

C₂×C₄⋊Q₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

#2857

—

maximal, Sylow

C₈₉

#2857

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₂²⋊Q₈

—

C₂²×C₄²

D₈₉

yes

characteristic

100

#391

—

101

#389

—

102

#384

—

103

#380

—

104

C₂²×Q₈

—

105

C₂³×C₄

C₂×D₈₉

yes

—

106

C₂³×C₄

C₂×D₈₉

yes

characteristic

107

C₄⋊Q₈

—

178

—

108

C₂²⋊Q₈

—

178

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₂²⋊C₄

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂×D₈₉

yes

—

116

C₂×C₄²

C₂×D₈₉

yes

characteristic

117

C₂×C₄²

C₂×D₈₉

yes

characteristic

118

#95

—

119

C₂⁴

C₂²×D₈₉

yes

characteristic

120

C₂×Q₈

—

178

—

121

C₂×Q₈

—

178

—

122

C₂×Q₈

—

178

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

Dic₁₇₈

yes

complemented

130

C₂²×C₄

D₃₅₆

yes

—

131

C₂²×C₄

C₂²×D₈₉

yes

characteristic

132

C₂²×C₄

C₂²×D₈₉

yes

—

133

C₂²×C₄

C₂²×D₈₉

yes

characteristic

134

C₂²×C₄

C₂²×D₈₉

yes

—

135

C₄⋊C₄

—

178

—

136

C₄⋊C₄

—

178

—

137

C₄⋊C₄

—

178

—

138

C₄⋊C₄

—

178

—

139

C₂²⋊C₄

—

178

—

140

C₄²

—

141

C₄²

—

142

C₄²

C₂²⋊D₈₉

yes

—

143

C₂³

—

144

C₂³

—

145

C₂³

—

146

C₂³

C₂×D₃₅₆

yes

—

147

C₂³

C₂×Dic₁₇₈

yes

—

148

C₂³

C₂³×D₈₉

yes

characteristic

149

Q₈

—

356

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

178

—

157

C₂×C₄

—

178

—

158

C₂×C₄

—

178

—

159

C₂×C₄

—

178

—

160

C₂×C₄

—

178

—

161

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

162

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

163

C₂×C₄

C₂×D₃₅₆

yes

—

164

C₂×C₄

C₂×D₃₅₆

yes

—

165

C₂×C₄

C₂×Dic₁₇₈

yes

—

166

C₂×C₄

C₂×Dic₁₇₈

yes

—

167

C₂×C₄

#13083574

yes

—

168

C₂×C₄

#13083574

yes

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

#15668012

yes

complemented

174

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

175

C₂²

C₂²×D₃₅₆

yes

characteristic

176

C₂²

C₂²×Dic₁₇₈

yes

—

177

C₂²

C₂×#13083574

yes

—

178

C₂²

C₂×#13083574

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

356

—

182

C₄

—

356

—

183

C₄

#15667986

yes

—

184

C₄

#15667991

yes

complemented

185

C₂

—

186

C₂

C₂×#15667986

yes

characteristic

187

C₂

C₂×#15667991

yes

—

188

C₂

C₂×#15668012

yes

characteristic

189

C₂

#19152148

yes

characteristic

190

C₂

#19152155

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₂²×C₄²×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂³
hypercenter	C₂²×C₄²	D₈₉
hypocenter	C₈₉	#2857
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₂²×C₄²×C₈₉

C₂

prime radical

C₂⁴×C₈₉

C₂³

hypercenter

C₂²×C₄²

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₂²×C₄²

N	H ≅ G/N	type
1. C₂×#15668012	C₂	semidirect
3. #19151504	C₂	semidirect
4. #19151496	C₂	semidirect
5. #19151468	C₂	semidirect
39. C₂²×C₄×C₈₉	Q₈	semidirect
77. C₂²×C₈₉	C₄⋊Q₈	semidirect
86. C₄×C₈₉	C₂²⋊Q₈	semidirect
96. C₈₉	#2857	semidirect
129. C₂²×C₄	Dic₁₇₈	semidirect
173. C₂²	#15668012	semidirect
184. C₄	#15667991	semidirect

H ≅ G/N

type

1. C₂×#15668012

C₂

semidirect

3. #19151504

C₂

semidirect

4. #19151496

C₂

semidirect

5. #19151468

C₂

semidirect

39. C₂²×C₄×C₈₉

Q₈

semidirect

77. C₂²×C₈₉

C₄⋊Q₈

semidirect

86. C₄×C₈₉

C₂²⋊Q₈

semidirect

96. C₈₉

#2857

semidirect

129. C₂²×C₄

Dic₁₇₈

semidirect

173. C₂²

#15668012

semidirect

184. C₄

#15667991

semidirect

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

subgroup	index
C₂×#15668012	2
C₂×#15667991	2
#19151504	2
#19151496	2
#19151468	2
#19151458	2
C₂²×C₄²×C₈₉	2
#2857	89

subgroup

index

C₂²×C₄²×C₈₉

Subgroups — #26028604

Conjugacy classes of subgroups

Complements

Maximal subgroups