Group #36401318 — subgroups

Subgroups — C₈₉×#23470 #36401318

Order 22784.

1178 conjugacy classes of subgroups; only the 334 normal subgroups are shown.

normal subgroups only

This group has 1178 conjugacy classes of subgroups — too many to draw. Only its 334 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	C₈₉×#3458	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2310	C₂	1	3	yes	maximal
3	C₈₉×#2030	C₂	1	8	yes	maximal
4	C₈₉×#1740	C₂	1	3	yes	maximal
5	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
6	#544×C₈₉	—	2	6	no	—
7	#543×C₈₉	C₂²	1	4	yes	—
8	#533×C₈₉	—	2	6	no	—
9	C₂×#117×C₈₉	C₂²	1	3	yes	—
10	C₂×#117×C₈₉	C₂²	1	3	yes	—
11	#391×C₈₉	—	2	6	no	—
12	#391×C₈₉	—	2	6	no	—
13	#381×C₈₉	C₂²	1	12	yes	—
14	#342×C₈₉	C₂²	1	12	yes	—
15	C₂²×Q₈×C₈₉	—	2	3	no	—
16	C₂²×Q₈×C₈₉	—	2	3	no	—
17	C₂²×Q₈×C₈₉	—	2	6	no	—
18	C₂²×Q₈×C₈₉	Q₈	1	4	yes	complemented
19	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
20	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
21	C₄⋊Q₈×C₈₉	—	2	12	no	—
22	#125×C₈₉	—	2	12	no	—
23	C₂²⋊Q₈×C₈₉	—	2	24	no	—
24	C₄×Q₈×C₈₉	—	2	8	no	—
25	#117×C₈₉	—	2	6	no	—
26	#117×C₈₉	—	2	6	no	—
27	#117×C₈₉	—	2	12	no	—
28	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
29	C₂×C₄⋊C₄×C₈₉	D₄	1	6	yes	—
30	C₂×C₂²⋊C₄×C₈₉	C₂³	1	6	yes	—
31	C₂×C₂²⋊C₄×C₈₉	C₂³	1	6	yes	—
32	C₂×C₄²×C₈₉	—	2	3	no	—
33	C₂×C₄²×C₈₉	D₄	1	6	yes	—
34	#95×C₈₉	—	2	24	no	—
35	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
36	C₂×Q₈×C₈₉	—	2	8	no	—
37	C₂×Q₈×C₈₉	—	2	12	no	—
38	C₂×Q₈×C₈₉	—	2	12	no	—
39	C₂×Q₈×C₈₉	—	4	1	no	—
40	C₂×Q₈×C₈₉	—	4	3	no	—
41	C₂×Q₈×C₈₉	—	4	6	no	—
42	C₂×Q₈×C₈₉	—	4	6	no	—
43	C₂²×C₄×C₈₉	—	2	6	no	—
44	C₂²×C₄×C₈₉	—	2	6	no	—
45	C₂²×C₄×C₈₉	—	2	6	no	—
46	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	—
47	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
50	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₈₉	#41	1	6	yes	—
52	C₄⋊C₄×C₈₉	—	2	24	no	—
53	C₄⋊C₄×C₈₉	—	2	12	no	—
54	C₄⋊C₄×C₈₉	—	4	6	no	—
55	C₂²⋊C₄×C₈₉	—	2	12	no	—
56	C₂²⋊C₄×C₈₉	—	2	12	no	—
57	C₄²×C₈₉	—	2	6	no	—
58	C₄²×C₈₉	—	2	6	no	—
59	C₄²×C₈₉	—	4	6	no	—
60	C₂³×C₈₉	—	4	2	no	—
61	C₂³×C₈₉	C₂²×D₄	1	3	yes	—
62	C₂³×C₈₉	C₂²×Q₈	1	1	yes	characteristic
63	C₂³×C₈₉	C₂×#41	1	3	yes	—
64	Q₈×C₈₉	—	4	4	no	—
65	Q₈×C₈₉	—	4	12	no	—
66	Q₈×C₈₉	—	8	8	no	—
67	C₂×C₄×C₈₉	—	2	12	no	—
68	C₂×C₄×C₈₉	—	2	12	no	—
69	C₂×C₄×C₈₉	—	4	2	no	—
70	C₂×C₄×C₈₉	—	4	6	no	—
71	C₂×C₄×C₈₉	—	4	12	no	—
72	C₂×C₄×C₈₉	—	4	12	no	—
73	C₂×C₄×C₈₉	—	4	6	no	—
74	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#120	1	3	yes	—
76	C₂×C₄×C₈₉	C₂²⋊Q₈	1	6	yes	—
77	C₂×C₄×C₈₉	C₂²⋊Q₈	1	6	yes	—
78	C₂×C₄×C₈₉	#124	1	6	yes	—
79	C₂×C₄×C₈₉	#124	1	6	yes	—
80	C₂²×C₈₉	—	2	6	no	—
81	C₂²×C₈₉	—	4	1	no	—
82	C₂²×C₈₉	—	4	3	no	—
83	C₂²×C₈₉	C₂×#120	1	1	yes	characteristic
84	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	3	yes	—
85	C₂²×C₈₉	C₂×#124	1	3	yes	—
86	C₄×C₈₉	—	4	4	no	—
87	C₄×C₈₉	—	4	12	no	—
88	C₄×C₈₉	—	8	6	no	—
89	C₄×C₈₉	#393	1	2	yes	—
90	C₄×C₈₉	#393	1	6	yes	—
91	#23470	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂×C₈₉	—	2	3	no	—
93	C₂×C₈₉	—	2	3	no	—
94	C₂×C₈₉	C₂×#393	1	1	yes	characteristic
95	C₂×C₈₉	#3028	1	2	yes	—
96	#3458	C₂×C₈₉	1	1	yes	characteristic
97	#2310	C₂×C₈₉	1	3	yes	—
98	#2030	C₂×C₈₉	1	8	yes	—
99	#1740	C₂×C₈₉	1	3	yes	—
100	C₈₉	#23470	1	1	yes	characteristic, direct product factor, Sylow
101	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic
102	#544	—	2	6	no	—
103	#543	C₂²×C₈₉	1	4	yes	—
104	#533	—	2	6	no	—
105	C₂×#117	C₂²×C₈₉	1	3	yes	—
106	C₂×#117	C₂²×C₈₉	1	3	yes	—
107	#391	—	2	6	no	—
108	#391	—	2	6	no	—
109	#381	C₂²×C₈₉	1	12	yes	—
110	#342	C₂²×C₈₉	1	12	yes	—
111	C₂²×Q₈	—	2	3	no	—
112	C₂²×Q₈	—	2	3	no	—
113	C₂²×Q₈	—	2	6	no	—
114	C₂²×Q₈	Q₈×C₈₉	1	4	yes	complemented
115	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
116	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
117	C₄⋊Q₈	—	2	12	no	—
118	#125	—	2	12	no	—
119	C₂²⋊Q₈	—	2	24	no	—
120	C₄×Q₈	—	2	8	no	—
121	#117	—	2	6	no	—
122	#117	—	2	6	no	—
123	#117	—	2	12	no	—
124	C₂×C₄⋊C₄	—	2	3	no	—
125	C₂×C₄⋊C₄	D₄×C₈₉	1	6	yes	—
126	C₂×C₂²⋊C₄	C₂³×C₈₉	1	6	yes	—
127	C₂×C₂²⋊C₄	C₂³×C₈₉	1	6	yes	—
128	C₂×C₄²	—	2	3	no	—
129	C₂×C₄²	D₄×C₈₉	1	6	yes	—
130	#95	—	2	24	no	—
131	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
132	C₂×Q₈	—	2	8	no	—
133	C₂×Q₈	—	2	12	no	—
134	C₂×Q₈	—	2	12	no	—
135	C₂×Q₈	—	4	1	no	—
136	C₂×Q₈	—	4	3	no	—
137	C₂×Q₈	—	4	6	no	—
138	C₂×Q₈	—	4	6	no	—
139	C₂²×C₄	—	2	6	no	—
140	C₂²×C₄	—	2	6	no	—
141	C₂²×C₄	—	2	6	no	—
142	C₂²×C₄	C₂×D₄×C₈₉	1	12	yes	—
143	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
144	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
145	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
146	C₂²×C₄	C₂×Q₈×C₈₉	1	2	yes	—
147	C₂²×C₄	#41×C₈₉	1	6	yes	—
148	C₄⋊C₄	—	2	24	no	—
149	C₄⋊C₄	—	2	12	no	—
150	C₄⋊C₄	—	4	6	no	—
151	C₂²⋊C₄	—	2	12	no	—
152	C₂²⋊C₄	—	2	12	no	—
153	C₄²	—	2	6	no	—
154	C₄²	—	2	6	no	—
155	C₄²	—	4	6	no	—
156	C₂³	—	4	2	no	—
157	C₂³	C₂²×D₄×C₈₉	1	3	yes	—
158	C₂³	C₂²×Q₈×C₈₉	1	1	yes	characteristic
159	C₂³	C₂×#41×C₈₉	1	3	yes	—
160	Q₈	—	4	4	no	—
161	Q₈	—	4	12	no	—
162	Q₈	—	8	8	no	—
163	C₂×C₄	—	2	12	no	—
164	C₂×C₄	—	2	12	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	6	no	—
167	C₂×C₄	—	4	12	no	—
168	C₂×C₄	—	4	12	no	—
169	C₂×C₄	—	4	6	no	—
170	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
171	C₂×C₄	#120×C₈₉	1	3	yes	—
172	C₂×C₄	C₂²⋊Q₈×C₈₉	1	6	yes	—
173	C₂×C₄	C₂²⋊Q₈×C₈₉	1	6	yes	—
174	C₂×C₄	#124×C₈₉	1	6	yes	—
175	C₂×C₄	#124×C₈₉	1	6	yes	—
176	C₂²	—	2	6	no	—
177	C₂²	—	4	1	no	—
178	C₂²	—	4	3	no	—
179	C₂²	C₂×#120×C₈₉	1	1	yes	characteristic
180	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	3	yes	—
181	C₂²	C₂×#124×C₈₉	1	3	yes	—
182	C₄	—	4	4	no	—
183	C₄	—	4	12	no	—
184	C₄	—	8	6	no	—
185	C₄	#393×C₈₉	1	2	yes	—
186	C₄	#393×C₈₉	1	6	yes	—
187	C₂	—	2	3	no	—
188	C₂	—	2	3	no	—
189	C₂	C₂×#393×C₈₉	1	1	yes	characteristic
190	C₂	C₈₉×#3028	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉×#3458

C₂

yes

maximal, characteristic

C₈₉×#2310

C₂

yes

maximal

C₈₉×#2030

C₂

yes

maximal

C₈₉×#1740

C₂

yes

maximal

C₂³×Q₈×C₈₉

C₂²

yes

characteristic

#544×C₈₉

—

#543×C₈₉

C₂²

yes

—

#533×C₈₉

—

C₂×#117×C₈₉

C₂²

yes

—

C₂×#117×C₈₉

C₂²

yes

—

#391×C₈₉

—

#391×C₈₉

—

#381×C₈₉

C₂²

yes

—

#342×C₈₉

C₂²

yes

—

C₂²×Q₈×C₈₉

—

C₂²×Q₈×C₈₉

—

C₂²×Q₈×C₈₉

—

C₂²×Q₈×C₈₉

Q₈

yes

complemented

C₂³×C₄×C₈₉

C₂³

yes

—

C₂³×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊Q₈×C₈₉

—

#125×C₈₉

—

C₂²⋊Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

#117×C₈₉

—

#117×C₈₉

—

#117×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

—

C₂×C₂²⋊C₄×C₈₉

C₂³

yes

—

C₂×C₂²⋊C₄×C₈₉

C₂³

yes

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

D₄

yes

—

#95×C₈₉

—

C₂⁴×C₈₉

C₂⁴

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×Q₈

yes

—

C₂²×C₄×C₈₉

C₂×Q₈

yes

—

C₂²×C₄×C₈₉

#41

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂²×D₄

yes

—

C₂³×C₈₉

C₂²×Q₈

yes

characteristic

C₂³×C₈₉

C₂×#41

yes

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#120

yes

characteristic

C₂×C₄×C₈₉

#120

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

#124

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×#120

yes

characteristic

C₂²×C₈₉

C₂×C₂²⋊Q₈

yes

—

C₂²×C₈₉

C₂×#124

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#393

yes

—

C₄×C₈₉

#393

yes

—

#23470

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#393

yes

characteristic

C₂×C₈₉

#3028

yes

—

#3458

C₂×C₈₉

yes

characteristic

#2310

C₂×C₈₉

yes

—

#2030

C₂×C₈₉

yes

—

#1740

C₂×C₈₉

yes

—

100

C₈₉

#23470

yes

characteristic, direct product factor, Sylow

101

C₂³×Q₈

C₂²×C₈₉

yes

characteristic

102

#544

—

103

#543

C₂²×C₈₉

yes

—

104

#533

—

105

C₂×#117

C₂²×C₈₉

yes

—

106

C₂×#117

C₂²×C₈₉

yes

—

107

#391

—

108

#391

—

109

#381

C₂²×C₈₉

yes

—

110

#342

C₂²×C₈₉

yes

—

111

C₂²×Q₈

—

112

C₂²×Q₈

—

113

C₂²×Q₈

—

114

C₂²×Q₈

Q₈×C₈₉

yes

complemented

115

C₂³×C₄

C₂³×C₈₉

yes

—

116

C₂³×C₄

C₂³×C₈₉

yes

characteristic

117

C₄⋊Q₈

—

118

#125

—

119

C₂²⋊Q₈

—

120

C₄×Q₈

—

121

#117

—

122

#117

—

123

#117

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

D₄×C₈₉

yes

—

126

C₂×C₂²⋊C₄

C₂³×C₈₉

yes

—

127

C₂×C₂²⋊C₄

C₂³×C₈₉

yes

—

128

C₂×C₄²

—

129

C₂×C₄²

D₄×C₈₉

yes

—

130

#95

—

131

C₂⁴

C₂⁴×C₈₉

yes

characteristic

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

C₂×D₄×C₈₉

yes

—

143

C₂²×C₄

C₂×D₄×C₈₉

yes

—

144

C₂²×C₄

C₂×D₄×C₈₉

yes

—

145

C₂²×C₄

C₂×Q₈×C₈₉

yes

—

146

C₂²×C₄

C₂×Q₈×C₈₉

yes

—

147

C₂²×C₄

#41×C₈₉

yes

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₂²⋊C₄

—

152

C₂²⋊C₄

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₂³

—

157

C₂³

C₂²×D₄×C₈₉

yes

—

158

C₂³

C₂²×Q₈×C₈₉

yes

characteristic

159

C₂³

C₂×#41×C₈₉

yes

—

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#120×C₈₉

yes

characteristic

171

C₂×C₄

#120×C₈₉

yes

—

172

C₂×C₄

C₂²⋊Q₈×C₈₉

yes

—

173

C₂×C₄

C₂²⋊Q₈×C₈₉

yes

—

174

C₂×C₄

#124×C₈₉

yes

—

175

C₂×C₄

#124×C₈₉

yes

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

C₂×#120×C₈₉

yes

characteristic

180

C₂²

C₂×C₂²⋊Q₈×C₈₉

yes

—

181

C₂²

C₂×#124×C₈₉

yes

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

#393×C₈₉

yes

—

186

C₄

#393×C₈₉

yes

—

187

C₂

—

188

C₂

—

189

C₂

C₂×#393×C₈₉

yes

characteristic

190

C₂

C₈₉×#3028

yes

—

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₈₉
prime radical	C₂⁴×C₈₉	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂³×C₄×C₈₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₈₉

C₂×#120

commutator, Frattini, cosocle

C₂⁴

C₂⁴×C₈₉

prime radical

C₂⁴×C₈₉

C₂⁴

LCS

G ▷ C₂⁴ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈₉ ◁ C₂³×C₄×C₈₉ ◁ G

N	H ≅ G/N	type
18. C₂²×Q₈×C₈₉	Q₈	semidirect
91. #23470	C₈₉	direct (N × H)
100. C₈₉	#23470	direct (N × H)
114. C₂²×Q₈	Q₈×C₈₉	semidirect

H ≅ G/N

type

18. C₂²×Q₈×C₈₉

Q₈

semidirect

91. #23470

C₈₉

direct (N × H)

100. C₈₉

#23470

direct (N × H)

114. C₂²×Q₈

Q₈×C₈₉

semidirect

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

subgroup	index
C₈₉×#3458	2
C₈₉×#2310	2
C₈₉×#2030	2
C₈₉×#1740	2
#23470	89

subgroup

index

C₈₉×#3458

C₈₉×#2310

C₈₉×#2030

C₈₉×#1740

#23470

Subgroups — C₈₉×#23470 #36401318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#23470 #36401318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#23470 #36401318