Group #36454255 — subgroups

Subgroups — #36454255

Order 22784.

526 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 163 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×#19151978	C₂	1	1	yes	maximal, characteristic
2	#26021212	C₂	1	2	yes	maximal
3	#26021212	C₂	1	2	yes	maximal
4	C₂×#19151320	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₆	—	2	2	no	—
7	C₂×#15668012	C₂²	1	2	yes	—
8	#19151978	—	2	2	no	—
9	#19151978	C₄	1	4	yes	complemented
10	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
12	#19151360	—	2	4	no	—
13	#19151360	—	2	4	no	—
14	#19151320	—	2	2	no	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#384×C₈₉	C₂²	1	2	yes	—
17	#340×C₈₉	—	2	2	no	—
18	C₂²×Dic₁₇₈	—	2	2	no	—
19	C₂×Dic₃₅₆	—	2	8	no	—
20	C₂×Dic₃₅₆	—	4	8	no	—
21	C₂²×#10623494	D₄	1	2	yes	—
22	C₂²×#10623494	D₄	1	2	yes	—
23	#15668012	—	2	2	no	—
24	#15668012	—	2	2	no	—
25	#15668012	—	2	2	no	—
26	#15668012	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊Dic₈₉	—	2	2	no	—
28	C₂×C₄⋊Dic₈₉	—	2	2	no	—
29	C₄×#10623494	—	2	1	no	—
30	C₄×#10623494	—	2	1	no	—
31	C₄×#10623494	—	2	1	no	—
32	C₄×#10623494	—	2	1	no	—
33	C₂²×C₈×C₈₉	D₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
37	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₈₉	D₄	1	2	yes	—
40	C₂×Dic₁₇₈	—	2	4	no	—
41	C₂×Dic₁₇₈	—	4	8	no	—
42	C₂×Dic₁₇₈	—	4	2	no	—
43	C₂²×Dic₈₉	—	4	2	no	—
44	Dic₃₅₆	—	4	16	no	—
45	C₂×#10623494	—	2	2	no	—
46	C₂×#10623494	—	2	2	no	—
47	C₂×#10623494	—	2	2	no	—
48	C₂×#10623494	—	2	4	no	—
49	C₂×#10623494	Q₁₆	1	4	yes	complemented
50	C₄⋊Dic₈₉	—	2	4	no	—
51	C₄⋊Dic₈₉	—	2	4	no	—
52	C₄⋊Dic₈₉	—	4	2	no	—
53	C₄⋊Dic₈₉	—	4	2	no	—
54	C₂²×C₄×C₈₉	—	4	2	no	—
55	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₈₉	—	2	4	no	—
59	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
60	C₄⋊C₄×C₈₉	—	4	4	no	—
61	C₄⋊C₄×C₈₉	—	4	4	no	—
62	C₄⋊C₄×C₈₉	—	4	4	no	—
63	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₈₉	#41	1	1	yes	characteristic
66	C₄²×C₈₉	#41	1	1	yes	characteristic
67	C₂³×C₈₉	#127	1	1	yes	characteristic
68	C₂×C₄×C₈₉	—	8	2	no	—
69	C₂×C₄×C₈₉	—	8	2	no	—
70	C₂×C₄×C₈₉	—	8	2	no	—
71	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
76	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
78	C₂×C₄×C₈₉	#135	1	2	yes	—
79	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×C₄×C₈₉	#137	1	4	yes	—
81	C₂×Dic₈₉	—	4	4	no	—
82	C₂×Dic₈₉	—	8	4	no	—
83	Dic₁₇₈	—	4	8	no	—
84	Dic₁₇₈	—	8	4	no	—
85	C₈×C₈₉	—	2	4	no	—
86	#10623494	—	2	4	no	—
87	#10623494	—	4	2	no	—
88	C₂²×C₈₉	#390	1	1	yes	characteristic
89	C₂²×C₈₉	#494	1	1	yes	characteristic
90	C₂²×C₈₉	#495	1	1	yes	characteristic
91	C₂²×C₈₉	#496	1	2	yes	—
92	C₂²×C₈₉	#497	1	2	yes	—
93	C₄×C₈₉	—	16	2	no	—
94	C₄×C₈₉	#418	1	2	yes	—
95	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
96	C₄×C₈₉	#441	1	2	yes	—
97	C₄×C₈₉	#467	1	2	yes	—
98	C₄×C₈₉	#470	1	2	yes	—
99	C₄×C₈₉	#490	1	2	yes	—
100	Dic₈₉	—	8	4	no	—
101	#9835	—	89	1	no	maximal, Sylow
102	C₂×C₈₉	#1720	1	2	yes	—
103	C₂×C₈₉	#1723	1	2	yes	—
104	C₂×C₈₉	#1961	1	1	yes	characteristic
105	C₂×C₈₉	#1969	1	1	yes	characteristic
106	C₂×C₈₉	#1971	1	1	yes	characteristic
107	C₂×#494	—	89	1	no	—
108	#1990	—	89	2	no	—
109	#1990	—	89	2	no	—
110	#1834	D₈₉	1	1	yes	characteristic
111	C₂×#322	—	89	1	no	—
112	C₈₉	#9835	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₁₆	—	178	2	no	—
114	C₂×C₄⋊Q₈	—	89	2	no	—
115	#494	—	89	4	no	—
116	#494	—	178	2	no	—
117	C₂×C₄×C₈	—	89	1	no	—
118	C₂×C₄×C₈	—	89	1	no	—
119	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
120	#384	C₂×D₈₉	1	2	yes	—
121	#340	—	2	2	no	—
122	#340	—	178	4	no	—
123	#340	—	178	4	no	—
124	#322	—	178	2	no	—
125	C₂²×Q₈	—	178	2	no	—
126	C₂×Q₁₆	—	356	8	no	—
127	C₂×Q₁₆	—	178	8	no	—
128	C₂²×C₈	—	89	2	no	—
129	C₂²×C₈	—	89	2	no	—
130	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
131	C₄⋊Q₈	—	89	4	no	—
132	C₄⋊Q₈	—	178	2	no	—
133	C₄⋊Q₈	—	178	2	no	—
134	C₄⋊Q₈	—	178	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	178	2	no	—
139	C₂×C₄⋊C₄	—	178	2	no	—
140	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
141	C₄×C₈	—	178	1	no	—
142	C₄×C₈	—	178	1	no	—
143	C₄×C₈	—	178	1	no	—
144	C₄×C₈	—	178	1	no	—
145	C₄×C₈	C₄×D₈₉	1	2	yes	—
146	C₄×C₈	D₃₅₆	1	2	yes	—
147	C₂×Q₈	—	356	2	no	—
148	C₂×Q₈	—	356	8	no	—
149	C₂×Q₈	—	178	4	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	—	356	2	no	—
152	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
153	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
155	Q₁₆	—	356	16	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	89	4	no	—
158	C₂×C₈	—	178	2	no	—
159	C₂×C₈	—	178	4	no	—
160	C₂×C₈	—	178	2	no	—
161	C₂×C₈	—	178	2	no	—
162	C₂×C₈	#13083551	1	4	yes	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	356	2	no	—
167	C₄⋊C₄	—	356	2	no	—
168	C₄⋊C₄	—	178	4	no	—
169	C₄⋊C₄	—	178	4	no	—
170	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
171	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
172	C₄²	#13083575	1	1	yes	characteristic
173	C₄²	#13083575	1	1	yes	characteristic
174	C₂³	#15668008	1	1	yes	characteristic
175	Q₈	—	356	8	no	—
176	Q₈	—	712	4	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	356	4	no	—
181	C₂×C₄	—	712	4	no	—
182	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
183	C₂×C₄	#15667974	1	1	yes	characteristic
184	C₂×C₄	#15667974	1	1	yes	characteristic
185	C₂×C₄	#15667987	1	1	yes	characteristic
186	C₂×C₄	#15667987	1	1	yes	characteristic
187	C₂×C₄	#15668001	1	1	yes	characteristic
188	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
189	C₂×C₄	#15668031	1	2	yes	—
190	C₂×C₄	#15668037	1	4	yes	—
191	C₂×C₄	#15668041	1	4	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	356	2	no	—
194	C₈	—	178	4	no	—
195	C₂²	#19151499	1	1	yes	characteristic
196	C₂²	#19151976	1	1	yes	characteristic
197	C₂²	#19151981	1	1	yes	characteristic
198	C₂²	#19151988	1	2	yes	—
199	C₂²	#19151989	1	2	yes	—
200	C₄	—	16	2	no	—
201	C₄	—	712	4	no	—
202	C₄	#19151602	1	2	yes	—
203	C₄	#19151692	1	2	yes	—
204	C₄	#19151704	1	2	yes	—
205	C₄	#19151840	1	2	yes	—
206	C₄	#19151857	1	2	yes	—
207	C₄	#19151964	1	2	yes	—
208	C₂	#26019911	1	2	yes	—
209	C₂	#26019931	1	2	yes	—
210	C₂	#26021075	1	1	yes	characteristic
211	C₂	#26021111	1	1	yes	characteristic
212	C₂	#26021121	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151978

C₂

yes

maximal, characteristic

#26021212

C₂

yes

maximal

#26021212

C₂

yes

maximal

C₂×#19151320

C₂

yes

maximal, characteristic

C₈₉×#1834

C₂

yes

maximal, characteristic

C₂²×Dic₃₅₆

—

C₂×#15668012

C₂²

yes

—

#19151978

—

#19151978

C₄

yes

complemented

C₂×C₄×#10623494

C₂²

yes

characteristic

C₂×C₄×#10623494

C₂²

yes

—

—

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#384×C₈₉

C₂²

yes

—

#340×C₈₉

—

C₂²×Dic₁₇₈

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂²×#10623494

D₄

yes

—

C₂²×#10623494

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

—

—

—

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

Dic₃₅₆

—

—

—

—

—

Q₁₆

yes

complemented

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

C₄²×C₈₉

yes

C₄²×C₈₉

yes

C₂³×C₈₉

yes

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

C₂×Q₁₆

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#497

yes

—

C₄×C₈₉

—

C₄×C₈₉

#418

yes

—

C₄×C₈₉

C₄×Q₁₆

yes

—

C₄×C₈₉

#441

yes

—

C₄×C₈₉

#467

yes

—

C₄×C₈₉

#470

yes

—

C₄×C₈₉

#490

yes

—

100

Dic₈₉

—

101

#9835

—

maximal, Sylow

102

C₂×C₈₉

#1720

yes

—

103

C₂×C₈₉

#1723

yes

—

104

C₂×C₈₉

#1961

yes

characteristic

105

C₂×C₈₉

#1969

yes

characteristic

106

C₂×C₈₉

#1971

yes

characteristic

107

C₂×#494

—

108

#1990

—

109

#1990

—

110

#1834

D₈₉

yes

characteristic

111

C₂×#322

—

112

C₈₉

#9835

yes

characteristic, complemented, Sylow

113

C₂²×Q₁₆

—

178

—

114

C₂×C₄⋊Q₈

—

115

#494

—

116

#494

—

178

—

117

C₂×C₄×C₈

—

118

C₂×C₄×C₈

—

119

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

120

#384

C₂×D₈₉

yes

—

121

#340

—

122

#340

—

178

—

123

#340

—

178

—

124

#322

—

178

—

125

C₂²×Q₈

—

178

—

126

C₂×Q₁₆

—

356

—

127

C₂×Q₁₆

—

178

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₂²×C₈

C₂²⋊D₈₉

yes

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

—

178

—

133

C₄⋊Q₈

—

178

—

134

C₄⋊Q₈

—

178

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄⋊C₄

—

138

C₂×C₄⋊C₄

—

178

—

139

C₂×C₄⋊C₄

—

178

—

140

C₂×C₄²

C₂²×D₈₉

yes

characteristic

141

C₄×C₈

—

178

—

142

C₄×C₈

—

178

—

143

C₄×C₈

—

178

—

144

C₄×C₈

—

178

—

145

C₄×C₈

C₄×D₈₉

yes

—

146

C₄×C₈

D₃₅₆

yes

—

147

C₂×Q₈

—

356

—

148

C₂×Q₈

—

356

—

149

C₂×Q₈

—

178

—

150

C₂²×C₄

—

151

C₂²×C₄

—

356

—

152

C₂²×C₄

D₄×D₈₉

yes

characteristic

153

C₂²×C₄

D₄×D₈₉

yes

characteristic

154

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

155

Q₁₆

—

356

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

178

—

159

C₂×C₈

—

178

—

160

C₂×C₈

—

178

—

161

C₂×C₈

—

178

—

162

C₂×C₈

#13083551

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

356

—

167

C₄⋊C₄

—

356

—

168

C₄⋊C₄

—

178

—

169

C₄⋊C₄

—

178

—

170

C₄²

C₂×C₄×D₈₉

yes

characteristic

171

C₄²

C₂×D₃₅₆

yes

characteristic

172

C₄²

#13083575

yes

characteristic

173

C₄²

#13083575

yes

characteristic

174

C₂³

#15668008

yes

characteristic

175

Q₈

—

356

—

176

Q₈

—

712

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

356

—

181

C₂×C₄

—

712

—

182

C₂×C₄

C₂×#13083551

yes

characteristic

183

C₂×C₄

#15667974

yes

characteristic

184

C₂×C₄

#15667974

yes

characteristic

185

C₂×C₄

#15667987

yes

characteristic

186

C₂×C₄

#15667987

yes

characteristic

187

C₂×C₄

#15668001

yes

characteristic

188

C₂×C₄

Q₁₆×D₈₉

yes

—

189

C₂×C₄

#15668031

yes

—

190

C₂×C₄

#15668037

yes

—

191

C₂×C₄

#15668041

yes

—

192

C₈

—

193

C₈

—

356

—

194

C₈

—

178

—

195

C₂²

#19151499

yes

characteristic

196

C₂²

#19151976

yes

characteristic

197

C₂²

#19151981

yes

characteristic

198

C₂²

#19151988

yes

—

199

C₂²

#19151989

yes

—

200

C₄

—

201

C₄

—

712

—

202

C₄

#19151602

yes

—

203

C₄

#19151692

yes

—

204

C₄

#19151704

yes

—

205

C₄

#19151840

yes

—

206

C₄

#19151857

yes

—

207

C₄

#19151964

yes

—

208

C₂

#26019911

yes

—

209

C₂

#26019931

yes

—

210

C₂

#26021075

yes

characteristic

211

C₂

#26021111

yes

characteristic

212

C₂

#26021121

yes

characteristic

	subgroup	G/H
center	C₂³	#15668008
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1834	C₂
socle, prime radical	C₂³×C₈₉	#127
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1834	D₈₉
hypocenter	C₈₉	#9835
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#15668008

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1834

C₂

socle, prime radical

C₂³×C₈₉

#127

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

N	H ≅ G/N	type
9. #19151978	C₄	semidirect
49. C₂×#10623494	Q₁₆	semidirect
112. C₈₉	#9835	semidirect

H ≅ G/N

type

9. #19151978

C₄

semidirect

49. C₂×#10623494

Q₁₆

semidirect

112. C₈₉

#9835

semidirect

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

subgroup	index
C₂×#19151978	2
#26021212	2
C₂×#19151320	2
C₈₉×#1834	2
#9835	89

subgroup

index

C₈₉×#1834

Subgroups — #36454255

Conjugacy classes of subgroups

Complements

Maximal subgroups