Group #36453055 — subgroups

Subgroups — #36453055

Order 22784.

294 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 131 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	#26019004	C₂	1	2	yes	maximal
2	C₂×#19151351	C₂	1	1	yes	maximal, characteristic
3	#26018544	C₂	1	1	yes	maximal, characteristic
4	#26018103	C₂	1	2	yes	maximal
5	C₄×#108×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₂²	1	1	yes	characteristic
7	#19151651	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	#19151351	—	2	2	no	—
10	#19151351	—	2	2	no	—
11	#19151349	—	2	2	no	—
12	#19151320	—	2	2	no	—
13	C₂×#108×C₈₉	C₄	1	2	yes	—
14	C₄×M₁₆×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#341×C₈₉	C₄	1	1	yes	characteristic
17	#341×C₈₉	C₄	1	1	yes	characteristic
18	C₂²×#10623494	—	2	2	no	—
19	C₄⋊#10623494	—	2	2	no	—
20	C₄⋊#10623494	—	2	2	no	—
21	C₄⋊#10623494	—	2	1	no	—
22	C₄⋊#10623494	D₄	1	1	yes	characteristic
23	C₄⋊#10623494	D₄	1	1	yes	characteristic
24	C₂²⋊#10623494	—	2	1	no	—
25	C₂²⋊#10623494	Q₈	1	2	yes	—
26	C₄×#10623494	—	2	1	no	—
27	C₄×#10623494	—	2	1	no	—
28	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	#108×C₈₉	—	2	4	no	—
34	#97×C₈₉	—	2	2	no	—
35	C₄×C₈×C₈₉	—	2	2	no	—
36	C₄×C₈×C₈₉	D₄	1	2	yes	—
37	C₄×C₈×C₈₉	Q₈	1	2	yes	—
38	C₂×#10623494	—	2	1	no	—
39	C₂×#10623494	—	2	1	no	—
40	C₂×#10623494	—	2	2	no	—
41	C₂×#10623494	—	4	2	no	—
42	C₂×#10623494	—	8	2	no	—
43	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
46	M₁₆×C₈₉	—	2	2	no	—
47	M₁₆×C₈₉	—	2	2	no	—
48	M₁₆×C₈₉	—	2	2	no	—
49	M₁₆×C₈₉	M₁₆	1	4	yes	—
50	C₂×C₈×C₈₉	—	2	2	no	—
51	C₂×C₈×C₈₉	—	2	2	no	—
52	C₂×C₈×C₈₉	—	2	2	no	—
53	C₂×C₈×C₈₉	—	2	2	no	—
54	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
56	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₈₉	#41	1	1	yes	characteristic
60	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₈₉	—	2	1	no	—
62	C₂×C₄×C₈₉	—	2	1	no	—
63	C₂×C₄×C₈₉	—	4	1	no	—
64	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₈₉	#125	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₈₉	#131	1	2	yes	—
72	C₈×C₈₉	—	2	2	no	—
73	C₈×C₈₉	—	2	2	no	—
74	C₈×C₈₉	—	2	4	no	—
75	C₈×C₈₉	—	4	2	no	—
76	#10623494	—	4	2	no	—
77	#10623494	—	4	1	no	—
78	#10623494	—	4	1	no	—
79	#10623494	—	8	2	no	—
80	C₂²×C₈₉	—	2	1	no	—
81	C₂²×C₈₉	—	2	1	no	—
82	C₂²×C₈₉	#389	1	1	yes	characteristic
83	C₂²×C₈₉	#446	1	2	yes	—
84	C₄×C₈₉	—	2	1	no	—
85	C₄×C₈₉	—	2	1	no	—
86	C₄×C₈₉	—	2	1	no	—
87	C₄×C₈₉	#423	1	1	yes	characteristic
88	C₄×C₈₉	#424	1	1	yes	characteristic
89	C₄×C₈₉	#435	1	2	yes	—
90	C₄×C₈₉	#472	1	1	yes	characteristic
91	C₄×C₈₉	#473	1	1	yes	characteristic
92	#9610	—	89	1	no	maximal, Sylow
93	C₂×C₈₉	—	2	2	no	—
94	C₂×C₈₉	#1940	1	1	yes	characteristic
95	C₂×C₈₉	#1952	1	1	yes	characteristic
96	C₂×C₈₉	#1953	1	1	yes	characteristic
97	C₄×#108	D₈₉	1	1	yes	characteristic
98	#1565	—	89	2	no	—
99	C₂×#335	—	89	1	no	—
100	#1452	—	89	1	no	—
101	#1283	—	89	2	no	—
102	C₈₉	#9610	1	1	yes	characteristic, complemented, Sylow
103	#431	—	89	1	no	—
104	#431	—	89	1	no	—
105	C₂×#108	Dic₈₉	1	2	yes	—
106	C₂×C₄⋊C₈	—	89	2	no	—
107	C₄×M₁₆	C₂×D₈₉	1	2	yes	—
108	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
109	#341	Dic₈₉	1	1	yes	characteristic
110	#341	Dic₈₉	1	1	yes	characteristic
111	#335	—	178	2	no	—
112	#335	—	178	2	no	—
113	#334	—	178	2	no	—
114	#322	—	178	2	no	—
115	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
116	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
117	C₂²×C₈	—	178	2	no	—
118	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
119	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
121	#108	—	2	4	no	—
122	C₄⋊C₈	—	89	1	no	—
123	C₄⋊C₈	—	89	1	no	—
124	C₄⋊C₈	—	178	2	no	—
125	C₄⋊C₈	—	178	2	no	—
126	C₄⋊C₈	—	178	1	no	—
127	C₂²⋊C₈	—	89	2	no	—
128	C₂²⋊C₈	—	178	1	no	—
129	#97	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	178	1	no	—
132	C₄×C₈	—	178	1	no	—
133	C₄×C₈	Dic₁₇₈	1	2	yes	—
134	C₄×C₈	D₃₅₆	1	2	yes	—
135	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
136	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
137	C₂²×C₄	#13083575	1	1	yes	characteristic
138	M₁₆	—	2	2	no	—
139	M₁₆	—	2	2	no	—
140	M₁₆	—	2	2	no	—
141	M₁₆	#13083558	1	4	yes	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	356	2	no	—
147	C₂×C₈	—	178	1	no	—
148	C₂×C₈	—	178	1	no	—
149	C₂×C₈	—	178	2	no	—
150	C₂×C₈	—	712	2	no	—
151	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
152	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
153	C₄²	D₄×D₈₉	1	1	yes	characteristic
154	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
155	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
156	C₄²	#13083573	1	1	yes	characteristic
157	C₂³	Q₈×Dic₈₉	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
162	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
163	C₂×C₄	#15667987	1	1	yes	characteristic
164	C₂×C₄	#15667992	1	1	yes	characteristic
165	C₂×C₄	#15668003	1	1	yes	characteristic
166	C₂×C₄	#15668011	1	1	yes	characteristic
167	C₂×C₄	C₂×#13083558	1	2	yes	—
168	C₂×C₄	#15668019	1	2	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	2	4	no	—
172	C₈	—	4	2	no	—
173	C₈	—	356	2	no	—
174	C₈	—	356	1	no	—
175	C₈	—	356	1	no	—
176	C₈	—	712	2	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	#19151497	1	1	yes	characteristic
180	C₂²	#19151729	1	2	yes	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	#19151622	1	1	yes	characteristic
185	C₄	#19151625	1	1	yes	characteristic
186	C₄	#19151670	1	2	yes	—
187	C₄	#19151869	1	1	yes	characteristic
188	C₄	#19151874	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	#26020976	1	1	yes	characteristic
191	C₂	#26021038	1	1	yes	characteristic
192	C₂	#26021043	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26019004

C₂

yes

maximal

C₂×#19151351

C₂

yes

maximal, characteristic

#26018544

C₂

yes

maximal, characteristic

#26018103

C₂

yes

maximal

C₄×#108×C₈₉

C₂

yes

maximal, characteristic

#19151651

C₂²

yes

characteristic

#19151651

C₂²

yes

characteristic

C₂×C₄⋊#10623494

C₂²

yes

—

—

—

—

—

C₂×#108×C₈₉

C₄

yes

—

C₄×M₁₆×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#341×C₈₉

C₄

yes

characteristic

#341×C₈₉

C₄

yes

—

—

—

—

D₄

yes

characteristic

C₄⋊#10623494

D₄

yes

characteristic

C₂²⋊#10623494

—

C₂²⋊#10623494

Q₈

yes

—

C₄×#10623494

—

C₄×#10623494

—

C₂×M₁₆×C₈₉

C₂×C₄

yes

—

C₂×M₁₆×C₈₉

C₂×C₄

yes

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#108×C₈₉

—

#97×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

—

—

—

—

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

M₁₆

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

#41

yes

characteristic

C₂³×C₈₉

C₄×Q₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

#125

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

—

—

—

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#389

yes

characteristic

C₂²×C₈₉

#446

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

yes

characteristic

#1565

—

C₂×#335

—

100

#1452

—

101

#1283

—

102

C₈₉

#9610

yes

characteristic, complemented, Sylow

103

#431

—

104

#431

—

105

C₂×#108

Dic₈₉

yes

—

106

C₂×C₄⋊C₈

—

107

C₄×M₁₆

C₂×D₈₉

yes

—

108

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

109

#341

Dic₈₉

yes

characteristic

110

#341

Dic₈₉

yes

characteristic

111

#335

—

178

—

112

#335

—

178

—

113

#334

—

178

—

114

#322

—

178

—

115

C₂×M₁₆

C₂×Dic₈₉

yes

—

116

C₂×M₁₆

C₂×Dic₈₉

yes

—

117

C₂²×C₈

—

178

—

118

C₂²×C₈

C₂×Dic₈₉

yes

characteristic

119

C₂²×C₈

C₂×Dic₈₉

yes

characteristic

120

C₂×C₄²

C₂²×D₈₉

yes

characteristic

121

#108

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

178

—

125

C₄⋊C₈

—

178

—

126

C₄⋊C₈

—

178

—

127

C₂²⋊C₈

—

128

C₂²⋊C₈

—

178

—

129

#97

—

130

C₄×C₈

—

131

C₄×C₈

—

178

—

132

C₄×C₈

—

178

—

133

C₄×C₈

Dic₁₇₈

yes

—

134

C₄×C₈

D₃₅₆

yes

—

135

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

136

C₂²×C₄

Q₈×D₈₉

yes

characteristic

137

C₂²×C₄

#13083575

yes

characteristic

138

M₁₆

—

139

M₁₆

—

140

M₁₆

—

141

M₁₆

#13083558

yes

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

356

—

147

C₂×C₈

—

178

—

148

C₂×C₈

—

178

—

149

C₂×C₈

—

178

—

150

C₂×C₈

—

712

—

151

C₂×C₈

C₄⋊Dic₈₉

yes

—

152

C₂×C₈

C₄⋊Dic₈₉

yes

—

153

C₄²

D₄×D₈₉

yes

characteristic

154

C₄²

C₂×D₃₅₆

yes

characteristic

155

C₄²

C₂×Dic₁₇₈

yes

characteristic

156

C₄²

#13083573

yes

characteristic

157

C₂³

Q₈×Dic₈₉

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

characteristic

162

C₂×C₄

D₄×Dic₈₉

yes

characteristic

163

C₂×C₄

#15667987

yes

characteristic

164

C₂×C₄

#15667992

yes

characteristic

165

C₂×C₄

#15668003

yes

characteristic

166

C₂×C₄

#15668011

yes

characteristic

167

C₂×C₄

C₂×#13083558

yes

—

168

C₂×C₄

#15668019

yes

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

356

—

174

C₈

—

356

—

175

C₈

—

356

—

176

C₈

—

712

—

177

C₂²

—

178

C₂²

—

179

C₂²

#19151497

yes

characteristic

180

C₂²

#19151729

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#19151622

yes

characteristic

185

C₄

#19151625

yes

characteristic

186

C₄

#19151670

yes

—

187

C₄

#19151869

yes

characteristic

188

C₄

#19151874

yes

characteristic

189

C₂

—

190

C₂

#26020976

yes

characteristic

191

C₂

#26021038

yes

characteristic

192

C₂

#26021043

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#15667987
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#108×C₈₉	C₂
socle	C₂²×C₈₉	#389
prime radical	C₂³×C₈₉	C₄×Q₈
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#108	D₈₉
hypocenter	C₈₉	#9610
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

subgroup

G/H

center

C₂×C₄

#15667987

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₄×#108×C₈₉

C₂

socle

C₂²×C₈₉

#389

prime radical

C₂³×C₈₉

C₄×Q₈

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

N	H ≅ G/N	type
102. C₈₉	#9610	semidirect

H ≅ G/N

type

102. C₈₉

#9610

semidirect

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

subgroup	index
#26019004	2
C₂×#19151351	2
#26018544	2
#26018103	2
C₄×#108×C₈₉	2
#9610	89

subgroup

index

C₄×#108×C₈₉

Subgroups — #36453055

Conjugacy classes of subgroups

Complements

Maximal subgroups