Group #36457568 — subgroups

Subgroups — #36457568

Order 22784.

364 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 93 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	#26021233	C₂	1	1	yes	maximal, characteristic
2	#26018750	C₂	1	2	yes	maximal
3	#26018155	C₂	1	2	yes	maximal
4	#26018081	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2402	C₂	1	1	yes	maximal, characteristic
6	#19152155	C₂²	1	1	yes	characteristic
7	#19151651	C₂²	1	2	yes	—
8	#19151622	C₂²	1	1	yes	characteristic
9	#19151358	—	2	2	no	—
10	#19151334	—	2	2	no	—
11	#19151334	—	2	2	no	—
12	#400×C₈₉	—	2	2	no	—
13	#398×C₈₉	—	2	2	no	—
14	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
15	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
16	#376×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#13083558	—	2	1	no	—
18	#15668012	Q₈	1	2	yes	—
19	#15668004	—	2	1	no	—
20	#15667991	—	2	2	no	—
21	C₄×Dic₁₇₈	—	2	2	no	—
22	#15667971	—	2	1	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	—	2	2	no	—
25	C₂²⋊#10623494	—	2	2	no	—
26	C₄×#10623494	—	2	2	no	—
27	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂×C₄²×C₈₉	—	2	2	no	—
31	C₂×C₄²×C₈₉	D₄	1	2	yes	—
32	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	#95×C₈₉	—	2	2	no	—
34	#95×C₈₉	—	2	2	no	—
35	#95×C₈₉	—	2	2	no	—
36	#95×C₈₉	D₄	1	2	yes	—
37	#95×C₈₉	D₄	1	2	yes	—
38	C₂×Dic₁₇₈	—	2	2	no	—
39	#13083558	—	4	2	no	—
40	C₂×#10623494	—	4	1	no	—
41	C₂×#10623494	—	4	2	no	—
42	C₄⋊Dic₈₉	—	2	2	no	—
43	C₄⋊Dic₈₉	—	2	2	no	—
44	#13083553	—	4	2	no	—
45	C₂²⋊Dic₈₉	—	4	1	no	—
46	C₄×Dic₈₉	—	4	1	no	—
47	C₂²×C₄×C₈₉	—	2	2	no	—
48	C₂²×C₄×C₈₉	—	2	2	no	—
49	C₂²×C₄×C₈₉	—	2	2	no	—
50	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₄⋊C₄×C₈₉	—	4	4	no	—
53	C₄⋊C₄×C₈₉	—	4	4	no	—
54	C₄⋊C₄×C₈₉	—	4	4	no	—
55	C₄²×C₈₉	—	2	2	no	—
56	C₄²×C₈₉	—	2	2	no	—
57	C₄²×C₈₉	—	4	2	no	—
58	C₄²×C₈₉	—	4	2	no	—
59	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₈₉	#41	1	2	yes	—
61	C₄²×C₈₉	#41	1	1	yes	characteristic
62	C₂³×C₈₉	#120	1	1	yes	characteristic
63	C₂×C₄×C₈₉	—	2	4	no	—
64	C₂×C₄×C₈₉	—	4	2	no	—
65	C₂×C₄×C₈₉	—	4	2	no	—
66	C₂×C₄×C₈₉	—	4	2	no	—
67	C₂×C₄×C₈₉	—	4	2	no	—
68	C₂×C₄×C₈₉	—	4	2	no	—
69	C₂×C₄×C₈₉	—	4	2	no	—
70	C₂×C₄×C₈₉	—	4	2	no	—
71	C₂×C₄×C₈₉	—	4	2	no	—
72	C₂×C₄×C₈₉	—	4	2	no	—
73	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#124	1	2	yes	—
76	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
77	C₂×C₄×C₈₉	#135	1	2	yes	—
78	C₂×C₄×C₈₉	#137	1	2	yes	—
79	C₂×Dic₈₉	—	4	2	no	—
80	Dic₁₇₈	—	8	2	no	—
81	#10623494	—	8	2	no	—
82	#10623494	—	8	1	no	—
83	C₂²×C₈₉	—	2	2	no	—
84	C₂²×C₈₉	#393	1	1	yes	characteristic
85	C₂²×C₈₉	#452	1	2	yes	—
86	C₄×C₈₉	—	2	2	no	—
87	C₄×C₈₉	—	2	2	no	—
88	C₄×C₈₉	—	4	4	no	—
89	C₄×C₈₉	—	8	2	no	—
90	C₄×C₈₉	—	8	2	no	—
91	C₄×C₈₉	#479	1	2	yes	—
92	C₄×C₈₉	#488	1	1	yes	characteristic
93	C₄×C₈₉	#491	1	1	yes	characteristic
94	Dic₈₉	—	8	2	no	—
95	#10373	—	89	1	no	maximal, Sylow
96	C₂×C₈₉	—	2	2	no	—
97	C₂×C₈₉	#2026	1	1	yes	characteristic
98	C₂×C₈₉	#2031	1	2	yes	—
99	#2402	D₈₉	1	1	yes	characteristic
100	#1995	—	89	1	no	—
101	#1503	—	89	2	no	—
102	#1303	—	89	2	no	—
103	#1274	—	89	1	no	—
104	C₈₉	#10373	1	1	yes	characteristic, complemented, Sylow
105	#533	—	89	1	no	—
106	#431	—	89	2	no	—
107	#423	—	89	1	no	—
108	#400	—	2	2	no	—
109	#398	—	2	2	no	—
110	C₄×C₄⋊C₄	—	2	2	no	—
111	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
112	#376	C₂×D₈₉	1	1	yes	characteristic
113	#339	—	178	2	no	—
114	#328	—	178	2	no	—
115	#328	—	178	2	no	—
116	C₂×M₁₆	—	178	1	no	—
117	C₄⋊Q₈	—	89	2	no	—
118	#125	—	178	1	no	—
119	C₂²⋊Q₈	—	178	2	no	—
120	C₄×Q₈	—	178	2	no	—
121	#117	—	178	1	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
127	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
128	C₄⋊C₈	—	178	2	no	—
129	C₄⋊C₈	—	178	2	no	—
130	C₂²⋊C₈	—	178	2	no	—
131	C₄×C₈	—	178	2	no	—
132	#95	—	2	2	no	—
133	#95	—	2	2	no	—
134	#95	—	2	2	no	—
135	#95	C₂²⋊D₈₉	1	2	yes	—
136	#95	C₂²⋊D₈₉	1	2	yes	—
137	C₂×Q₈	—	178	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
143	M₁₆	—	356	2	no	—
144	C₂×C₈	—	356	2	no	—
145	C₂×C₈	—	356	1	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	356	2	no	—
150	C₄⋊C₄	—	178	2	no	—
151	C₄⋊C₄	—	178	2	no	—
152	C₂²⋊C₄	—	356	1	no	—
153	C₄²	—	2	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	4	2	no	—
156	C₄²	—	4	2	no	—
157	C₄²	—	356	1	no	—
158	C₄²	Q₈×D₈₉	1	1	yes	characteristic
159	C₄²	#13083575	1	1	yes	characteristic
160	C₄²	#13083575	1	2	yes	—
161	C₂³	#15667981	1	1	yes	characteristic
162	Q₈	—	712	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	356	2	no	—
174	C₂×C₄	#15667988	1	2	yes	—
175	C₂×C₄	#15667988	1	1	yes	characteristic
176	C₂×C₄	#15668001	1	2	yes	—
177	C₂×C₄	#15668001	1	1	yes	characteristic
178	C₂×C₄	#15668032	1	2	yes	—
179	C₂×C₄	#15668042	1	2	yes	—
180	C₈	—	712	2	no	—
181	C₈	—	712	1	no	—
182	C₂²	—	2	2	no	—
183	C₂²	#19151509	1	1	yes	characteristic
184	C₂²	#19151760	1	2	yes	—
185	C₄	—	2	2	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	4	no	—
188	C₄	—	8	2	no	—
189	C₄	—	8	2	no	—
190	C₄	—	712	2	no	—
191	C₄	#19151902	1	2	yes	—
192	C₄	#19151955	1	1	yes	characteristic
193	C₄	#19151968	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	#26021400	1	1	yes	characteristic
196	C₂	#26021421	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26021233

C₂

yes

maximal, characteristic

#26018750

C₂

yes

maximal

#26018155

C₂

yes

maximal

#26018081

C₂

yes

maximal, characteristic

C₈₉×#2402

C₂

yes

maximal, characteristic

#19152155

C₂²

yes

characteristic

#19151651

C₂²

yes

—

#19151622

C₂²

yes

—

—

—

#400×C₈₉

—

#398×C₈₉

—

C₄×C₄⋊C₄×C₈₉

—

C₄×C₄⋊C₄×C₈₉

C₂²

yes

—

#376×C₈₉

C₂²

yes

characteristic

C₂×#13083558

—

#15668012

Q₈

yes

—

#15668004

—

#15667991

—

C₄×Dic₁₇₈

—

—

—

—

—

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#95×C₈₉

—

#95×C₈₉

—

#95×C₈₉

—

#95×C₈₉

D₄

yes

—

#95×C₈₉

D₄

yes

—

C₂×Dic₁₇₈

—

#13083558

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₄²×C₈₉

yes

C₂³×C₈₉

yes

C₂×C₄×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

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

Dic₁₇₈

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#393

yes

characteristic

C₂²×C₈₉

#452

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#479

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

#10373

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#2026

yes

characteristic

C₂×C₈₉

#2031

yes

—

#2402

D₈₉

yes

characteristic

100

#1995

—

101

#1503

—

102

#1303

—

103

#1274

—

104

C₈₉

#10373

yes

characteristic, complemented, Sylow

105

#533

—

106

#431

—

107

#423

—

108

#400

—

109

#398

—

110

C₄×C₄⋊C₄

—

111

C₄×C₄⋊C₄

C₂×D₈₉

yes

—

112

#376

C₂×D₈₉

yes

characteristic

113

#339

—

178

—

114

#328

—

178

—

115

#328

—

178

—

116

C₂×M₁₆

—

178

—

117

C₄⋊Q₈

—

118

#125

—

178

—

119

C₂²⋊Q₈

—

178

—

120

C₄×Q₈

—

178

—

121

#117

—

178

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

—

126

C₂×C₄²

C₂²⋊D₈₉

yes

—

127

C₂×C₄²

C₂²×D₈₉

yes

characteristic

128

C₄⋊C₈

—

178

—

129

C₄⋊C₈

—

178

—

130

C₂²⋊C₈

—

178

—

131

C₄×C₈

—

178

—

132

#95

—

133

#95

—

134

#95

—

135

#95

C₂²⋊D₈₉

yes

—

136

#95

C₂²⋊D₈₉

yes

—

137

C₂×Q₈

—

178

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

—

142

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

143

M₁₆

—

356

—

144

C₂×C₈

—

356

—

145

C₂×C₈

—

356

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

356

—

150

C₄⋊C₄

—

178

—

151

C₄⋊C₄

—

178

—

152

C₂²⋊C₄

—

356

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

356

—

158

C₄²

Q₈×D₈₉

yes

characteristic

159

C₄²

#13083575

yes

characteristic

160

C₄²

#13083575

yes

—

161

C₂³

#15667981

yes

characteristic

162

Q₈

—

712

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

356

—

174

C₂×C₄

#15667988

yes

—

175

C₂×C₄

#15667988

yes

characteristic

176

C₂×C₄

#15668001

yes

—

177

C₂×C₄

#15668001

yes

characteristic

178

C₂×C₄

#15668032

yes

—

179

C₂×C₄

#15668042

yes

—

180

C₈

—

712

—

181

C₈

—

712

—

182

C₂²

—

183

C₂²

#19151509

yes

characteristic

184

C₂²

#19151760

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

712

—

191

C₄

#19151902

yes

—

192

C₄

#19151955

yes

characteristic

193

C₄

#19151968

yes

characteristic

194

C₂

—

195

C₂

#26021400

yes

characteristic

196

C₂

#26021421

yes

—

	subgroup	G/H
center	C₂×C₄	#15668001
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#2402	C₂
socle	C₂²×C₈₉	#393
prime radical	C₂³×C₈₉	#120
hypercenter	#2402	D₈₉
hypocenter	C₈₉	#10373
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2402

subgroup

G/H

center

C₂×C₄

#15668001

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#2402

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2402

N	H ≅ G/N	type
104. C₈₉	#10373	semidirect

H ≅ G/N

type

104. C₈₉

#10373

semidirect

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

subgroup	index
#26021233	2
#26018750	2
#26018155	2
#26018081	2
C₈₉×#2402	2
#10373	89

subgroup

index

C₈₉×#2402

Subgroups — #36457568

Conjugacy classes of subgroups

Complements

Maximal subgroups