Group #36474302 — subgroups

Subgroups — #36474302

Order 22784.

284 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 87 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	#26021865	C₂	1	1	yes	maximal, characteristic
2	#26020317	C₂	1	2	yes	maximal
3	#26018360	C₂	1	2	yes	maximal, complemented
4	#26018346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#366×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668018	C₂²	1	1	yes	characteristic
7	C₂×#15667952	C₂²	1	1	yes	characteristic
8	C₂×#15667948	C₂²	1	2	yes	—
9	#19151597	—	2	2	no	—
10	#19151377	—	2	2	no	—
11	#19151367	—	2	2	no	—
12	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×#107×C₈₉	C₂²	1	2	yes	—
14	#366×C₈₉	—	2	2	no	—
15	C₂×#13083574	D₄	1	1	yes	characteristic
16	#15668018	—	2	2	no	—
17	#15668018	—	2	2	no	—
18	C₂×#13083558	—	2	2	no	—
19	C₂×#13083558	D₄	1	1	yes	characteristic
20	C₂×#13083557	—	2	1	no	—
21	C₂×C₈×D₈₉	—	2	1	no	—
22	#15667952	—	2	2	no	—
23	C₂×#13083549	—	2	1	no	—
24	#15667948	—	2	2	no	—
25	#15667948	—	2	2	no	—
26	#15667935	—	4	2	no	—
27	#15667932	—	4	2	no	—
28	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂×C₁₆×C₈₉	—	2	2	no	—
31	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
32	#107×C₈₉	—	2	2	no	—
33	#107×C₈₉	—	2	2	no	—
34	#13083574	—	2	2	no	—
35	#13083574	—	2	2	no	—
36	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
37	C₂×D₃₅₆	D₈	1	1	yes	characteristic
38	C₂×C₂²⋊D₈₉	—	2	1	no	—
39	C₂×C₄×D₈₉	—	2	1	no	—
40	#13083558	—	2	2	no	—
41	#13083558	—	4	2	no	—
42	#13083558	—	4	2	no	—
43	#13083557	—	4	2	no	—
44	C₂×#10623494	—	2	1	no	—
45	C₂×#10623494	—	4	2	no	—
46	C₈×D₈₉	—	4	1	no	—
47	C₈×D₈₉	—	4	1	no	—
48	#13083549	—	4	1	no	—
49	C₂²×C₄×C₈₉	—	4	2	no	—
50	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	#41	1	2	yes	—
53	C₂×C₈×C₈₉	#41	1	2	yes	—
54	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
55	C₄⋊C₄×C₈₉	—	4	2	no	—
56	C₄⋊C₄×C₈₉	—	4	2	no	—
57	C₁₆×C₈₉	—	2	2	no	—
58	C₂³×C₈₉	#135	1	1	yes	characteristic
59	C₂²×D₈₉	—	2	1	no	—
60	C₂×C₄×C₈₉	—	8	2	no	—
61	C₂×C₄×C₈₉	—	16	2	no	—
62	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
63	C₂×C₄×C₈₉	#123	1	2	yes	—
64	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₈₉	#137	1	2	yes	—
66	C₂²⋊D₈₉	—	4	2	no	—
67	C₂×Dic₈₉	—	2	1	no	—
68	D₃₅₆	—	2	2	no	—
69	C₄×D₈₉	—	4	1	no	—
70	C₄×D₈₉	—	4	1	no	—
71	Dic₁₇₈	—	2	2	no	—
72	C₈×C₈₉	#123	1	1	yes	characteristic
73	C₈×C₈₉	#124	1	1	yes	characteristic
74	C₈×C₈₉	#126	1	2	yes	—
75	#10623494	—	4	1	no	—
76	#10623494	—	8	2	no	—
77	C₂²×C₈₉	—	2	1	no	—
78	C₂²×C₈₉	—	2	1	no	—
79	C₂²×C₈₉	#466	1	1	yes	characteristic
80	C₂²×C₈₉	#485	1	2	yes	—
81	C₂×D₈₉	—	4	1	no	—
82	C₂×D₈₉	—	4	1	no	—
83	C₂×D₈₉	—	4	1	no	—
84	C₄×C₈₉	—	16	2	no	—
85	C₄×C₈₉	#397	1	1	yes	characteristic
86	C₄×C₈₉	#470	1	1	yes	characteristic
87	C₄×C₈₉	#480	1	2	yes	—
88	Dic₈₉	—	4	1	no	—
89	#13495	—	89	1	no	maximal, Sylow
90	C₂×C₈₉	—	2	2	no	—
91	C₂×C₈₉	#2075	1	1	yes	characteristic
92	C₂×C₈₉	#2221	1	1	yes	characteristic
93	C₂×C₈₉	#2222	1	1	yes	characteristic
94	D₈₉	—	8	1	no	—
95	C₂×#366	D₈₉	1	1	yes	characteristic, complemented
96	#2113	—	89	1	no	—
97	#1816	—	89	2	no	—
98	#1387	—	89	2	no	—
99	#1381	—	89	1	no	—
100	C₈₉	#13495	1	1	yes	characteristic, complemented, Sylow
101	C₂×#131	—	89	1	no	—
102	C₂×M₃₂	—	89	1	no	—
103	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
104	C₂×#108	—	89	2	no	—
105	C₂×#107	C₂×D₈₉	1	2	yes	—
106	#417	—	178	2	no	—
107	#366	—	2	2	no	—
108	C₂²⋊C₁₆	—	178	2	no	—
109	#343	—	178	2	no	—
110	C₂×#41	—	89	1	no	—
111	#131	—	178	2	no	—
112	#131	—	178	2	no	—
113	C₂×M₁₆	—	89	1	no	—
114	C₂×M₁₆	—	178	2	no	—
115	C₂×M₁₆	—	178	1	no	—
116	C₂²×C₈	—	178	1	no	—
117	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	2	2	no	—
119	M₃₂	—	178	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	178	1	no	—
122	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
123	#108	—	178	2	no	—
124	#108	—	178	2	no	—
125	#107	—	2	2	no	—
126	#107	—	2	2	no	—
127	#103	—	356	2	no	—
128	#102	—	356	2	no	—
129	#41	—	178	2	no	—
130	#41	—	178	2	no	—
131	C₂×Q₈	—	89	1	no	—
132	C₂×D₄	—	89	1	no	—
133	C₂×D₄	—	178	1	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	178	1	no	—
136	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
137	M₁₆	—	356	2	no	—
138	M₁₆	—	356	2	no	—
139	M₁₆	—	356	2	no	—
140	M₁₆	—	178	2	no	—
141	C₂×C₈	—	356	1	no	—
142	C₂×C₈	—	356	1	no	—
143	C₂×C₈	—	356	2	no	—
144	C₂×C₈	—	178	1	no	—
145	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
146	C₂×C₈	#13083573	1	1	yes	characteristic
147	C₂×C₈	#13083574	1	2	yes	—
148	C₂×C₈	#13083575	1	2	yes	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₁₆	—	2	2	no	—
152	C₁₆	—	356	1	no	—
153	C₂³	—	178	1	no	—
154	C₂³	#15668031	1	1	yes	characteristic
155	Q₈	—	178	2	no	—
156	D₄	—	356	2	no	—
157	D₄	—	178	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	16	2	no	—
160	C₂×C₄	—	356	1	no	—
161	C₂×C₄	—	356	1	no	—
162	C₂×C₄	—	178	1	no	—
163	C₂×C₄	#15667983	1	1	yes	characteristic
164	C₂×C₄	#15667997	1	2	yes	—
165	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
166	C₂×C₄	#15668044	1	2	yes	—
167	C₈	—	356	1	no	—
168	C₈	—	712	2	no	—
169	C₈	#15667995	1	1	yes	characteristic
170	C₈	#15668001	1	1	yes	characteristic
171	C₈	#15668007	1	2	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	356	1	no	—
175	C₂²	—	356	1	no	—
176	C₂²	—	356	1	no	—
177	C₂²	#19151831	1	1	yes	characteristic
178	C₂²	#19151941	1	2	yes	—
179	C₄	—	16	2	no	—
180	C₄	—	356	1	no	—
181	C₄	#19151524	1	1	yes	characteristic
182	C₄	#19151853	1	1	yes	characteristic
183	C₄	#19151913	1	2	yes	—
184	C₂	—	2	2	no	—
185	C₂	—	712	1	no	—
186	C₂	#26021676	1	1	yes	characteristic
187	C₂	#26022369	1	1	yes	characteristic
188	C₂	#26022374	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26021865

C₂

yes

maximal, characteristic

#26020317

C₂

yes

maximal

#26018360

C₂

yes

maximal, complemented

#26018346

C₂

yes

maximal, characteristic, complemented

C₂×#366×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#15668018

C₂²

yes

characteristic

C₂×#15667952

C₂²

yes

characteristic

C₂×#15667948

C₂²

yes

—

#19151597

—

#19151377

—

#19151367

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#107×C₈₉

C₂²

yes

—

#366×C₈₉

—

C₂×#13083574

D₄

yes

—

—

—

D₄

yes

characteristic

C₂×#13083557

—

C₂×C₈×D₈₉

—

—

—

—

—

—

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

—

#107×C₈₉

—

#107×C₈₉

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

D₈

yes

characteristic

C₂×D₃₅₆

D₈

yes

characteristic

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

—

—

—

—

—

—

C₈×D₈₉

—

C₈×D₈₉

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

—

C₂³×C₈₉

#135

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

#123

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#137

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₈×C₈₉

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#466

yes

characteristic

C₂²×C₈₉

#485

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

#13495

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

C₂×#366

D₈₉

yes

characteristic, complemented

—

—

—

—

100

C₈₉

#13495

yes

characteristic, complemented, Sylow

101

C₂×#131

—

102

C₂×M₃₂

—

103

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

104

C₂×#108

—

105

C₂×#107

C₂×D₈₉

yes

—

106

#417

—

178

—

107

#366

—

108

C₂²⋊C₁₆

—

178

—

109

#343

—

178

—

110

C₂×#41

—

111

#131

—

178

—

112

#131

—

178

—

113

C₂×M₁₆

—

114

C₂×M₁₆

—

178

—

115

C₂×M₁₆

—

178

—

116

C₂²×C₈

—

178

—

117

C₂²×C₈

C₂²×D₈₉

yes

characteristic

118

C₂×C₄⋊C₄

—

119

M₃₂

—

178

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

178

—

122

C₂×C₁₆

C₂²⋊D₈₉

yes

—

123

#108

—

178

—

124

#108

—

178

—

125

#107

—

126

#107

—

127

#103

—

356

—

128

#102

—

356

—

129

#41

—

178

—

130

#41

—

178

—

131

C₂×Q₈

—

132

C₂×D₄

—

133

C₂×D₄

—

178

—

134

C₂²×C₄

—

135

C₂²×C₄

—

178

—

136

C₂²×C₄

D₄×D₈₉

yes

characteristic

137

M₁₆

—

356

—

138

M₁₆

—

356

—

139

M₁₆

—

356

—

140

M₁₆

—

178

—

141

C₂×C₈

—

356

—

142

C₂×C₈

—

356

—

143

C₂×C₈

—

356

—

144

C₂×C₈

—

178

—

145

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

146

C₂×C₈

#13083573

yes

characteristic

147

C₂×C₈

#13083574

yes

—

148

C₂×C₈

#13083575

yes

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₁₆

—

152

C₁₆

—

356

—

153

C₂³

—

178

—

154

C₂³

#15668031

yes

characteristic

155

Q₈

—

178

—

156

D₄

—

356

—

157

D₄

—

178

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

356

—

161

C₂×C₄

—

356

—

162

C₂×C₄

—

178

—

163

C₂×C₄

#15667983

yes

characteristic

164

C₂×C₄

#15667997

yes

—

165

C₂×C₄

D₈×D₈₉

yes

characteristic

166

C₂×C₄

#15668044

yes

—

167

C₈

—

356

—

168

C₈

—

712

—

169

C₈

#15667995

yes

characteristic

170

C₈

#15668001

yes

characteristic

171

C₈

#15668007

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

356

—

175

C₂²

—

356

—

176

C₂²

—

356

—

177

C₂²

#19151831

yes

characteristic

178

C₂²

#19151941

yes

—

179

C₄

—

180

C₄

—

356

—

181

C₄

#19151524

yes

characteristic

182

C₄

#19151853

yes

characteristic

183

C₄

#19151913

yes

—

184

C₂

—

185

C₂

—

712

—

186

C₂

#26021676

yes

characteristic

187

C₂

#26022369

yes

characteristic

188

C₂

#26022374

yes

characteristic

	subgroup	G/H
center	C₂²	#19151831
commutator, cosocle	C₂²×C₈×C₈₉	C₂³
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#366×C₈₉	C₂
socle	C₂²×C₈₉	#466
prime radical	C₂×#13083574	D₄
hypercenter	C₂×#366	D₈₉
hypocenter	C₈₉	#13495
LCS	G ▷ C₂²×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

subgroup

G/H

center

C₂²

#19151831

commutator, cosocle

C₂²×C₈×C₈₉

C₂³

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂×#366×C₈₉

C₂

C₂²×C₈₉

D₄

D₈₉

C₈₉

G ▷ C₂²×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

N	H ≅ G/N	type
3. #26018360	C₂	semidirect
4. #26018346	C₂	semidirect
5. C₂×#366×C₈₉	C₂	semidirect
95. C₂×#366	D₈₉	semidirect
100. C₈₉	#13495	semidirect

H ≅ G/N

type

3. #26018360

C₂

semidirect

4. #26018346

C₂

semidirect

5. C₂×#366×C₈₉

C₂

semidirect

95. C₂×#366

D₈₉

semidirect

100. C₈₉

#13495

semidirect

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

subgroup	index
#26021865	2
#26020317	2
#26018360	2
#26018346	2
C₂×#366×C₈₉	2
#13495	89

subgroup

index

C₂×#366×C₈₉

Subgroups — #36474302

Conjugacy classes of subgroups

Complements

Maximal subgroups