Group #36470620 — subgroups

Subgroups — #36470620

Order 22784.

354 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 111 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×#19152016	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151427	C₂	1	2	yes	maximal, complemented
3	#26020299	C₂	1	2	yes	maximal
4	#26018356	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1385	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668018	C₂²	1	1	yes	characteristic
7	#19152016	—	2	4	no	—
8	C₂×#15667951	C₄	1	2	yes	—
9	C₂²×#13083549	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×D₈₉	C₄	1	2	yes	—
11	C₂×#15667948	C₂²	1	2	yes	—
12	#19151580	—	2	2	no	—
13	#19151427	—	2	4	no	—
14	#19151356	—	2	2	no	—
15	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#108×C₈₉	C₂²	1	2	yes	—
17	C₂×#13083574	D₄	1	1	yes	characteristic
18	#15668018	—	2	2	no	—
19	#15668018	—	2	2	no	—
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	D₄	1	1	yes	characteristic
22	C₂×#13083557	Q₈	1	2	yes	—
23	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
24	#15667951	—	2	4	no	—
25	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
26	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
27	C₂×#13083549	D₄	1	4	yes	—
28	C₁₆×D₈₉	—	2	4	no	—
29	#15667948	—	2	2	no	—
30	#15667948	—	2	2	no	—
31	#15667930	—	4	2	no	—
32	#15667928	—	4	2	no	—
33	C₂×M₁₆×C₈₉	—	2	2	no	—
34	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	M₃₂×C₈₉	—	2	2	no	—
36	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
37	#108×C₈₉	—	2	2	no	—
38	#108×C₈₉	—	2	2	no	—
39	#13083574	—	2	2	no	—
40	#13083574	—	2	2	no	—
41	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
42	C₂×D₃₅₆	D₈	1	1	yes	characteristic
43	C₂×C₂²⋊D₈₉	Q₁₆	1	2	yes	—
44	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
45	#13083558	—	2	2	no	—
46	#13083558	—	4	2	no	—
47	#13083558	—	4	2	no	—
48	#13083557	—	2	4	no	—
49	C₂×#10623494	—	4	2	no	—
50	C₂×#10623494	C₄⋊C₄	1	2	yes	—
51	C₈×D₈₉	—	2	2	no	—
52	C₈×D₈₉	—	2	2	no	—
53	#13083549	C₂²⋊C₄	1	2	yes	—
54	#13083549	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	M₁₆×C₈₉	—	4	2	no	—
57	M₁₆×C₈₉	—	4	2	no	—
58	C₂×C₈×C₈₉	—	4	2	no	—
59	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₈₉	#41	1	2	yes	—
63	C₁₆×C₈₉	—	2	2	no	—
64	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₂²×D₈₉	#107	1	2	yes	—
66	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#121	1	2	yes	—
68	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₈₉	#135	1	2	yes	—
70	C₂²⋊D₈₉	—	2	4	no	—
71	C₂×Dic₈₉	#107	1	2	yes	—
72	D₃₅₆	—	2	2	no	—
73	C₄×D₈₉	—	2	2	no	—
74	C₄×D₈₉	—	2	2	no	—
75	Dic₁₇₈	—	2	2	no	—
76	C₈×C₈₉	—	8	2	no	—
77	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₈×C₈₉	#117	1	1	yes	characteristic
79	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
80	#10623494	—	2	2	no	—
81	#10623494	—	8	2	no	—
82	C₂²×C₈₉	—	4	1	no	—
83	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
84	C₂²×C₈₉	#467	1	2	yes	—
85	C₂×D₈₉	—	2	2	no	—
86	C₂×D₈₉	—	4	2	no	—
87	C₄×C₈₉	#380	1	1	yes	characteristic
88	C₄×C₈₉	#427	1	1	yes	characteristic
89	C₄×C₈₉	#466	1	2	yes	—
90	Dic₈₉	—	2	2	no	—
91	#12840	—	89	1	no	maximal, Sylow
92	C₂×C₈₉	—	2	1	no	—
93	C₂×C₈₉	—	2	2	no	—
94	C₂×C₈₉	#1808	1	1	yes	characteristic
95	D₈₉	—	4	2	no	—
96	C₂×#504	—	89	1	no	—
97	C₂×#368	—	89	2	no	—
98	#1813	—	89	2	no	—
99	#1385	—	89	1	no	—
100	#1385	D₈₉	1	1	yes	characteristic, complemented
101	C₈₉	#12840	1	1	yes	characteristic, complemented, Sylow
102	C₂×#131	—	89	1	no	—
103	#504	—	178	4	no	—
104	C₂×M₃₂	—	89	2	no	—
105	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
106	C₂²×C₁₆	—	89	1	no	—
107	C₂²×C₁₆	—	89	2	no	—
108	C₂×#108	—	89	2	no	—
109	C₂×#108	C₂×D₈₉	1	2	yes	—
110	#413	—	178	2	no	—
111	#368	—	178	4	no	—
112	#338	—	178	2	no	—
113	C₂×#41	—	89	1	no	—
114	#131	—	178	2	no	—
115	#131	—	178	2	no	—
116	C₂×M₁₆	—	2	2	no	—
117	C₂×M₁₆	—	89	1	no	—
118	C₂×M₁₆	—	89	2	no	—
119	C₂×M₁₆	—	178	2	no	—
120	C₂²×C₈	—	89	2	no	—
121	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	—	178	4	no	—
124	C₂×C₁₆	—	89	1	no	—
125	C₂×C₁₆	—	89	1	no	—
126	C₂×C₁₆	—	89	4	no	—
127	C₂×C₁₆	—	178	4	no	—
128	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
129	#108	—	2	2	no	—
130	#108	—	2	2	no	—
131	#108	—	178	2	no	—
132	#108	—	178	2	no	—
133	#101	—	356	2	no	—
134	#100	—	356	2	no	—
135	#41	—	178	2	no	—
136	#41	—	178	2	no	—
137	C₂×Q₈	—	89	1	no	—
138	C₂×D₄	—	89	2	no	—
139	C₂×D₄	—	89	1	no	—
140	C₂²×C₄	—	89	2	no	—
141	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	356	2	no	—
145	M₁₆	—	356	2	no	—
146	M₁₆	—	178	2	no	—
147	M₁₆	—	178	4	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	89	2	no	—
150	C₂×C₈	—	356	2	no	—
151	C₂×C₈	—	178	2	no	—
152	C₂×C₈	—	178	2	no	—
153	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
154	C₂×C₈	D₄×D₈₉	1	2	yes	—
155	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
156	C₂×C₈	#13083574	1	2	yes	—
157	C₁₆	—	2	2	no	—
158	C₁₆	—	89	2	no	—
159	C₁₆	—	89	2	no	—
160	C₂³	—	89	2	no	—
161	C₂³	Q₁₆×D₈₉	1	1	yes	characteristic
162	Q₈	—	178	2	no	—
163	D₄	—	178	4	no	—
164	D₄	—	178	2	no	—
165	C₂×C₄	—	89	2	no	—
166	C₂×C₄	—	178	2	no	—
167	C₂×C₄	—	178	2	no	—
168	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
169	C₂×C₄	#15667985	1	2	yes	—
170	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
171	C₂×C₄	#15668033	1	2	yes	—
172	C₈	—	8	2	no	—
173	C₈	—	178	2	no	—
174	C₈	—	712	2	no	—
175	C₈	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
176	C₈	#15667969	1	1	yes	characteristic
177	C₈	#15667990	1	2	yes	—
178	C₂²	—	4	1	no	—
179	C₂²	—	356	2	no	—
180	C₂²	—	178	2	no	—
181	C₂²	#107×D₈₉	1	1	yes	characteristic
182	C₂²	#19151838	1	2	yes	—
183	C₄	—	178	2	no	—
184	C₄	#19151457	1	1	yes	characteristic
185	C₄	#19151633	1	1	yes	characteristic
186	C₄	#19151833	1	2	yes	—
187	C₂	—	2	1	no	—
188	C₂	—	2	2	no	—
189	C₂	—	356	2	no	—
190	C₂	#26020280	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19152016

C₂

yes

maximal, characteristic

C₂×#19151427

C₂

yes

maximal, complemented

#26020299

C₂

yes

maximal

#26018356

C₂

yes

maximal, characteristic, complemented

C₈₉×#1385

C₂

yes

maximal, characteristic, complemented

C₂×#15668018

C₂²

yes

characteristic

#19152016

—

C₂×#15667951

C₄

yes

—

C₂²×#13083549

C₂²

yes

characteristic

C₂×C₁₆×D₈₉

C₄

yes

—

C₂×#15667948

C₂²

yes

—

#19151580

—

#19151427

—

#19151356

—

C₂×M₃₂×C₈₉

C₂²

yes

characteristic

C₂×#108×C₈₉

C₂²

yes

—

C₂×#13083574

D₄

yes

—

—

—

D₄

yes

characteristic

C₂×#13083557

Q₈

yes

—

C₂×C₈×D₈₉

C₂×C₄

yes

—

#15667951

—

C₂×#13083549

C₂×C₄

yes

characteristic

C₂×#13083549

C₂×C₄

yes

characteristic

C₂×#13083549

D₄

yes

—

C₁₆×D₈₉

—

—

—

—

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

#108×C₈₉

—

#108×C₈₉

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

D₈

yes

characteristic

C₂×D₃₅₆

D₈

yes

characteristic

C₂×C₂²⋊D₈₉

Q₁₆

yes

—

C₂×C₄×D₈₉

C₄⋊C₄

yes

—

—

—

—

—

—

C₄⋊C₄

yes

—

C₈×D₈₉

—

C₈×D₈₉

—

#13083549

C₂²⋊C₄

yes

—

#13083549

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×Q₁₆

yes

characteristic

C₂²×D₈₉

#107

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#135

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

#107

yes

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₈₉

#117

yes

characteristic

C₈×C₈₉

C₂²⋊Q₈

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×#107

yes

characteristic

C₂²×C₈₉

#467

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

#12840

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1808

yes

characteristic

D₈₉

—

—

—

—

—

100

#1385

D₈₉

yes

characteristic, complemented

101

C₈₉

#12840

yes

characteristic, complemented, Sylow

102

C₂×#131

—

103

#504

—

178

—

104

C₂×M₃₂

—

105

C₂×M₃₂

C₂×D₈₉

yes

characteristic

106

C₂²×C₁₆

—

107

C₂²×C₁₆

—

108

C₂×#108

—

109

C₂×#108

C₂×D₈₉

yes

—

110

#413

—

178

—

111

#368

—

178

—

112

#338

—

178

—

113

C₂×#41

—

114

#131

—

178

—

115

#131

—

178

—

116

C₂×M₁₆

—

117

C₂×M₁₆

—

118

C₂×M₁₆

—

119

C₂×M₁₆

—

178

—

120

C₂²×C₈

—

121

C₂²×C₈

C₂²×D₈₉

yes

characteristic

122

M₃₂

—

123

M₃₂

—

178

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

178

—

128

C₂×C₁₆

C₄×D₈₉

yes

—

129

#108

—

130

#108

—

131

#108

—

178

—

132

#108

—

178

—

133

#101

—

356

—

134

#100

—

356

—

135

#41

—

178

—

136

#41

—

178

—

137

C₂×Q₈

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

D₄×D₈₉

yes

characteristic

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

356

—

145

M₁₆

—

356

—

146

M₁₆

—

178

—

147

M₁₆

—

178

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

356

—

151

C₂×C₈

—

178

—

152

C₂×C₈

—

178

—

153

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

154

C₂×C₈

D₄×D₈₉

yes

—

155

C₂×C₈

Q₈×D₈₉

yes

characteristic

156

C₂×C₈

#13083574

yes

—

157

C₁₆

—

158

C₁₆

—

159

C₁₆

—

160

C₂³

—

161

C₂³

Q₁₆×D₈₉

yes

characteristic

162

Q₈

—

178

—

163

D₄

—

178

—

164

D₄

—

178

—

165

C₂×C₄

—

166

C₂×C₄

—

178

—

167

C₂×C₄

—

178

—

168

C₂×C₄

C₄⋊C₄×D₈₉

yes

characteristic

169

C₂×C₄

#15667985

yes

—

170

C₂×C₄

D₈×D₈₉

yes

characteristic

171

C₂×C₄

#15668033

yes

—

172

C₈

—

173

C₈

—

178

—

174

C₈

—

712

—

175

C₈

C₂²⋊C₄×D₈₉

yes

characteristic

176

C₈

#15667969

yes

characteristic

177

C₈

#15667990

yes

—

178

C₂²

—

179

C₂²

—

356

—

180

C₂²

—

178

—

181

C₂²

#107×D₈₉

yes

characteristic

182

C₂²

#19151838

yes

—

183

C₄

—

178

—

184

C₄

#19151457

yes

characteristic

185

C₄

#19151633

yes

characteristic

186

C₄

#19151833

yes

—

187

C₂

—

188

C₂

—

189

C₂

—

356

—

190

C₂

#26020280

yes

characteristic

	subgroup	G/H
center	C₄	#19151633
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1385	C₂
socle	C₂×C₈₉	#1808
prime radical	C₂×#13083574	D₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1385	D₈₉
hypocenter	C₈₉	#12840
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#19151633

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1385

C₂

C₂×C₈₉

D₄

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
2. C₂×#19151427	C₂	semidirect
4. #26018356	C₂	semidirect
5. C₈₉×#1385	C₂	semidirect
100. #1385	D₈₉	semidirect
101. C₈₉	#12840	semidirect

H ≅ G/N

type

2. C₂×#19151427

C₂

semidirect

4. #26018356

C₂

semidirect

5. C₈₉×#1385

C₂

semidirect

100. #1385

D₈₉

semidirect

101. C₈₉

#12840

semidirect

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

subgroup	index
C₂×#19152016	2
C₂×#19151427	2
#26020299	2
#26018356	2
C₈₉×#1385	2
#12840	89

subgroup

index

C₈₉×#1385

Subgroups — #36470620

Conjugacy classes of subgroups

Complements

Maximal subgroups