Group #36470645 — subgroups

Subgroups — #36470645

Order 22784.

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

normal subgroups only

This group has 478 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	#26021863	C₂	1	2	yes	maximal
2	#26018336	C₂	1	4	yes	maximal, complemented
3	C₂²×#97×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667952	C₂²	1	4	yes	—
5	C₂²⋊#13083549	—	2	4	no	—
6	C₂³×C₈×C₈₉	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₈₉	C₄	1	1	yes	characteristic
9	C₂×#97×C₈₉	—	2	4	no	—
10	C₂×#97×C₈₉	C₄	1	2	yes	—
11	C₂×#97×C₈₉	C₂²	1	2	yes	—
12	#15667952	—	2	8	no	—
13	C₂×#13083549	—	2	4	no	—
14	C₂³×C₄×C₈₉	C₈	1	2	yes	—
15	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₈₉	—	2	2	no	—
17	C₂²×C₈×C₈₉	—	2	4	no	—
18	C₂²×C₈×C₈₉	—	2	4	no	—
19	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₈₉	D₄	1	4	yes	—
22	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₈₉	—	2	2	no	—
24	C₂×C₄²×C₈₉	—	2	2	no	—
25	C₂×C₄²×C₈₉	C₈	1	2	yes	—
26	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
27	#97×C₈₉	—	2	4	no	—
28	#97×C₈₉	—	2	4	no	—
29	#13083549	—	4	4	no	—
30	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₈₉	—	2	2	no	—
32	C₂²×C₄×C₈₉	—	2	2	no	—
33	C₂²×C₄×C₈₉	—	2	2	no	—
34	C₂²×C₄×C₈₉	—	2	2	no	—
35	C₂²×C₄×C₈₉	—	4	4	no	—
36	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	C₂×C₈	1	4	yes	—
39	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
40	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	—	2	8	no	—
42	C₂×C₈×C₈₉	—	2	2	no	—
43	C₂×C₈×C₈₉	—	2	2	no	—
44	C₂×C₈×C₈₉	—	2	2	no	—
45	C₂×C₈×C₈₉	—	2	8	no	—
46	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
47	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
48	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
49	C₂×C₈×C₈₉	#41	1	4	yes	—
50	C₄²×C₈₉	—	2	4	no	—
51	C₄²×C₈₉	—	2	2	no	—
52	C₄²×C₈₉	—	2	2	no	—
53	C₂³×C₈₉	—	2	4	no	—
54	C₂³×C₈₉	—	2	2	no	—
55	C₂³×C₈₉	C₂²⋊C₈	1	2	yes	—
56	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₈₉	—	2	2	no	—
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	—	2	4	no	—
61	C₂×C₄×C₈₉	—	2	4	no	—
62	C₂×C₄×C₈₉	—	4	4	no	—
63	C₂×C₄×C₈₉	—	4	4	no	—
64	C₂×C₄×C₈₉	—	4	4	no	—
65	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
66	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#117	1	2	yes	—
68	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₈₉	—	2	2	no	—
71	C₈×C₈₉	—	2	4	no	—
72	C₈×C₈₉	#123	1	4	yes	—
73	C₂²×C₈₉	—	2	8	no	—
74	C₂²×C₈₉	—	2	2	no	—
75	C₂²×C₈₉	—	2	2	no	—
76	C₂²×C₈₉	—	2	2	no	—
77	C₂²×C₈₉	#349	1	4	yes	—
78	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
79	C₂²×C₈₉	#431	1	2	yes	—
80	C₄×C₈₉	—	2	2	no	—
81	C₄×C₈₉	—	4	4	no	—
82	C₄×C₈₉	#381	1	1	yes	characteristic
83	C₄×C₈₉	#407	1	1	yes	characteristic
84	C₄×C₈₉	#432	1	2	yes	—
85	#12846	—	89	1	no	maximal, Sylow
86	C₂×C₈₉	—	2	4	no	—
87	C₂×C₈₉	—	2	2	no	—
88	C₂×C₈₉	#1817	1	1	yes	characteristic
89	C₂×C₈₉	C₂×#349	1	2	yes	—
90	C₂²×#97	D₈₉	1	1	yes	characteristic
91	#2112	—	89	2	no	—
92	#1377	—	89	4	no	—
93	C₈₉	#12846	1	1	yes	characteristic, complemented, Sylow
94	C₂³×C₈	Dic₈₉	1	1	yes	characteristic
95	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
96	C₂²×C₄²	Dic₈₉	1	1	yes	characteristic
97	C₂×M₃₂	—	89	4	no	—
98	C₂×#97	—	2	4	no	—
99	C₂×#97	Dic₈₉	1	2	yes	—
100	C₂×#97	C₂×D₈₉	1	2	yes	—
101	C₂²⋊C₁₆	—	178	4	no	—
102	C₂³×C₄	#10623494	1	2	yes	—
103	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	2	4	no	—
106	C₂²×C₈	—	2	4	no	—
107	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
108	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
109	C₂²×C₈	C₂²⋊D₈₉	1	4	yes	—
110	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
111	C₂×C₄²	—	2	2	no	—
112	C₂×C₄²	—	2	2	no	—
113	C₂×C₄²	#10623494	1	2	yes	—
114	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
115	M₃₂	—	178	8	no	—
116	C₂×C₁₆	—	178	4	no	—
117	#97	—	2	4	no	—
118	#97	—	2	4	no	—
119	C₂⁴	C₂×#10623494	1	1	yes	characteristic
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	4	4	no	—
125	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
126	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
127	C₂²×C₄	C₂×#10623494	1	4	yes	—
128	C₂²×C₄	#13083558	1	2	yes	—
129	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
130	C₂×C₈	—	2	8	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	8	no	—
135	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
136	C₂×C₈	#13083558	1	4	yes	—
137	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
138	C₂×C₈	#13083574	1	4	yes	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	4	no	—
141	C₄²	—	2	2	no	—
142	C₁₆	—	356	4	no	—
143	C₂³	—	2	4	no	—
144	C₂³	—	2	2	no	—
145	C₂³	C₂²⋊#10623494	1	2	yes	—
146	C₂³	C₂²×#10623494	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	C₂²⋊#10623494	1	2	yes	—
156	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
157	C₂×C₄	#15667971	1	2	yes	—
158	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
159	C₂×C₄	C₂×#13083558	1	2	yes	—
160	C₈	—	2	2	no	—
161	C₈	—	2	4	no	—
162	C₈	#15667995	1	4	yes	—
163	C₂²	—	2	8	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	#19151380	1	4	yes	—
168	C₂²	C₂×C₂²⋊#10623494	1	1	yes	characteristic
169	C₂²	#19151651	1	2	yes	—
170	C₄	—	2	2	no	—
171	C₄	—	4	4	no	—
172	C₄	#19151461	1	1	yes	characteristic
173	C₄	#19151562	1	1	yes	characteristic
174	C₄	#19151655	1	2	yes	—
175	C₂	—	2	4	no	—
176	C₂	—	2	2	no	—
177	C₂	#26020320	1	1	yes	characteristic
178	C₂	C₂×#19151380	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26021863

C₂

yes

maximal

#26018336

C₂

yes

maximal, complemented

C₂²×#97×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667952

C₂²

yes

—

C₂²⋊#13083549

—

C₂³×C₈×C₈₉

C₄

yes

characteristic

C₂³×C₈×C₈₉

C₂²

yes

characteristic

C₂²×C₄²×C₈₉

C₄

yes

characteristic

C₂×#97×C₈₉

—

C₂×#97×C₈₉

C₄

yes

—

C₂×#97×C₈₉

C₂²

yes

—

#15667952

—

C₂×#13083549

—

C₂³×C₄×C₈₉

C₈

yes

—

C₂³×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

—

C₂²×C₈×C₈₉

—

C₂²×C₈×C₈₉

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₈

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

#97×C₈₉

—

#97×C₈₉

—

#13083549

—

C₂⁴×C₈₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₈

yes

—

C₂²×C₄×C₈₉

M₁₆

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

M₁₆

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂²⋊C₈

yes

—

C₂³×C₈₉

C₂²×C₈

yes

characteristic

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⋊C₈

yes

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#117

yes

—

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×C₄×C₈₉

C₂×M₁₆

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#123

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#349

yes

—

C₂²×C₈₉

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₈₉

#431

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

#12846

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1817

yes

characteristic

C₂×C₈₉

C₂×#349

yes

—

C₂²×#97

D₈₉

yes

characteristic

#2112

—

#1377

—

C₈₉

#12846

yes

characteristic, complemented, Sylow

C₂³×C₈

Dic₈₉

yes

characteristic

C₂³×C₈

C₂×D₈₉

yes

characteristic

C₂²×C₄²

Dic₈₉

yes

characteristic

C₂×M₃₂

—

C₂×#97

—

C₂×#97

Dic₈₉

yes

—

100

C₂×#97

C₂×D₈₉

yes

—

101

C₂²⋊C₁₆

—

178

—

102

C₂³×C₄

#10623494

yes

—

103

C₂³×C₄

C₂×Dic₈₉

yes

characteristic

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂²×C₈

C₂×Dic₈₉

yes

characteristic

108

C₂²×C₈

C₂×Dic₈₉

yes

—

109

C₂²×C₈

C₂²⋊D₈₉

yes

—

110

C₂²×C₈

C₂²×D₈₉

yes

characteristic

111

C₂×C₄²

—

112

C₂×C₄²

—

113

C₂×C₄²

#10623494

yes

—

114

C₂×C₄²

C₂×Dic₈₉

yes

—

115

M₃₂

—

178

—

116

C₂×C₁₆

—

178

—

117

#97

—

118

#97

—

119

C₂⁴

C₂×#10623494

yes

characteristic

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂²⋊Dic₈₉

yes

—

126

C₂²×C₄

C₂×#10623494

yes

characteristic

127

C₂²×C₄

C₂×#10623494

yes

—

128

C₂²×C₄

#13083558

yes

—

129

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

C₂²⋊Dic₈₉

yes

—

136

C₂×C₈

#13083558

yes

—

137

C₂×C₈

C₂×C₂²⋊D₈₉

yes

—

138

C₂×C₈

#13083574

yes

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₁₆

—

356

—

143

C₂³

—

144

C₂³

—

145

C₂³

C₂²⋊#10623494

yes

—

146

C₂³

C₂²×#10623494

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

C₂²⋊#10623494

yes

—

156

C₂×C₄

C₂×C₂²⋊Dic₈₉

yes

characteristic

157

C₂×C₄

#15667971

yes

—

158

C₂×C₄

C₂×#13083558

yes

characteristic

159

C₂×C₄

C₂×#13083558

yes

—

160

C₈

—

161

C₈

—

162

C₈

#15667995

yes

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

#19151380

yes

—

168

C₂²

C₂×C₂²⋊#10623494

yes

characteristic

169

C₂²

#19151651

yes

—

170

C₄

—

171

C₄

—

172

C₄

#19151461

yes

characteristic

173

C₄

#19151562

yes

characteristic

174

C₄

#19151655

yes

—

175

C₂

—

176

C₂

—

177

C₂

#26020320

yes

characteristic

178

C₂

C₂×#19151380

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₈₉
commutator	C₂³×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂²×#97×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₈₉	C₂×C₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂²×#97	D₈₉
hypocenter	C₈₉	#12846
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×#97

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊Dic₈₉

commutator

C₂³×C₈₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂²×#97×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₈₉

C₂×C₈

cosocle

C₂²×C₈×C₈₉

C₂³

hypercenter

C₂²×#97

D₈₉

hypocenter

C₈₉

#12846

LCS

G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×#97

N	H ≅ G/N	type
2. #26018336	C₂	semidirect
93. C₈₉	#12846	semidirect

H ≅ G/N

type

2. #26018336

C₂

semidirect

93. C₈₉

#12846

semidirect

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

subgroup	index
#26021863	2
#26018336	2
C₂²×#97×C₈₉	2
#12846	89

subgroup

index

#26021863

#26018336

C₂²×#97×C₈₉

#12846

Subgroups — #36470645

Conjugacy classes of subgroups

Complements

Maximal subgroups