Group #36562809 — subgroups

Subgroups — C₈×#15668009 #36562809

Order 22784.

1350 conjugacy classes of subgroups; only the 395 normal subgroups are shown.

normal subgroups only

This group has 1350 conjugacy classes of subgroups — too many to draw. Only its 395 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×C₈×D₃₅₆	C₂	1	6	yes	maximal, complemented
2	C₄×#15668009	C₂	1	1	yes	maximal, characteristic
3	#26021046	C₂	1	6	yes	maximal, complemented
4	#26020404	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₈₉	C₂²	1	4	yes	—
7	C₂×#15668009	C₄	1	1	yes	characteristic
8	C₂×C₄×D₃₅₆	C₂²	1	6	yes	—
9	C₈×D₃₅₆	—	2	24	no	—
10	C₂×C₄⋊#10623494	C₂²	1	6	yes	—
11	C₂×#15667924	C₂²	1	12	yes	—
12	#19151500	C₄	1	6	yes	—
13	#19151468	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
16	C₄³×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×D₃₅₆	C₂×C₄	1	6	yes	—
18	C₂²×C₄×D₈₉	C₂³	1	4	yes	—
19	C₂²×#10623494	C₂³	1	4	yes	—
20	C₂×C₈×D₈₉	—	2	24	no	—
21	#15668009	C₈	1	8	yes	direct product factor
22	C₄×D₃₅₆	—	2	12	no	—
23	C₄×D₃₅₆	—	2	12	no	—
24	C₂×C₄⋊Dic₈₉	C₂×C₄	1	6	yes	—
25	C₂×#13083551	C₂×C₄	1	12	yes	—
26	C₄⋊#10623494	—	2	12	no	—
27	#15667924	—	2	24	no	—
28	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
29	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	C₂×C₄	1	3	yes	—
31	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
33	C₄×C₈×C₈₉	—	2	6	no	—
34	C₄×C₈×C₈₉	D₄	1	12	yes	complemented
35	C₂³×D₈₉	C₂²×C₄	1	4	yes	—
36	C₂×D₃₅₆	—	2	12	no	—
37	C₂×D₃₅₆	—	2	12	no	—
38	C₂×D₃₅₆	C₂×C₈	1	24	yes	complemented
39	C₂²×Dic₈₉	C₂²×C₄	1	4	yes	—
40	C₂×C₄×D₈₉	—	2	12	no	—
41	C₂×C₄×D₈₉	—	2	12	no	—
42	C₂×#10623494	—	2	12	no	—
43	C₈×D₈₉	—	4	16	no	—
44	C₄⋊Dic₈₉	—	2	12	no	—
45	#13083551	—	2	24	no	—
46	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
49	C₂×C₈×C₈₉	—	2	6	no	—
50	C₂×C₈×C₈₉	C₂×D₄	1	6	yes	—
51	C₂×C₈×C₈₉	C₂×D₄	1	6	yes	—
52	C₄²×C₈₉	—	2	6	no	—
53	C₄²×C₈₉	C₂×C₈	1	4	yes	complemented
54	C₄²×C₈₉	C₂×D₄	1	6	yes	—
55	C₄²×C₈₉	#41	1	6	yes	—
56	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
57	C₂²×D₈₉	—	2	12	no	—
58	C₂²×D₈₉	—	2	12	no	—
59	C₂²×D₈₉	C₂²×C₈	1	8	yes	—
60	C₂×C₄×C₈₉	—	2	6	no	—
61	C₂×C₄×C₈₉	C₄×D₄	1	6	yes	—
62	C₂×C₄×C₈₉	C₄×D₄	1	6	yes	—
63	C₂×C₄×C₈₉	C₂²×C₈	1	6	yes	—
64	C₂×C₄×C₈₉	#131	1	6	yes	—
65	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
66	C₂×C₄×C₈₉	C₂×#41	1	3	yes	—
67	C₂×Dic₈₉	—	2	12	no	—
68	D₃₅₆	—	2	48	no	—
69	C₄×D₈₉	—	4	4	no	—
70	C₄×D₈₉	—	4	12	no	—
71	C₈×C₈₉	—	2	6	no	—
72	C₈×C₈₉	#127	1	4	yes	complemented
73	#10623494	—	4	4	no	—
74	C₂²×C₈₉	C₂×C₄×D₄	1	3	yes	—
75	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
76	C₂²×C₈₉	C₂×#131	1	3	yes	—
77	C₂×D₈₉	—	2	24	no	—
78	C₂×D₈₉	—	4	4	no	—
79	C₂×D₈₉	—	4	12	no	—
80	C₄×C₈₉	—	2	6	no	—
81	C₄×C₈₉	C₈×D₄	1	12	yes	complemented
82	C₄×C₈₉	C₂×#127	1	1	yes	characteristic
83	C₄×C₈₉	#532	1	3	yes	—
84	Dic₈₉	—	4	4	no	—
85	C₈×#127	—	89	1	no	maximal, Sylow
86	C₂×C₈₉	C₄×#127	1	1	yes	characteristic
87	C₂×C₈₉	C₂×C₈×D₄	1	3	yes	—
88	C₂×C₈₉	#2931	1	3	yes	—
89	D₈₉	—	4	8	no	—
90	C₂×C₈×D₄	—	89	6	no	—
91	C₄×#127	—	89	1	no	—
92	#1954	—	89	6	no	—
93	#1842	—	89	1	no	—
94	C₄²×C₈	D₈₉	1	1	yes	characteristic, complemented
95	C₈₉	C₈×#127	1	1	yes	characteristic, complemented, Sylow
96	C₂³×C₈	—	89	4	no	—
97	C₂×#127	—	89	1	no	—
98	C₂×C₄×D₄	—	89	6	no	—
99	C₈×D₄	—	178	24	no	—
100	C₂×C₄⋊C₈	—	89	6	no	—
101	C₂×C₂²⋊C₈	—	89	12	no	—
102	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
103	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
104	#390	—	89	6	no	—
105	#384	—	89	1	no	—
106	C₄³	C₂×D₈₉	1	1	yes	characteristic
107	C₂²×D₄	—	89	6	no	—
108	C₂³×C₄	—	89	4	no	—
109	C₂²×C₈	—	89	4	no	—
110	C₂²×C₈	—	178	24	no	—
111	C₂²×C₈	C₂²×D₈₉	1	3	yes	—
112	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
113	#127	—	89	8	no	—
114	C₄×D₄	—	178	12	no	—
115	C₄×D₄	—	178	12	no	—
116	C₂×C₄⋊C₄	—	89	6	no	—
117	C₂×C₂²⋊C₄	—	89	12	no	—
118	C₂×C₄²	C₄×D₈₉	1	3	yes	—
119	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
121	C₄⋊C₈	—	178	12	no	—
122	C₂²⋊C₈	—	178	24	no	—
123	C₄×C₈	—	2	6	no	—
124	C₄×C₈	D₃₅₆	1	12	yes	complemented
125	C₂⁴	—	89	4	no	—
126	C₂×D₄	—	89	24	no	—
127	C₂×D₄	—	178	12	no	—
128	C₂×D₄	—	178	12	no	—
129	C₂²×C₄	—	89	4	no	—
130	C₂²×C₄	—	178	12	no	—
131	C₂²×C₄	—	178	12	no	—
132	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
133	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
134	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	—	356	16	no	—
137	C₂×C₈	—	178	12	no	—
138	C₂×C₈	C₂×D₃₅₆	1	6	yes	—
139	C₂×C₈	C₂×D₃₅₆	1	6	yes	—
140	C₄⋊C₄	—	178	12	no	—
141	C₂²⋊C₄	—	178	24	no	—
142	C₄²	—	2	6	no	—
143	C₄²	C₈×D₈₉	1	4	yes	complemented
144	C₄²	C₂×D₃₅₆	1	6	yes	—
145	C₄²	#13083574	1	6	yes	—
146	C₂³	—	89	8	no	—
147	C₂³	—	178	12	no	—
148	C₂³	—	178	12	no	—
149	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
150	D₄	—	178	48	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	356	4	no	—
153	C₂×C₄	—	356	12	no	—
154	C₂×C₄	—	178	12	no	—
155	C₂×C₄	C₄×D₃₅₆	1	6	yes	—
156	C₂×C₄	C₄×D₃₅₆	1	6	yes	—
157	C₂×C₄	C₂×C₈×D₈₉	1	6	yes	—
158	C₂×C₄	#15668018	1	6	yes	—
159	C₂×C₄	C₂²×D₃₅₆	1	3	yes	—
160	C₂×C₄	C₂×#13083574	1	3	yes	—
161	C₈	—	2	6	no	—
162	C₈	—	356	4	no	—
163	C₈	#15668009	1	4	yes	direct product factor
164	C₂²	—	356	4	no	—
165	C₂²	—	356	12	no	—
166	C₂²	—	178	24	no	—
167	C₂²	C₂×C₄×D₃₅₆	1	3	yes	—
168	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
169	C₂²	C₂×#15668018	1	3	yes	—
170	C₄	—	2	6	no	—
171	C₄	—	356	4	no	—
172	C₄	C₈×D₃₅₆	1	12	yes	complemented
173	C₄	C₂×#15668009	1	1	yes	characteristic
174	C₄	#19152148	1	3	yes	—
175	C₂	—	356	8	no	—
176	C₂	C₄×#15668009	1	1	yes	characteristic
177	C₂	C₂×C₈×D₃₅₆	1	3	yes	—
178	C₂	#26028965	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×D₃₅₆

C₂

yes

maximal, complemented

C₄×#15668009

C₂

yes

maximal, characteristic

#26021046

C₂

yes

maximal, complemented

#26020404

C₂

yes

maximal, characteristic, complemented

C₄²×C₈×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₈₉

C₂²

yes

—

C₂×#15668009

C₄

yes

characteristic

C₂×C₄×D₃₅₆

C₂²

yes

—

C₈×D₃₅₆

—

C₂×C₄⋊#10623494

C₂²

yes

—

C₂×#15667924

C₂²

yes

—

#19151500

C₄

yes

—

#19151468

C₄

yes

characteristic

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

characteristic

C₂²×D₃₅₆

C₂×C₄

yes

—

C₂²×C₄×D₈₉

C₂³

yes

—

C₂²×#10623494

C₂³

yes

—

C₂×C₈×D₈₉

—

#15668009

C₈

yes

direct product factor

C₄×D₃₅₆

—

C₄×D₃₅₆

—

C₂×C₄⋊Dic₈₉

C₂×C₄

yes

—

C₂×#13083551

C₂×C₄

yes

—

C₄⋊#10623494

—

#15667924

—

C₂²×C₈×C₈₉

C₂³

yes

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

C₂³

yes

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

complemented

C₂³×D₈₉

C₂²×C₄

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂×C₈

yes

complemented

C₂²×Dic₈₉

C₂²×C₄

yes

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₂×#10623494

—

C₈×D₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×C₈

yes

complemented

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂³×C₄

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

C₂²×C₈

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

C₂²×C₈

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₂×C₄×C₈₉

C₂²×D₄

yes

—

C₂×C₄×C₈₉

C₂×#41

yes

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#127

yes

complemented

#10623494

—

C₂²×C₈₉

C₂×C₄×D₄

yes

—

C₂²×C₈₉

C₂³×C₈

yes

characteristic

C₂²×C₈₉

C₂×#131

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₈×D₄

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

C₂×C₈×D₄

yes

—

C₂×C₈₉

#2931

yes

—

D₈₉

—

C₂×C₈×D₄

—

C₄×#127

—

#1954

—

#1842

—

C₄²×C₈

D₈₉

yes

characteristic, complemented

C₈₉

C₈×#127

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂×#127

—

C₂×C₄×D₄

—

C₈×D₄

—

178

—

100

C₂×C₄⋊C₈

—

101

C₂×C₂²⋊C₈

—

102

C₂×C₄×C₈

C₂×D₈₉

yes

—

103

C₂×C₄×C₈

C₂×D₈₉

yes

—

104

#390

—

105

#384

—

106

C₄³

C₂×D₈₉

yes

characteristic

107

C₂²×D₄

—

108

C₂³×C₄

—

109

C₂²×C₈

—

110

C₂²×C₈

—

178

—

111

C₂²×C₈

C₂²×D₈₉

yes

—

112

C₂²×C₈

C₂²×D₈₉

yes

characteristic

113

#127

—

114

C₄×D₄

—

178

—

115

C₄×D₄

—

178

—

116

C₂×C₄⋊C₄

—

117

C₂×C₂²⋊C₄

—

118

C₂×C₄²

C₄×D₈₉

yes

—

119

C₂×C₄²

C₄×D₈₉

yes

characteristic

120

C₂×C₄²

C₂²×D₈₉

yes

—

121

C₄⋊C₈

—

178

—

122

C₂²⋊C₈

—

178

—

123

C₄×C₈

—

124

C₄×C₈

D₃₅₆

yes

complemented

125

C₂⁴

—

126

C₂×D₄

—

127

C₂×D₄

—

178

—

128

C₂×D₄

—

178

—

129

C₂²×C₄

—

130

C₂²×C₄

—

178

—

131

C₂²×C₄

—

178

—

132

C₂²×C₄

C₂×C₄×D₈₉

yes

—

133

C₂²×C₄

C₂×C₄×D₈₉

yes

—

134

C₂²×C₄

C₂³×D₈₉

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

356

—

137

C₂×C₈

—

178

—

138

C₂×C₈

C₂×D₃₅₆

yes

—

139

C₂×C₈

C₂×D₃₅₆

yes

—

140

C₄⋊C₄

—

178

—

141

C₂²⋊C₄

—

178

—

142

C₄²

—

143

C₄²

C₈×D₈₉

yes

complemented

144

C₄²

C₂×D₃₅₆

yes

—

145

C₄²

#13083574

yes

—

146

C₂³

—

147

C₂³

—

178

—

148

C₂³

—

178

—

149

C₂³

C₂²×C₄×D₈₉

yes

characteristic

150

D₄

—

178

—

151

C₂×C₄

—

152

C₂×C₄

—

356

—

153

C₂×C₄

—

356

—

154

C₂×C₄

—

178

—

155

C₂×C₄

C₄×D₃₅₆

yes

—

156

C₂×C₄

C₄×D₃₅₆

yes

—

157

C₂×C₄

C₂×C₈×D₈₉

yes

—

158

C₂×C₄

#15668018

yes

—

159

C₂×C₄

C₂²×D₃₅₆

yes

—

160

C₂×C₄

C₂×#13083574

yes

—

161

C₈

—

162

C₈

—

356

—

163

C₈

#15668009

yes

direct product factor

164

C₂²

—

356

—

165

C₂²

—

356

—

166

C₂²

—

178

—

167

C₂²

C₂×C₄×D₃₅₆

yes

—

168

C₂²

C₂²×C₈×D₈₉

yes

characteristic

169

C₂²

C₂×#15668018

yes

—

170

C₄

—

171

C₄

—

356

—

172

C₄

C₈×D₃₅₆

yes

complemented

173

C₄

C₂×#15668009

yes

characteristic

174

C₄

#19152148

yes

—

175

C₂

—

356

—

176

C₂

C₄×#15668009

yes

characteristic

177

C₂

C₂×C₈×D₃₅₆

yes

—

178

C₂

#26028965

yes

—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄²×C₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂³×C₄
prime radical	C₂×#15668009	C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄²×C₈	D₈₉
hypocenter	C₈₉	C₈×#127
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

subgroup

G/H

center

C₂²×C₈

C₂²×D₈₉

commutator

C₂²×C₈₉

C₂³×C₈

Frattini

C₂²×C₄

C₂³×D₈₉

Fitting

C₄²×C₈×C₈₉

C₂

socle

C₂³×C₈₉

C₂³×C₄

prime radical

C₂×#15668009

C₄

cosocle

C₂²×C₄×C₈₉

C₂⁴

hypercenter

C₄²×C₈

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
1. C₂×C₈×D₃₅₆	C₂	semidirect
3. #26021046	C₂	semidirect
4. #26020404	C₂	semidirect
5. C₄²×C₈×C₈₉	C₂	semidirect
21. #15668009	C₈	direct (N × H)
34. C₄×C₈×C₈₉	D₄	semidirect
38. C₂×D₃₅₆	C₂×C₈	semidirect
53. C₄²×C₈₉	C₂×C₈	semidirect
72. C₈×C₈₉	#127	semidirect
81. C₄×C₈₉	C₈×D₄	semidirect
94. C₄²×C₈	D₈₉	semidirect
95. C₈₉	C₈×#127	semidirect
124. C₄×C₈	D₃₅₆	semidirect
143. C₄²	C₈×D₈₉	semidirect
163. C₈	#15668009	direct (N × H)
172. C₄	C₈×D₃₅₆	semidirect

H ≅ G/N

type

1. C₂×C₈×D₃₅₆

C₂

semidirect

3. #26021046

C₂

semidirect

4. #26020404

C₂

semidirect

5. C₄²×C₈×C₈₉

C₂

semidirect

21. #15668009

C₈

direct (N × H)

34. C₄×C₈×C₈₉

D₄

semidirect

38. C₂×D₃₅₆

C₂×C₈

semidirect

53. C₄²×C₈₉

C₂×C₈

semidirect

72. C₈×C₈₉

#127

semidirect

81. C₄×C₈₉

C₈×D₄

semidirect

94. C₄²×C₈

D₈₉

semidirect

95. C₈₉

C₈×#127

semidirect

124. C₄×C₈

D₃₅₆

semidirect

143. C₄²

C₈×D₈₉

semidirect

163. C₈

#15668009

direct (N × H)

172. C₄

C₈×D₃₅₆

semidirect

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

subgroup	index
C₂×C₈×D₃₅₆	2
C₄×#15668009	2
#26021046	2
#26020404	2
C₄²×C₈×C₈₉	2
C₈×#127	89

subgroup

index

C₂×C₈×D₃₅₆

C₄²×C₈×C₈₉

Subgroups — C₈×#15668009 #36562809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×#15668009 #36562809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#15668009 #36562809