Group #36388533 — subgroups

Subgroups — C₈₉×#10685 #36388533

Order 22784.

442 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 178 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	C₈₉×#1990	C₂	1	2	yes	maximal
2	C₈₉×#1956	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1844	C₂	1	2	yes	maximal
4	C₈₉×#1844	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
7	C₂×#106×C₈₉	—	2	2	no	—
8	C₂×#106×C₈₉	—	2	4	no	—
9	C₂×#103×C₈₉	—	2	2	no	—
10	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
12	#384×C₈₉	C₂²	1	1	yes	characteristic
13	#384×C₈₉	C₂²	1	2	yes	—
14	#340×C₈₉	—	2	4	no	—
15	C₂²×Q₈×C₈₉	—	2	1	no	—
16	C₂²×C₈×C₈₉	D₄	1	2	yes	—
17	C₂²×C₈×C₈₉	D₄	1	4	yes	—
18	C₄⋊Q₈×C₈₉	—	2	2	no	—
19	C₄⋊Q₈×C₈₉	—	2	1	no	—
20	C₄⋊Q₈×C₈₉	Q₈	1	2	yes	—
21	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
22	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
23	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
27	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
28	#106×C₈₉	—	4	4	no	—
29	#106×C₈₉	—	4	8	no	—
30	#103×C₈₉	—	4	4	no	—
31	C₄×C₈×C₈₉	—	2	2	no	—
32	C₄×C₈×C₈₉	—	2	2	no	—
33	C₄×C₈×C₈₉	—	2	2	no	—
34	C₂×Q₈×C₈₉	—	2	2	no	—
35	C₂×Q₈×C₈₉	—	4	1	no	—
36	C₂×Q₈×C₈₉	—	4	2	no	—
37	C₂×Q₈×C₈₉	—	4	2	no	—
38	C₂²×C₄×C₈₉	—	4	1	no	—
39	C₂²×C₄×C₈₉	—	4	1	no	—
40	C₂²×C₄×C₈₉	—	4	2	no	—
41	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
43	C₂×C₈×C₈₉	—	2	2	no	—
44	C₂×C₈×C₈₉	—	2	4	no	—
45	C₂×C₈×C₈₉	—	2	2	no	—
46	C₂×C₈×C₈₉	—	2	4	no	—
47	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
48	C₂×C₈×C₈₉	SD₁₆	1	8	yes	—
49	C₄⋊C₄×C₈₉	—	2	4	no	—
50	C₄⋊C₄×C₈₉	—	4	2	no	—
51	C₄⋊C₄×C₈₉	—	4	2	no	—
52	C₄⋊C₄×C₈₉	—	4	4	no	—
53	C₄⋊C₄×C₈₉	—	4	4	no	—
54	C₄⋊C₄×C₈₉	—	4	4	no	—
55	C₄⋊C₄×C₈₉	—	4	4	no	—
56	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₈₉	#41	1	2	yes	—
58	C₄²×C₈₉	#41	1	1	yes	characteristic
59	C₂³×C₈₉	#127	1	1	yes	characteristic
60	Q₈×C₈₉	—	4	4	no	—
61	Q₈×C₈₉	—	8	2	no	—
62	C₂×C₄×C₈₉	—	4	2	no	—
63	C₂×C₄×C₈₉	—	8	1	no	—
64	C₂×C₄×C₈₉	—	8	1	no	—
65	C₂×C₄×C₈₉	—	8	1	no	—
66	C₂×C₄×C₈₉	—	8	2	no	—
67	C₂×C₄×C₈₉	—	8	2	no	—
68	C₂×C₄×C₈₉	—	8	2	no	—
69	C₂×C₄×C₈₉	—	8	2	no	—
70	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
73	C₂×C₄×C₈₉	#123	1	2	yes	—
74	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
75	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₈₉	#137	1	4	yes	—
78	C₈×C₈₉	—	2	4	no	—
79	C₈×C₈₉	—	2	8	no	—
80	C₂²×C₈₉	#399	1	1	yes	characteristic
81	C₂²×C₈₉	#492	1	1	yes	characteristic
82	C₂²×C₈₉	#492	1	2	yes	—
83	C₂²×C₈₉	#496	1	1	yes	characteristic
84	C₂²×C₈₉	#497	1	2	yes	—
85	C₄×C₈₉	—	8	2	no	—
86	C₄×C₈₉	—	16	1	no	—
87	C₄×C₈₉	—	16	2	no	—
88	C₄×C₈₉	#475	1	4	yes	—
89	C₄×C₈₉	#476	1	2	yes	—
90	C₄×C₈₉	#481	1	2	yes	—
91	C₄×C₈₉	#483	1	4	yes	—
92	#10685	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂×C₈₉	#1705	1	1	yes	characteristic
94	C₂×C₈₉	#1714	1	2	yes	—
95	C₂×C₈₉	#1716	1	1	yes	characteristic
96	C₂×C₈₉	#2094	1	1	yes	characteristic
97	C₂×C₈₉	#2096	1	2	yes	—
98	#1990	C₂×C₈₉	1	2	yes	—
99	#1956	C₂×C₈₉	1	1	yes	characteristic
100	#1844	C₂×C₈₉	1	2	yes	—
101	#1844	C₂×C₈₉	1	1	yes	characteristic
102	#1273	C₂×C₈₉	1	1	yes	characteristic
103	C₈₉	#10685	1	1	yes	characteristic, direct product factor, Sylow
104	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
105	C₂×#106	—	2	2	no	—
106	C₂×#106	—	2	4	no	—
107	C₂×#103	—	2	2	no	—
108	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
110	#384	C₂²×C₈₉	1	1	yes	characteristic
111	#384	C₂²×C₈₉	1	2	yes	—
112	#340	—	2	4	no	—
113	C₂²×Q₈	—	2	1	no	—
114	C₂²×C₈	D₄×C₈₉	1	4	yes	—
115	C₂²×C₈	D₄×C₈₉	1	2	yes	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	—	2	1	no	—
118	C₄⋊Q₈	Q₈×C₈₉	1	2	yes	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	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	C₂³×C₈₉	1	1	yes	characteristic
126	#106	—	4	4	no	—
127	#106	—	4	8	no	—
128	#103	—	4	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	—	4	1	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	SD₁₆×C₈₉	1	4	yes	—
146	C₂×C₈	SD₁₆×C₈₉	1	8	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄⋊C₄	—	4	4	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄²	C₂×Q₈×C₈₉	1	1	yes	characteristic
155	C₄²	#41×C₈₉	1	2	yes	—
156	C₄²	#41×C₈₉	1	1	yes	characteristic
157	C₂³	#127×C₈₉	1	1	yes	characteristic
158	Q₈	—	4	4	no	—
159	Q₈	—	8	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	C₂²⋊Q₈×C₈₉	1	2	yes	—
169	C₂×C₄	C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
170	C₂×C₄	#123×C₈₉	1	1	yes	characteristic
171	C₂×C₄	#123×C₈₉	1	2	yes	—
172	C₂×C₄	C₂×SD₁₆×C₈₉	1	4	yes	—
173	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
174	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
175	C₂×C₄	#137×C₈₉	1	4	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	2	8	no	—
178	C₂²	#399×C₈₉	1	1	yes	characteristic
179	C₂²	#492×C₈₉	1	1	yes	characteristic
180	C₂²	#492×C₈₉	1	2	yes	—
181	C₂²	#496×C₈₉	1	1	yes	characteristic
182	C₂²	#497×C₈₉	1	2	yes	—
183	C₄	—	8	2	no	—
184	C₄	—	16	1	no	—
185	C₄	—	16	2	no	—
186	C₄	#475×C₈₉	1	4	yes	—
187	C₄	#476×C₈₉	1	2	yes	—
188	C₄	#481×C₈₉	1	2	yes	—
189	C₄	#483×C₈₉	1	4	yes	—
190	C₂	C₈₉×#1705	1	1	yes	characteristic
191	C₂	C₈₉×#1714	1	2	yes	—
192	C₂	C₈₉×#1716	1	1	yes	characteristic
193	C₂	C₈₉×#2094	1	1	yes	characteristic
194	C₂	C₈₉×#2096	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉×#1990

C₂

yes

maximal

C₈₉×#1956

C₂

yes

maximal, characteristic

C₈₉×#1844

C₂

yes

maximal

C₈₉×#1844

C₂

yes

maximal, characteristic

C₈₉×#1273

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₂×#106×C₈₉

—

C₂×#106×C₈₉

—

C₂×#103×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₈₉

C₂²

yes

—

#384×C₈₉

C₂²

yes

characteristic

#384×C₈₉

C₂²

yes

—

#340×C₈₉

—

C₂²×Q₈×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂²×C₈×C₈₉

D₄

yes

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

Q₈

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#106×C₈₉

—

#106×C₈₉

—

#103×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₄²×C₈₉

yes

C₂³×C₈₉

yes

Q₈×C₈₉

—

Q₈×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

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#496

yes

characteristic

C₂²×C₈₉

#497

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#475

yes

—

C₄×C₈₉

#476

yes

—

C₄×C₈₉

#481

yes

—

C₄×C₈₉

#483

yes

—

#10685

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

#1705

yes

characteristic

C₂×C₈₉

#1714

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

#1990

C₂×C₈₉

yes

—

#1956

C₂×C₈₉

yes

characteristic

100

#1844

C₂×C₈₉

yes

—

101

#1844

C₂×C₈₉

yes

characteristic

102

#1273

C₂×C₈₉

yes

characteristic

103

C₈₉

#10685

yes

characteristic, direct product factor, Sylow

104

C₂×C₄⋊Q₈

C₂²×C₈₉

yes

characteristic

105

C₂×#106

—

106

C₂×#106

—

107

C₂×#103

—

108

C₂×C₄×C₈

C₂²×C₈₉

yes

characteristic

109

C₂×C₄×C₈

C₂²×C₈₉

yes

—

110

#384

C₂²×C₈₉

yes

characteristic

111

#384

C₂²×C₈₉

yes

—

112

#340

—

113

C₂²×Q₈

—

114

C₂²×C₈

D₄×C₈₉

yes

—

115

C₂²×C₈

D₄×C₈₉

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

Q₈×C₈₉

yes

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

C₂³×C₈₉

yes

characteristic

126

#106

—

127

#106

—

128

#103

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

C₂×D₄×C₈₉

yes

characteristic

140

C₂²×C₄

C₂×D₄×C₈₉

yes

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

SD₁₆×C₈₉

yes

—

146

C₂×C₈

SD₁₆×C₈₉

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄²

C₂×Q₈×C₈₉

yes

characteristic

155

C₄²

#41×C₈₉

yes

—

156

C₄²

#41×C₈₉

yes

characteristic

157

C₂³

#127×C₈₉

yes

characteristic

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

C₂²⋊Q₈×C₈₉

yes

—

169

C₂×C₄

C₂²⋊Q₈×C₈₉

yes

characteristic

170

C₂×C₄

#123×C₈₉

yes

characteristic

171

C₂×C₄

#123×C₈₉

yes

—

172

C₂×C₄

C₂×SD₁₆×C₈₉

yes

—

173

C₂×C₄

C₂×SD₁₆×C₈₉

yes

—

174

C₂×C₄

AGL(1,ℤ/8)×C₈₉

yes

—

175

C₂×C₄

#137×C₈₉

yes

—

176

C₈

—

177

C₈

—

178

C₂²

#399×C₈₉

yes

characteristic

179

C₂²

#492×C₈₉

yes

characteristic

180

C₂²

#492×C₈₉

yes

—

181

C₂²

#496×C₈₉

yes

characteristic

182

C₂²

#497×C₈₉

yes

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

#475×C₈₉

yes

—

187

C₄

#476×C₈₉

yes

—

188

C₄

#481×C₈₉

yes

—

189

C₄

#483×C₈₉

yes

—

190

C₂

C₈₉×#1705

yes

characteristic

191

C₂

C₈₉×#1714

yes

—

192

C₂

C₈₉×#1716

yes

characteristic

193

C₂

C₈₉×#2094

yes

characteristic

194

C₂

C₈₉×#2096

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₈₉

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₈₉

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

N	H ≅ G/N	type
92. #10685	C₈₉	direct (N × H)
103. C₈₉	#10685	direct (N × H)

H ≅ G/N

type

92. #10685

C₈₉

direct (N × H)

103. C₈₉

#10685

direct (N × H)

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

subgroup	index
C₈₉×#1990	2
C₈₉×#1956	2
C₈₉×#1844	2
C₈₉×#1273	2
#10685	89

subgroup

index

C₈₉×#1990

C₈₉×#1956

C₈₉×#1844

C₈₉×#1273

#10685

Subgroups — C₈₉×#10685 #36388533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#10685 #36388533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#10685 #36388533