Group #31891096 — subgroups

Subgroups — #31891096

Order 18176.

310 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 123 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	#22588027	C₂	1	1	yes	maximal, characteristic
2	#22587086	C₂	1	2	yes	maximal
3	#22586184	C₂	1	2	yes	maximal
4	#22586169	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1274	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	2	yes	—
7	C₂×#14556039	C₄	1	1	yes	characteristic
8	C₂×#14556039	C₄	1	1	yes	characteristic
9	C₂×#14556039	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
13	#17985751	—	2	2	no	—
14	#17985749	—	2	2	no	—
15	#431×C₇₁	C₂²	1	2	yes	—
16	#376×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
22	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
23	#14556039	—	2	2	no	—
24	#14556039	—	2	2	no	—
25	#14556039	—	2	2	no	—
26	C₄×C₇₁⋊C₈	—	2	2	no	—
27	C₄×C₇₁⋊C₈	—	2	2	no	—
28	C₄×C₇₁⋊C₈	C₈	1	4	yes	complemented
29	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
30	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
31	C₄×C₇₁⋊C₈	Q₈	1	1	yes	characteristic
32	C₄×C₇₁⋊C₈	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₇₁	—	2	2	no	—
37	C₂²⋊C₈×C₇₁	—	2	2	no	—
38	C₄×C₈×C₇₁	—	2	2	no	—
39	#95×C₇₁	—	2	2	no	—
40	C₂×C₇₁⋊C₈	—	2	1	no	—
41	C₂×C₇₁⋊C₈	—	2	1	no	—
42	C₂×C₇₁⋊C₈	—	2	1	no	—
43	C₂×C₇₁⋊C₈	—	2	1	no	—
44	C₂×C₇₁⋊C₈	—	2	1	no	—
45	C₂×C₇₁⋊C₈	—	2	1	no	—
46	C₂×C₇₁⋊C₈	—	4	2	no	—
47	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
52	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
53	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
54	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
55	C₂²×C₄×C₇₁	—	2	1	no	—
56	C₂²×C₄×C₇₁	—	2	1	no	—
57	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	#41	1	2	yes	—
59	C₂×C₈×C₇₁	—	4	2	no	—
60	C₄²×C₇₁	—	2	1	no	—
61	C₄²×C₇₁	—	2	1	no	—
62	C₄²×C₇₁	C₂×C₈	1	2	yes	—
63	C₄²×C₇₁	M₁₆	1	2	yes	—
64	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₇₁	#41	1	2	yes	—
67	C₂³×C₇₁	#117	1	1	yes	characteristic
68	C₂×C₄×C₇₁	—	2	2	no	—
69	C₂×C₄×C₇₁	—	2	2	no	—
70	C₂×C₄×C₇₁	—	4	2	no	—
71	C₂×C₄×C₇₁	—	4	2	no	—
72	C₂×C₄×C₇₁	—	4	2	no	—
73	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
75	C₂×C₄×C₇₁	#124	1	2	yes	—
76	C₂×C₄×C₇₁	#125	1	2	yes	—
77	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
81	C₈×C₇₁	—	8	2	no	—
82	C₇₁⋊C₈	—	2	1	no	—
83	C₇₁⋊C₈	—	2	1	no	—
84	C₇₁⋊C₈	—	2	2	no	—
85	C₇₁⋊C₈	—	2	2	no	—
86	C₇₁⋊C₈	—	8	2	no	—
87	C₇₁⋊C₈	C₄⋊C₈	1	2	yes	—
88	C₇₁⋊C₈	C₄⋊C₈	1	2	yes	—
89	C₂²×C₇₁	—	2	2	no	—
90	C₂²×C₇₁	#382	1	1	yes	characteristic
91	C₂²×C₇₁	#431	1	1	yes	characteristic
92	C₂²×C₇₁	#432	1	1	yes	characteristic
93	C₄×C₇₁	—	2	2	no	—
94	C₄×C₇₁	—	2	2	no	—
95	C₄×C₇₁	—	4	2	no	—
96	C₄×C₇₁	—	4	2	no	—
97	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₄×C₇₁	#423	1	1	yes	characteristic
99	C₄×C₇₁	#431	1	1	yes	characteristic
100	C₄×C₇₁	#432	1	1	yes	characteristic
101	#8780	—	71	1	no	maximal, Sylow
102	C₂×C₇₁	—	2	1	no	—
103	C₂×C₇₁	—	2	1	no	—
104	C₂×C₇₁	#1831	1	1	yes	characteristic
105	C₂×C₇₁	#1833	1	2	yes	—
106	#1726	—	71	1	no	—
107	#1564	—	71	2	no	—
108	#1281	—	71	2	no	—
109	#1274	—	71	1	no	—
110	#1274	D₇₁	1	1	yes	characteristic
111	C₇₁	#8780	1	1	yes	characteristic, complemented, Sylow
112	#431	—	71	2	no	—
113	#431	C₂×D₇₁	1	2	yes	—
114	C₂×#97	—	71	1	no	—
115	C₂×#97	—	71	1	no	—
116	C₂×#97	—	71	1	no	—
117	C₂×C₄×C₈	—	71	1	no	—
118	C₂×C₄×C₈	—	71	1	no	—
119	C₂×C₄×C₈	—	71	1	no	—
120	#376	C₂×D₇₁	1	1	yes	characteristic
121	#335	—	142	2	no	—
122	#334	—	142	2	no	—
123	C₂²×C₈	—	71	1	no	—
124	C₂²×C₈	—	71	1	no	—
125	C₂²×C₈	—	71	1	no	—
126	C₂²×C₈	—	71	1	no	—
127	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
128	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
129	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	142	2	no	—
132	C₂²⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	142	2	no	—
134	#97	—	142	2	no	—
135	#97	—	142	2	no	—
136	#97	—	142	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	71	4	no	—
139	C₄×C₈	—	71	1	no	—
140	C₄×C₈	—	71	1	no	—
141	C₄×C₈	—	71	1	no	—
142	C₄×C₈	—	71	1	no	—
143	C₄×C₈	—	142	2	no	—
144	C₄×C₈	—	142	2	no	—
145	#95	—	2	2	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
149	C₂²×C₄	#11765864	1	2	yes	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	284	2	no	—
152	C₂×C₈	—	71	1	no	—
153	C₂×C₈	—	71	1	no	—
154	C₂×C₈	—	71	2	no	—
155	C₂×C₈	—	71	2	no	—
156	C₂×C₈	—	71	1	no	—
157	C₂×C₈	—	71	1	no	—
158	C₂×C₈	—	71	2	no	—
159	C₂×C₈	—	71	2	no	—
160	C₂×C₈	—	142	1	no	—
161	C₂×C₈	—	142	1	no	—
162	C₂×C₈	—	142	1	no	—
163	C₂×C₈	—	142	1	no	—
164	C₂×C₈	—	142	1	no	—
165	C₂×C₈	—	142	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	C₈×D₇₁	1	2	yes	—
169	C₄²	#11765848	1	2	yes	—
170	C₄²	D₄×D₇₁	1	1	yes	characteristic
171	C₄²	Q₈×D₇₁	1	1	yes	characteristic
172	C₄²	#11765865	1	2	yes	—
173	C₂³	#14556084	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
180	C₂×C₄	#14556085	1	1	yes	characteristic
181	C₂×C₄	#14556114	1	2	yes	—
182	C₂×C₄	#14556118	1	2	yes	—
183	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
184	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
185	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
186	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
187	C₈	—	8	2	no	—
188	C₈	—	568	2	no	—
189	C₈	—	71	2	no	—
190	C₈	—	71	2	no	—
191	C₈	—	142	2	no	—
192	C₈	—	142	2	no	—
193	C₈	—	142	1	no	—
194	C₈	—	142	1	no	—
195	C₂²	—	2	2	no	—
196	C₂²	#17985863	1	1	yes	characteristic
197	C₂²	#17986049	1	1	yes	characteristic
198	C₂²	#17986053	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	2	no	—
202	C₄	—	4	2	no	—
203	C₄	C₄⋊C₈×D₇₁	1	1	yes	characteristic
204	C₄	#17986021	1	1	yes	characteristic
205	C₄	#17986050	1	1	yes	characteristic
206	C₄	#17986054	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	#22588455	1	1	yes	characteristic
210	C₂	#22588462	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588027

C₂

yes

maximal, characteristic

#22587086

C₂

yes

maximal

#22586184

C₂

yes

maximal

#22586169

C₂

yes

maximal, characteristic

C₇₁×#1274

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

C₂×#14556039

C₄

yes

characteristic

C₂×#14556039

C₄

yes

characteristic

C₂×#14556039

C₂²

yes

characteristic

C₂×C₄×C₇₁⋊C₈

C₄

yes

characteristic

C₂×C₄×C₇₁⋊C₈

C₄

yes

characteristic

C₂×C₄×C₇₁⋊C₈

C₂²

yes

characteristic

#17985751

—

#17985749

—

#431×C₇₁

C₂²

yes

—

#376×C₇₁

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

#14556039

—

#14556039

—

#14556039

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

C₈

yes

complemented

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₄×C₇₁⋊C₈

Q₈

yes

characteristic

C₄×C₇₁⋊C₈

Q₈

yes

characteristic

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

—

#95×C₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₈

yes

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

—

C₂×C₇₁⋊C₈

M₁₆

yes

—

C₂×C₇₁⋊C₈

M₁₆

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×C₈

yes

—

C₄²×C₇₁

M₁₆

yes

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

#117

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

#125

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₄⋊C₈

yes

—

C₇₁⋊C₈

C₄⋊C₈

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₈

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

100

C₄×C₇₁

yes

101

—

102

C₂×C₇₁

—

103

C₂×C₇₁

—

104

C₂×C₇₁

#1831

yes

characteristic

105

C₂×C₇₁

#1833

yes

—

106

#1726

—

107

#1564

—

108

#1281

—

109

#1274

—

110

#1274

D₇₁

yes

characteristic

111

C₇₁

#8780

yes

characteristic, complemented, Sylow

112

#431

—

113

#431

C₂×D₇₁

yes

—

114

C₂×#97

—

115

C₂×#97

—

116

C₂×#97

—

117

C₂×C₄×C₈

—

118

C₂×C₄×C₈

—

119

C₂×C₄×C₈

—

120

#376

C₂×D₇₁

yes

characteristic

121

#335

—

142

—

122

#334

—

142

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₂×C₄²

C₄×D₇₁

yes

characteristic

128

C₂×C₄²

C₄×D₇₁

yes

characteristic

129

C₂×C₄²

C₂²×D₇₁

yes

characteristic

130

C₄⋊C₈

—

131

C₄⋊C₈

—

142

—

132

C₂²⋊C₈

—

133

C₂²⋊C₈

—

142

—

134

#97

—

142

—

135

#97

—

142

—

136

#97

—

142

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

142

—

144

C₄×C₈

—

142

—

145

#95

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

149

C₂²×C₄

#11765864

yes

—

150

C₂×C₈

—

151

C₂×C₈

—

284

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

142

—

161

C₂×C₈

—

142

—

162

C₂×C₈

—

142

—

163

C₂×C₈

—

142

—

164

C₂×C₈

—

142

—

165

C₂×C₈

—

142

—

166

C₄²

—

167

C₄²

—

168

C₄²

C₈×D₇₁

yes

—

169

C₄²

#11765848

yes

—

170

C₄²

D₄×D₇₁

yes

characteristic

171

C₄²

Q₈×D₇₁

yes

characteristic

172

C₄²

#11765865

yes

—

173

C₂³

#14556084

yes

characteristic

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

180

C₂×C₄

#14556085

yes

characteristic

181

C₂×C₄

#14556114

yes

—

182

C₂×C₄

#14556118

yes

—

183

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

184

C₂×C₄

M₁₆×D₇₁

yes

characteristic

185

C₂×C₄

M₁₆×D₇₁

yes

characteristic

186

C₂×C₄

C₂×#11765848

yes

characteristic

187

C₈

—

188

C₈

—

568

—

189

C₈

—

190

C₈

—

191

C₈

—

142

—

192

C₈

—

142

—

193

C₈

—

142

—

194

C₈

—

142

—

195

C₂²

—

196

C₂²

#17985863

yes

characteristic

197

C₂²

#17986049

yes

characteristic

198

C₂²

#17986053

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

C₄⋊C₈×D₇₁

yes

characteristic

204

C₄

#17986021

yes

characteristic

205

C₄

#17986050

yes

characteristic

206

C₄

#17986054

yes

characteristic

207

C₂

—

208

C₂

—

209

C₂

#22588455

yes

characteristic

210

C₂

#22588462

yes

—

	subgroup	G/H
center	C₂×C₄	#14556085
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1274	C₂
socle	C₂²×C₇₁	#382
prime radical	C₂³×C₇₁	#117
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1274	D₇₁
hypocenter	C₇₁	#8780
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

subgroup

G/H

center

C₂×C₄

#14556085

commutator

C₂×C₄×C₇₁

C₂²×C₈

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1274

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

N	H ≅ G/N	type
28. C₄×C₇₁⋊C₈	C₈	semidirect
111. C₇₁	#8780	semidirect

H ≅ G/N

type

28. C₄×C₇₁⋊C₈

C₈

semidirect

111. C₇₁

#8780

semidirect

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

subgroup	index
#22588027	2
#22587086	2
#22586184	2
#22586169	2
C₇₁×#1274	2
#8780	71

subgroup

index

C₇₁×#1274

Subgroups — #31891096

Conjugacy classes of subgroups

Complements

Maximal subgroups