Group #31897274 — subgroups

Subgroups — #31897274

Order 18176.

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

normal subgroups only

This group has 494 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	C₂×#17986404	C₂	1	1	yes	maximal, characteristic
2	#22589300	C₂	1	2	yes	maximal
3	#22589141	C₂	1	2	yes	maximal
4	C₂×#17985721	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556128	C₂²	1	2	yes	—
7	#17986404	—	2	2	no	—
8	#17986404	C₄	1	4	yes	complemented
9	C₂×#14556062	—	2	2	no	—
10	C₂×#14556051	—	2	4	no	—
11	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
13	#17985761	—	2	4	no	—
14	#17985721	—	2	2	no	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	#384×C₇₁	C₂²	1	2	yes	—
17	#340×C₇₁	—	2	2	no	—
18	C₂²×Dic₁₄₂	—	2	2	no	—
19	C₂²×C₇₁⋊C₈	D₄	1	2	yes	—
20	C₂²×C₇₁⋊C₈	D₄	1	2	yes	—
21	#14556128	—	2	2	no	—
22	#14556128	—	2	2	no	—
23	#14556128	—	2	2	no	—
24	#14556128	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₇₁	—	2	2	no	—
26	C₂×C₄⋊Dic₇₁	—	2	2	no	—
27	#14556062	—	2	8	no	—
28	#14556051	—	4	8	no	—
29	C₄×C₇₁⋊C₈	—	2	1	no	—
30	C₄×C₇₁⋊C₈	—	2	1	no	—
31	C₄×C₇₁⋊C₈	—	2	1	no	—
32	C₄×C₇₁⋊C₈	—	2	1	no	—
33	C₂²×C₈×C₇₁	D₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
37	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₁	D₄	1	2	yes	—
40	C₂×Dic₁₄₂	—	2	4	no	—
41	C₂×Dic₁₄₂	—	4	4	no	—
42	C₂×Dic₁₄₂	—	4	4	no	—
43	C₂×Dic₁₄₂	—	4	2	no	—
44	C₂²×Dic₇₁	—	4	2	no	—
45	C₂×C₇₁⋊C₈	—	2	2	no	—
46	C₂×C₇₁⋊C₈	—	2	2	no	—
47	C₂×C₇₁⋊C₈	—	2	2	no	—
48	C₂×C₇₁⋊C₈	—	2	4	no	—
49	C₂×C₇₁⋊C₈	Q₁₆	1	4	yes	—
50	C₄⋊Dic₇₁	—	2	4	no	—
51	C₄⋊Dic₇₁	—	2	4	no	—
52	C₄⋊Dic₇₁	—	4	2	no	—
53	C₄⋊Dic₇₁	—	4	2	no	—
54	C₂²×C₄×C₇₁	—	4	2	no	—
55	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	—	2	4	no	—
59	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
60	C₄⋊C₄×C₇₁	—	4	4	no	—
61	C₄⋊C₄×C₇₁	—	4	4	no	—
62	C₄⋊C₄×C₇₁	—	4	4	no	—
63	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₇₁	#41	1	1	yes	characteristic
66	C₄²×C₇₁	#41	1	1	yes	characteristic
67	C₂³×C₇₁	#127	1	1	yes	characteristic
68	C₂×C₄×C₇₁	—	8	2	no	—
69	C₂×C₄×C₇₁	—	8	2	no	—
70	C₂×C₄×C₇₁	—	8	2	no	—
71	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
75	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
76	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
78	C₂×C₄×C₇₁	#135	1	2	yes	—
79	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×C₄×C₇₁	#137	1	4	yes	—
81	C₂×Dic₇₁	—	4	4	no	—
82	C₂×Dic₇₁	—	8	2	no	—
83	C₂×Dic₇₁	—	8	2	no	—
84	Dic₁₄₂	—	4	8	no	—
85	Dic₁₄₂	—	8	4	no	—
86	C₈×C₇₁	—	2	4	no	—
87	C₇₁⋊C₈	—	2	4	no	—
88	C₇₁⋊C₈	—	4	2	no	—
89	C₂²×C₇₁	#390	1	1	yes	characteristic
90	C₂²×C₇₁	#494	1	1	yes	characteristic
91	C₂²×C₇₁	#495	1	1	yes	characteristic
92	C₂²×C₇₁	#496	1	2	yes	—
93	C₂²×C₇₁	#497	1	2	yes	—
94	C₄×C₇₁	—	16	2	no	—
95	C₄×C₇₁	#419	1	2	yes	—
96	C₄×C₇₁	C₄×Q₁₆	1	2	yes	—
97	C₄×C₇₁	#442	1	2	yes	—
98	C₄×C₇₁	#467	1	2	yes	—
99	C₄×C₇₁	#469	1	2	yes	—
100	C₄×C₇₁	#491	1	2	yes	—
101	Dic₇₁	—	8	4	no	—
102	#9834	—	71	1	no	maximal, Sylow
103	C₂×C₇₁	#1719	1	2	yes	—
104	C₂×C₇₁	#1723	1	2	yes	—
105	C₂×C₇₁	#1960	1	1	yes	characteristic
106	C₂×C₇₁	#1969	1	1	yes	characteristic
107	C₂×C₇₁	#1972	1	1	yes	characteristic
108	C₂×#500	—	71	1	no	—
109	#1990	—	71	2	no	—
110	#1956	—	71	2	no	—
111	#1834	D₇₁	1	1	yes	characteristic
112	C₂×#322	—	71	1	no	—
113	C₇₁	#9834	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₄⋊Q₈	—	71	2	no	—
115	#500	—	71	4	no	—
116	#500	—	142	2	no	—
117	C₂×#107	—	142	2	no	—
118	C₂×#103	—	142	4	no	—
119	C₂×C₄×C₈	—	71	1	no	—
120	C₂×C₄×C₈	—	71	1	no	—
121	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
122	#384	C₂×D₇₁	1	2	yes	—
123	#340	—	2	2	no	—
124	#340	—	142	4	no	—
125	#322	—	142	2	no	—
126	C₂²×Q₈	—	142	2	no	—
127	C₂²×C₈	—	71	2	no	—
128	C₂²×C₈	—	71	2	no	—
129	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
130	C₄⋊Q₈	—	71	4	no	—
131	C₄⋊Q₈	—	142	2	no	—
132	C₄⋊Q₈	—	142	2	no	—
133	C₄⋊Q₈	—	142	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	142	2	no	—
138	C₂×C₄⋊C₄	—	142	2	no	—
139	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
140	#107	—	142	8	no	—
141	#103	—	284	8	no	—
142	C₄×C₈	—	142	1	no	—
143	C₄×C₈	—	142	1	no	—
144	C₄×C₈	—	142	1	no	—
145	C₄×C₈	—	142	1	no	—
146	C₄×C₈	C₄×D₇₁	1	2	yes	—
147	C₄×C₈	D₂₈₄	1	2	yes	—
148	C₂×Q₈	—	284	2	no	—
149	C₂×Q₈	—	284	4	no	—
150	C₂×Q₈	—	284	4	no	—
151	C₂×Q₈	—	142	4	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	—	284	2	no	—
154	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
155	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	71	4	no	—
159	C₂×C₈	—	142	2	no	—
160	C₂×C₈	—	142	4	no	—
161	C₂×C₈	—	142	2	no	—
162	C₂×C₈	—	142	2	no	—
163	C₂×C₈	(C₂×D₇₁)⋊C₄	1	4	yes	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	284	2	no	—
168	C₄⋊C₄	—	284	2	no	—
169	C₄⋊C₄	—	142	4	no	—
170	C₄⋊C₄	—	142	4	no	—
171	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
172	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
173	C₄²	#11765866	1	1	yes	characteristic
174	C₄²	#11765866	1	1	yes	characteristic
175	C₂³	#14556124	1	1	yes	characteristic
176	Q₈	—	284	8	no	—
177	Q₈	—	568	4	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	284	4	no	—
182	C₂×C₄	—	568	2	no	—
183	C₂×C₄	—	568	2	no	—
184	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
185	C₂×C₄	#14556090	1	1	yes	characteristic
186	C₂×C₄	#14556090	1	1	yes	characteristic
187	C₂×C₄	#14556103	1	1	yes	characteristic
188	C₂×C₄	#14556103	1	1	yes	characteristic
189	C₂×C₄	#14556117	1	1	yes	characteristic
190	C₂×C₄	Q₁₆×D₇₁	1	2	yes	—
191	C₂×C₄	#14556147	1	2	yes	—
192	C₂×C₄	#14556155	1	4	yes	—
193	C₂×C₄	#14556158	1	4	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	284	2	no	—
196	C₈	—	142	4	no	—
197	C₂²	#17985900	1	1	yes	characteristic
198	C₂²	#17986377	1	1	yes	characteristic
199	C₂²	#17986382	1	1	yes	characteristic
200	C₂²	#17986385	1	2	yes	—
201	C₂²	#17986394	1	2	yes	—
202	C₄	—	16	2	no	—
203	C₄	—	568	4	no	—
204	C₄	#17986007	1	2	yes	—
205	C₄	#17986093	1	2	yes	—
206	C₄	#17986110	1	2	yes	—
207	C₄	#17986241	1	2	yes	—
208	C₄	#17986253	1	2	yes	—
209	C₄	#17986369	1	2	yes	—
210	C₂	#22587992	1	2	yes	—
211	C₂	#22588020	1	2	yes	—
212	C₂	#22589159	1	1	yes	characteristic
213	C₂	#22589199	1	1	yes	characteristic
214	C₂	#22589214	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986404

C₂

yes

maximal, characteristic

#22589300

C₂

yes

maximal

#22589141

C₂

yes

maximal

C₂×#17985721

C₂

yes

maximal, characteristic

C₇₁×#1834

C₂

yes

maximal, characteristic

C₂×#14556128

C₂²

yes

—

#17986404

—

#17986404

C₄

yes

complemented

C₂×#14556062

—

C₂×#14556051

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₇₁⋊C₈

C₂²

yes

characteristic

#17985761

—

#17985721

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#384×C₇₁

C₂²

yes

—

#340×C₇₁

—

C₂²×Dic₁₄₂

—

C₂²×C₇₁⋊C₈

D₄

yes

—

C₂²×C₇₁⋊C₈

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

#14556062

—

#14556051

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₂²×C₈×C₇₁

D₄

yes

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

Q₁₆

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

C₄²×C₇₁

yes

C₄²×C₇₁

yes

C₂³×C₇₁

yes

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₂×Q₁₆

yes

—

C₂×C₄×C₇₁

#135

yes

—

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₁

#137

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₂²×C₇₁

#497

yes

—

C₄×C₇₁

—

C₄×C₇₁

#419

yes

—

C₄×C₇₁

C₄×Q₁₆

yes

—

C₄×C₇₁

#442

yes

—

C₄×C₇₁

#467

yes

—

C₄×C₇₁

#469

yes

—

100

C₄×C₇₁

#491

yes

—

101

Dic₇₁

—

102

#9834

—

maximal, Sylow

103

C₂×C₇₁

#1719

yes

—

104

C₂×C₇₁

#1723

yes

—

105

C₂×C₇₁

#1960

yes

characteristic

106

C₂×C₇₁

#1969

yes

characteristic

107

C₂×C₇₁

#1972

yes

characteristic

108

C₂×#500

—

109

#1990

—

110

#1956

—

111

#1834

D₇₁

yes

characteristic

112

C₂×#322

—

113

C₇₁

#9834

yes

characteristic, complemented, Sylow

114

C₂×C₄⋊Q₈

—

115

#500

—

116

#500

—

142

—

117

C₂×#107

—

142

—

118

C₂×#103

—

142

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

—

121

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

122

#384

C₂×D₇₁

yes

—

123

#340

—

124

#340

—

142

—

125

#322

—

142

—

126

C₂²×Q₈

—

142

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

C₂²×C₈

C₂²⋊D₇₁

yes

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

142

—

132

C₄⋊Q₈

—

142

—

133

C₄⋊Q₈

—

142

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄⋊C₄

—

142

—

138

C₂×C₄⋊C₄

—

142

—

139

C₂×C₄²

C₂²×D₇₁

yes

characteristic

140

#107

—

142

—

141

#103

—

284

—

142

C₄×C₈

—

142

—

143

C₄×C₈

—

142

—

144

C₄×C₈

—

142

—

145

C₄×C₈

—

142

—

146

C₄×C₈

C₄×D₇₁

yes

—

147

C₄×C₈

D₂₈₄

yes

—

148

C₂×Q₈

—

284

—

149

C₂×Q₈

—

284

—

150

C₂×Q₈

—

284

—

151

C₂×Q₈

—

142

—

152

C₂²×C₄

—

153

C₂²×C₄

—

284

—

154

C₂²×C₄

D₄×D₇₁

yes

characteristic

155

C₂²×C₄

D₄×D₇₁

yes

characteristic

156

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

142

—

160

C₂×C₈

—

142

—

161

C₂×C₈

—

142

—

162

C₂×C₈

—

142

—

163

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

284

—

168

C₄⋊C₄

—

284

—

169

C₄⋊C₄

—

142

—

170

C₄⋊C₄

—

142

—

171

C₄²

C₂×C₄×D₇₁

yes

characteristic

172

C₄²

C₂×D₂₈₄

yes

characteristic

173

C₄²

#11765866

yes

characteristic

174

C₄²

#11765866

yes

characteristic

175

C₂³

#14556124

yes

characteristic

176

Q₈

—

284

—

177

Q₈

—

568

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

284

—

182

C₂×C₄

—

568

—

183

C₂×C₄

—

568

—

184

C₂×C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

185

C₂×C₄

#14556090

yes

characteristic

186

C₂×C₄

#14556090

yes

characteristic

187

C₂×C₄

#14556103

yes

characteristic

188

C₂×C₄

#14556103

yes

characteristic

189

C₂×C₄

#14556117

yes

characteristic

190

C₂×C₄

Q₁₆×D₇₁

yes

—

191

C₂×C₄

#14556147

yes

—

192

C₂×C₄

#14556155

yes

—

193

C₂×C₄

#14556158

yes

—

194

C₈

—

195

C₈

—

284

—

196

C₈

—

142

—

197

C₂²

#17985900

yes

characteristic

198

C₂²

#17986377

yes

characteristic

199

C₂²

#17986382

yes

characteristic

200

C₂²

#17986385

yes

—

201

C₂²

#17986394

yes

—

202

C₄

—

203

C₄

—

568

—

204

C₄

#17986007

yes

—

205

C₄

#17986093

yes

—

206

C₄

#17986110

yes

—

207

C₄

#17986241

yes

—

208

C₄

#17986253

yes

—

209

C₄

#17986369

yes

—

210

C₂

#22587992

yes

—

211

C₂

#22588020

yes

—

212

C₂

#22589159

yes

characteristic

213

C₂

#22589199

yes

characteristic

214

C₂

#22589214

yes

characteristic

	subgroup	G/H
center	C₂³	#14556124
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1834	C₂
socle, prime radical	C₂³×C₇₁	#127
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1834	D₇₁
hypocenter	C₇₁	#9834
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#14556124

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1834

C₂

socle, prime radical

C₂³×C₇₁

#127

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

N	H ≅ G/N	type
8. #17986404	C₄	semidirect
113. C₇₁	#9834	semidirect

H ≅ G/N

type

8. #17986404

C₄

semidirect

113. C₇₁

#9834

semidirect

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

subgroup	index
C₂×#17986404	2
#22589300	2
#22589141	2
C₂×#17985721	2
C₇₁×#1834	2
#9834	71

subgroup

index

C₇₁×#1834

Subgroups — #31897274

Conjugacy classes of subgroups

Complements

Maximal subgroups