Group #31897070 — subgroups

Subgroups — #31897070

Order 18176.

442 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 139 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	#22589185	C₂	1	2	yes	maximal
2	#22587099	C₂	1	1	yes	maximal, characteristic
3	#22587012	C₂	1	2	yes	maximal
4	C₈×#11765849	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	1	yes	characteristic
7	#17986023	C₂²	1	2	yes	—
8	C₂×#14556047	—	2	4	no	—
9	C₄×#11765849	C₂²	1	1	yes	characteristic
10	#17985752	—	2	2	no	—
11	#17985745	—	2	4	no	—
12	#17985721	—	2	1	no	—
13	C₈×C₇₁⋊C₈	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
15	#500×C₇₁	—	2	1	no	—
16	#500×C₇₁	—	2	1	no	—
17	#500×C₇₁	—	2	2	no	—
18	C₂×#107×C₇₁	C₄	1	4	yes	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×#11765849	—	2	2	no	—
21	C₂×#11765849	D₄	1	2	yes	—
22	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₁⋊C₈)	D₄	1	2	yes	—
25	#14556047	—	4	8	no	—
26	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
27	#14556039	—	2	1	no	—
28	C₄×C₇₁⋊C₈	—	2	1	no	—
29	C₄×C₇₁⋊C₈	—	2	1	no	—
30	C₂²×Q₈×C₇₁	—	2	2	no	—
31	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈×C₇₁	—	2	2	no	—
33	C₄⋊Q₈×C₇₁	—	2	2	no	—
34	C₄⋊Q₈×C₇₁	—	2	2	no	—
35	C₄⋊Q₈×C₇₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
39	#107×C₇₁	—	2	4	no	—
40	#107×C₇₁	—	2	4	no	—
41	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
42	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
43	C₄×C₈×C₇₁	D₄	1	2	yes	—
44	#11765849	—	2	2	no	—
45	#11765849	—	4	4	no	—
46	#11765849	D₈	1	4	yes	—
47	C₂×C₇₁⋊C₈	—	2	2	no	—
48	C₂×C₇₁⋊C₈	—	2	2	no	—
49	C₂×C₇₁⋊C₈	—	4	2	no	—
50	C₂×Q₈×C₇₁	—	4	2	no	—
51	C₂×Q₈×C₇₁	—	4	2	no	—
52	C₂×Q₈×C₇₁	—	4	4	no	—
53	C₂×Q₈×C₇₁	—	4	4	no	—
54	C₂²×C₄×C₇₁	—	2	4	no	—
55	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
58	C₂×C₈×C₇₁	—	2	4	no	—
59	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
61	C₄⋊C₄×C₇₁	—	2	4	no	—
62	C₄⋊C₄×C₇₁	—	2	4	no	—
63	C₄⋊C₄×C₇₁	—	4	2	no	—
64	C₄⋊C₄×C₇₁	—	4	2	no	—
65	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₇₁	#41	1	1	yes	characteristic
69	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
70	Q₈×C₇₁	—	8	4	no	—
71	Q₈×C₇₁	—	8	2	no	—
72	Q₈×C₇₁	—	8	2	no	—
73	C₂×C₄×C₇₁	—	2	2	no	—
74	C₂×C₄×C₇₁	—	2	2	no	—
75	C₂×C₄×C₇₁	—	4	4	no	—
76	C₂×C₄×C₇₁	—	4	8	no	—
77	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
80	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
81	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
82	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
83	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₇₁	#135	1	2	yes	—
85	C₈×C₇₁	—	2	2	no	—
86	C₈×C₇₁	#101	1	4	yes	complemented
87	C₇₁⋊C₈	—	2	4	no	—
88	C₇₁⋊C₈	—	4	2	no	—
89	C₇₁⋊C₈	—	8	2	no	—
90	C₂²×C₇₁	—	2	1	no	—
91	C₂²×C₇₁	—	2	1	no	—
92	C₂²×C₇₁	#390	1	1	yes	characteristic
93	C₂²×C₇₁	C₄×D₈	1	2	yes	—
94	C₄×C₇₁	—	2	1	no	—
95	C₄×C₇₁	—	2	1	no	—
96	C₄×C₇₁	—	8	4	no	—
97	C₄×C₇₁	C₂×#101	1	2	yes	—
98	C₄×C₇₁	#466	1	2	yes	—
99	C₄×C₇₁	#473	1	2	yes	—
100	C₄×C₇₁	#493	1	1	yes	characteristic
101	C₄×C₇₁	#495	1	1	yes	characteristic
102	#9799	—	71	1	no	maximal, Sylow
103	C₂×C₇₁	—	2	2	no	—
104	C₂×C₇₁	#1966	1	1	yes	characteristic
105	C₂×C₇₁	#1970	1	1	yes	characteristic
106	C₂×C₇₁	#1979	1	1	yes	characteristic
107	C₂×#500	D₇₁	1	1	yes	characteristic
108	#1966	—	71	2	no	—
109	#1567	—	71	1	no	—
110	#1546	—	71	2	no	—
111	C₈×M₁₆	—	71	1	no	—
112	C₇₁	#9799	1	1	yes	characteristic, complemented, Sylow
113	C₂×C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
114	#500	—	2	1	no	—
115	#500	—	2	1	no	—
116	#500	—	2	2	no	—
117	#431	—	71	1	no	—
118	C₂×#107	Dic₇₁	1	4	yes	—
119	#423	—	71	2	no	—
120	C₂×#101	—	142	4	no	—
121	C₄×M₁₆	—	71	1	no	—
122	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
123	#335	—	142	2	no	—
124	#332	—	142	4	no	—
125	#322	—	142	1	no	—
126	C₈²	—	142	1	no	—
127	C₂²×Q₈	—	2	2	no	—
128	C₂×M₁₆	—	71	2	no	—
129	C₂×M₁₆	—	142	2	no	—
130	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	C₂×Dic₇₁	1	4	yes	—
137	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
138	#107	—	2	4	no	—
139	#107	—	2	4	no	—
140	C₄⋊C₈	—	71	2	no	—
141	C₄⋊C₈	—	142	2	no	—
142	C₄⋊C₈	—	142	2	no	—
143	#101	—	284	8	no	—
144	C₂²⋊C₈	—	142	1	no	—
145	#97	—	142	1	no	—
146	C₄×C₈	—	142	1	no	—
147	C₄×C₈	—	142	1	no	—
148	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂×Q₈	—	4	4	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
157	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
158	C₂²×C₄	#11765864	1	1	yes	characteristic
159	M₁₆	—	284	4	no	—
160	M₁₆	—	71	4	no	—
161	M₁₆	—	142	2	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	284	2	no	—
164	C₂×C₈	—	142	2	no	—
165	C₂×C₈	—	142	2	no	—
166	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
167	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄²	D₄×D₇₁	1	1	yes	characteristic
173	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
174	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
175	C₄²	#11765864	1	1	yes	characteristic
176	C₂³	D₄×Dic₇₁	1	1	yes	characteristic
177	Q₈	—	8	2	no	—
178	Q₈	—	8	2	no	—
179	Q₈	—	8	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
185	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
186	C₂×C₄	#14556099	1	1	yes	characteristic
187	C₂×C₄	#14556099	1	1	yes	characteristic
188	C₂×C₄	#14556115	1	1	yes	characteristic
189	C₂×C₄	#14556124	1	1	yes	characteristic
190	C₂×C₄	D₈×D₇₁	1	2	yes	—
191	C₂×C₄	#14556147	1	2	yes	—
192	C₈	—	2	2	no	—
193	C₈	—	284	2	no	—
194	C₈	—	568	2	no	—
195	C₈	—	142	4	no	—
196	C₈	#14556047	1	4	yes	complemented
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#17985902	1	1	yes	characteristic
200	C₂²	D₈×Dic₇₁	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	8	4	no	—
204	C₄	C₂×#14556047	1	2	yes	—
205	C₄	#17986232	1	2	yes	—
206	C₄	#17986271	1	2	yes	—
207	C₄	#17986374	1	1	yes	characteristic
208	C₄	#17986382	1	1	yes	characteristic
209	C₂	—	2	2	no	—
210	C₂	#22589185	1	1	yes	characteristic
211	C₂	#22589207	1	1	yes	characteristic
212	C₂	#22589256	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22589185

C₂

yes

maximal

#22587099

C₂

yes

maximal, characteristic

#22587012

C₂

yes

maximal

C₈×#11765849

C₂

yes

maximal, characteristic

C₂×#500×C₇₁

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

characteristic

#17986023

C₂²

yes

—

C₂×#14556047

—

C₄×#11765849

C₂²

yes

—

—

—

C₈×C₇₁⋊C₈

—

C₂×C₄⋊Q₈×C₇₁

C₂²

yes

—

#500×C₇₁

—

#500×C₇₁

—

#500×C₇₁

—

C₂×#107×C₇₁

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×#11765849

—

C₂×#11765849

D₄

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

D₄

yes

—

#14556047

—

C₂²⋊(C₇₁⋊C₈)

—

#14556039

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₂²×Q₈×C₇₁

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#107×C₇₁

—

#107×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

—

#11765849

—

#11765849

—

#11765849

D₈

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

#41

yes

characteristic

C₂³×C₇₁

C₄×D₄

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×C₄×C₇₁

#135

yes

—

C₈×C₇₁

—

C₈×C₇₁

#101

yes

complemented

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#390

yes

characteristic

C₂²×C₇₁

C₄×D₈

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#101

yes

—

C₄×C₇₁

#466

yes

—

C₄×C₇₁

#473

yes

—

100

C₄×C₇₁

#493

yes

characteristic

101

C₄×C₇₁

yes

102

—

103

C₂×C₇₁

—

104

C₂×C₇₁

#1966

yes

characteristic

105

C₂×C₇₁

#1970

yes

characteristic

106

C₂×C₇₁

#1979

yes

characteristic

107

C₂×#500

D₇₁

yes

characteristic

108

#1966

—

109

#1567

—

110

#1546

—

111

C₈×M₁₆

—

112

C₇₁

#9799

yes

characteristic, complemented, Sylow

113

C₂×C₄⋊Q₈

C₂×D₇₁

yes

—

114

#500

—

115

#500

—

116

#500

—

117

#431

—

118

C₂×#107

Dic₇₁

yes

—

119

#423

—

120

C₂×#101

—

142

—

121

C₄×M₁₆

—

122

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

123

#335

—

142

—

124

#332

—

142

—

125

#322

—

142

—

126

C₈²

—

142

—

127

C₂²×Q₈

—

128

C₂×M₁₆

—

129

C₂×M₁₆

—

142

—

130

C₂²×C₈

C₂×Dic₇₁

yes

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

—

133

C₄⋊Q₈

—

134

C₄⋊Q₈

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

C₂×Dic₇₁

yes

—

137

C₂×C₄²

C₂²×D₇₁

yes

characteristic

138

#107

—

139

#107

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

142

—

142

C₄⋊C₈

—

142

—

143

#101

—

284

—

144

C₂²⋊C₈

—

142

—

145

#97

—

142

—

146

C₄×C₈

—

142

—

147

C₄×C₈

—

142

—

148

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

149

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

150

C₄×C₈

C₂²⋊D₇₁

yes

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂²×C₄

—

156

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

157

C₂²×C₄

D₄×D₇₁

yes

characteristic

158

C₂²×C₄

#11765864

yes

characteristic

159

M₁₆

—

284

—

160

M₁₆

—

161

M₁₆

—

142

—

162

C₂×C₈

—

163

C₂×C₈

—

284

—

164

C₂×C₈

—

142

—

165

C₂×C₈

—

142

—

166

C₂×C₈

C₂²⋊Dic₇₁

yes

—

167

C₂×C₈

C₂²⋊Dic₇₁

yes

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄²

D₄×D₇₁

yes

characteristic

173

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

174

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

175

C₄²

#11765864

yes

characteristic

176

C₂³

D₄×Dic₇₁

yes

characteristic

177

Q₈

—

178

Q₈

—

179

Q₈

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

185

C₂×C₄

D₄×Dic₇₁

yes

characteristic

186

C₂×C₄

#14556099

yes

characteristic

187

C₂×C₄

#14556099

yes

characteristic

188

C₂×C₄

#14556115

yes

characteristic

189

C₂×C₄

#14556124

yes

characteristic

190

C₂×C₄

D₈×D₇₁

yes

—

191

C₂×C₄

#14556147

yes

—

192

C₈

—

193

C₈

—

284

—

194

C₈

—

568

—

195

C₈

—

142

—

196

C₈

#14556047

yes

complemented

197

C₂²

—

198

C₂²

—

199

C₂²

#17985902

yes

characteristic

200

C₂²

D₈×Dic₇₁

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

C₂×#14556047

yes

—

205

C₄

#17986232

yes

—

206

C₄

#17986271

yes

—

207

C₄

#17986374

yes

characteristic

208

C₄

#17986382

yes

characteristic

209

C₂

—

210

C₂

#22589185

yes

characteristic

211

C₂

#22589207

yes

characteristic

212

C₂

#22589256

yes

characteristic

	subgroup	G/H
center	C₂²	#17985902
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#500×C₇₁	C₂
socle	C₂²×C₇₁	#390
prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#500	D₇₁
hypocenter	C₇₁	#9799
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#17985902

commutator

C₂²×C₄×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₂×#500×C₇₁

C₂

socle

C₂²×C₇₁

#390

prime radical

C₂³×C₇₁

C₄×D₄

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
86. C₈×C₇₁	#101	semidirect
112. C₇₁	#9799	semidirect
196. C₈	#14556047	semidirect

H ≅ G/N

type

86. C₈×C₇₁

#101

semidirect

112. C₇₁

#9799

semidirect

196. C₈

#14556047

semidirect

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

subgroup	index
#22589185	2
#22587099	2
#22587012	2
C₈×#11765849	2
C₂×#500×C₇₁	2
#9799	71

subgroup

index

C₂×#500×C₇₁

Subgroups — #31897070

Conjugacy classes of subgroups

Complements

Maximal subgroups