Group #31915975 — subgroups

Subgroups — #31915975

Order 18176.

386 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 159 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×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#22586417	C₂	1	4	yes	maximal
4	C₂×#346×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₄₂	C₄	1	1	yes	characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
7	C₈×Dic₁₄₂	—	2	2	no	—
8	C₈×Dic₁₄₂	—	2	2	no	—
9	C₂×C₄⋊(C₇₁⋊C₈)	C₄	1	1	yes	characteristic
10	C₂×Dic₇₁⋊C₈	C₄	1	2	yes	—
11	C₂×C₈×Dic₇₁	C₂²	1	2	yes	—
12	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
13	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	#346×C₇₁	—	2	2	no	—
16	C₂²×Dic₁₄₂	C₈	1	1	yes	characteristic
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₄₂	—	2	2	no	—
19	C₄×Dic₁₄₂	—	2	2	no	—
20	C₂×C₄⋊Dic₇₁	C₈	1	1	yes	characteristic
21	C₂×Dic₇₁⋊C₄	C₈	1	2	yes	—
22	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
23	C₂×C₇₁⋊C₁₆	—	2	2	no	—
24	C₂×C₇₁⋊C₁₆	—	2	4	no	—
25	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
26	Dic₇₁⋊C₈	—	2	4	no	—
27	C₈×Dic₇₁	—	2	2	no	—
28	C₈×Dic₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₁	—	2	2	no	—
33	C₂×C₁₆×C₇₁	—	2	2	no	—
34	C₂×C₁₆×C₇₁	Q₈	1	4	yes	complemented
35	C₄×C₈×C₇₁	D₄	1	2	yes	—
36	C₄×C₈×C₇₁	D₄	1	2	yes	—
37	C₂×Dic₁₄₂	—	2	2	no	—
38	C₂×Dic₁₄₂	—	2	4	no	—
39	C₂²×Dic₇₁	C₂×C₈	1	2	yes	—
40	C₂²×Dic₇₁	C₂×C₈	1	2	yes	—
41	C₂×C₇₁⋊C₈	—	2	2	no	—
42	C₂×C₇₁⋊C₈	—	2	4	no	—
43	C₄⋊Dic₇₁	—	2	2	no	—
44	Dic₇₁⋊C₄	—	2	4	no	—
45	C₄×Dic₇₁	—	2	2	no	—
46	C₄×Dic₇₁	—	2	2	no	—
47	C₇₁⋊C₁₆	—	4	2	no	—
48	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	—	2	2	no	—
52	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₇₁	#41	1	2	yes	—
57	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₇₁	M₁₆	1	2	yes	—
59	C₁₆×C₇₁	—	2	4	no	—
60	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₁	—	2	2	no	—
62	C₂×C₄×C₇₁	—	2	2	no	—
63	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
65	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#131	1	2	yes	—
69	C₂×Dic₇₁	—	2	2	no	—
70	C₂×Dic₇₁	—	2	2	no	—
71	C₂×Dic₇₁	—	2	2	no	—
72	C₂×Dic₇₁	—	2	4	no	—
73	C₂×Dic₇₁	M₃₂	1	4	yes	—
74	Dic₁₄₂	—	2	8	no	—
75	C₈×C₇₁	—	2	2	no	—
76	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₇₁	#124	1	1	yes	characteristic
78	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
79	C₇₁⋊C₈	—	4	2	no	—
80	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
81	C₂²×C₇₁	C₈×Q₈	1	2	yes	—
82	C₂²×C₇₁	C₂×M₃₂	1	2	yes	—
83	C₂²×C₇₁	#504	1	2	yes	—
84	C₄×C₇₁	—	2	2	no	—
85	C₄×C₇₁	—	2	2	no	—
86	C₄×C₇₁	#391	1	1	yes	characteristic
87	C₄×C₇₁	#408	1	1	yes	characteristic
88	C₄×C₇₁	#446	1	2	yes	—
89	Dic₇₁	—	2	4	no	—
90	Dic₇₁	—	4	2	no	—
91	#13264	—	71	1	no	maximal, Sylow
92	C₂×C₇₁	#1984	1	1	yes	characteristic
93	C₂×C₇₁	#2112	1	1	yes	characteristic
94	C₂×C₇₁	#2113	1	1	yes	characteristic
95	C₂×C₇₁	#2183	1	4	yes	—
96	C₂×C₈×Q₈	—	71	1	no	—
97	C₂×C₄⋊C₁₆	—	71	1	no	—
98	C₂×#346	D₇₁	1	1	yes	characteristic
99	#1374	—	71	4	no	—
100	C₇₁	#13264	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄×Q₈	—	71	1	no	—
102	C₂²×C₁₆	—	71	2	no	—
103	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
104	C₈×Q₈	—	142	2	no	—
105	C₈×Q₈	—	142	2	no	—
106	C₂×C₄⋊C₈	—	71	2	no	—
107	C₂×C₄⋊C₈	—	71	1	no	—
108	C₂×C₄×C₈	—	71	2	no	—
109	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
110	C₄⋊C₁₆	—	142	2	no	—
111	#346	—	2	2	no	—
112	C₂²×Q₈	—	71	1	no	—
113	C₂²×C₈	—	71	2	no	—
114	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
116	C₄×Q₈	—	142	2	no	—
117	C₄×Q₈	—	142	2	no	—
118	C₂×C₄⋊C₄	—	71	2	no	—
119	C₂×C₄⋊C₄	—	71	1	no	—
120	C₂×C₄²	—	71	2	no	—
121	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	142	2	no	—
125	C₂×C₁₆	—	142	4	no	—
126	C₂×C₁₆	Dic₁₄₂	1	4	yes	complemented
127	C₄⋊C₈	—	142	4	no	—
128	C₄⋊C₈	—	142	2	no	—
129	C₄×C₈	—	142	2	no	—
130	C₄×C₈	—	142	2	no	—
131	C₄×C₈	D₂₈₄	1	2	yes	—
132	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
133	C₂×Q₈	—	142	4	no	—
134	C₂×Q₈	—	142	2	no	—
135	C₂²×C₄	—	71	2	no	—
136	C₂²×C₄	—	71	2	no	—
137	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
138	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	142	2	no	—
142	C₂×C₈	—	142	4	no	—
143	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
144	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
145	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
146	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
147	C₂×C₈	#11765865	1	2	yes	—
148	C₄⋊C₄	—	142	4	no	—
149	C₄⋊C₄	—	142	2	no	—
150	C₄²	—	142	2	no	—
151	C₄²	—	142	2	no	—
152	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
153	C₄²	#11765848	1	2	yes	—
154	C₁₆	—	2	4	no	—
155	C₁₆	—	284	2	no	—
156	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
157	Q₈	—	142	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	71	4	no	—
161	C₂×C₄	—	142	4	no	—
162	C₂×C₄	—	142	2	no	—
163	C₂×C₄	—	142	2	no	—
164	C₂×C₄	—	142	2	no	—
165	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
166	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
167	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
169	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
170	C₂×C₄	#14556134	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	284	2	no	—
173	C₈	#14556107	1	2	yes	—
174	C₈	#14556116	1	1	yes	characteristic
175	C₈	#14556128	1	1	yes	characteristic
176	C₂²	C₂×(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
177	C₂²	C₈×Dic₁₄₂	1	2	yes	—
178	C₂²	M₃₂×D₇₁	1	2	yes	—
179	C₂²	#17986417	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	284	2	no	—
183	C₄	—	142	4	no	—
184	C₄	#17985905	1	1	yes	characteristic
185	C₄	#17985965	1	1	yes	characteristic
186	C₄	#17986129	1	2	yes	—
187	C₂	#22589276	1	1	yes	characteristic
188	C₂	#22589950	1	1	yes	characteristic
189	C₂	#22589955	1	1	yes	characteristic
190	C₂	#22590246	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₄₂

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₁⋊C₁₆)

C₂

yes

maximal, characteristic

#22586417

C₂

yes

maximal

C₂×#346×C₇₁

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₄₂

C₄

yes

characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

—

C₈×Dic₁₄₂

—

C₈×Dic₁₄₂

—

C₂×C₄⋊(C₇₁⋊C₈)

C₄

yes

characteristic

C₂×Dic₇₁⋊C₈

C₄

yes

—

C₂×C₈×Dic₇₁

C₂²

yes

—

C₄⋊(C₇₁⋊C₁₆)

—

C₂²×C₁₆×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#346×C₇₁

—

C₂²×Dic₁₄₂

C₈

yes

characteristic

C₂²×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₄₂

—

C₄×Dic₁₄₂

—

C₂×C₄⋊Dic₇₁

C₈

yes

characteristic

C₂×Dic₇₁⋊C₄

C₈

yes

—

C₂×C₄×Dic₇₁

C₂×C₄

yes

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₄⋊(C₇₁⋊C₈)

—

Dic₇₁⋊C₈

—

C₈×Dic₇₁

—

C₈×Dic₇₁

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

Q₈

yes

complemented

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

C₂×C₈

yes

—

C₂²×Dic₇₁

C₂×C₈

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄²×C₇₁

C₂²⋊C₄

yes

—

C₄²×C₇₁

M₁₆

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₈

yes

—

C₂×C₄×C₇₁

C₂²⋊C₈

yes

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×Q₈

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₁

#131

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

M₃₂

yes

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂²⋊Q₈

yes

—

C₈×C₇₁

#124

yes

characteristic

C₈×C₇₁

C₄⋊Q₈

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₇₁

C₈×Q₈

yes

—

C₂²×C₇₁

C₂×M₃₂

yes

—

C₂²×C₇₁

#504

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₈×Q₈

—

C₂×C₄⋊C₁₆

—

C₂×#346

D₇₁

yes

characteristic

#1374

—

100

C₇₁

#13264

yes

characteristic, complemented, Sylow

101

C₂×C₄×Q₈

—

102

C₂²×C₁₆

—

103

C₂²×C₁₆

C₂×D₇₁

yes

—

104

C₈×Q₈

—

142

—

105

C₈×Q₈

—

142

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

110

C₄⋊C₁₆

—

142

—

111

#346

—

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₄×D₇₁

yes

characteristic

115

C₂²×C₈

C₂²×D₇₁

yes

characteristic

116

C₄×Q₈

—

142

—

117

C₄×Q₈

—

142

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄²

—

121

C₂×C₄²

C₄×D₇₁

yes

characteristic

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

142

—

125

C₂×C₁₆

—

142

—

126

C₂×C₁₆

Dic₁₄₂

yes

complemented

127

C₄⋊C₈

—

142

—

128

C₄⋊C₈

—

142

—

129

C₄×C₈

—

142

—

130

C₄×C₈

—

142

—

131

C₄×C₈

D₂₈₄

yes

—

132

C₄×C₈

C₂²⋊D₇₁

yes

—

133

C₂×Q₈

—

142

—

134

C₂×Q₈

—

142

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₈×D₇₁

yes

characteristic

138

C₂²×C₄

C₈×D₇₁

yes

characteristic

139

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

—

142

C₂×C₈

—

142

—

143

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

144

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

145

C₂×C₈

C₂×D₂₈₄

yes

characteristic

146

C₂×C₈

C₂×Dic₁₄₂

yes

—

147

C₂×C₈

#11765865

yes

—

148

C₄⋊C₄

—

142

—

149

C₄⋊C₄

—

142

—

150

C₄²

—

142

—

151

C₄²

—

142

—

152

C₄²

(C₂×D₇₁)⋊C₄

yes

—

153

C₄²

#11765848

yes

—

154

C₁₆

—

155

C₁₆

—

284

—

156

C₂³

C₂×C₈×D₇₁

yes

characteristic

157

Q₈

—

142

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

142

—

162

C₂×C₄

—

142

—

163

C₂×C₄

—

142

—

164

C₂×C₄

—

142

—

165

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

—

166

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

—

167

C₂×C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

168

C₂×C₄

C₄×Dic₁₄₂

yes

—

169

C₂×C₄

C₂×#11765848

yes

characteristic

170

C₂×C₄

#14556134

yes

—

171

C₈

—

172

C₈

—

284

—

173

C₈

#14556107

yes

—

174

C₈

#14556116

yes

characteristic

175

C₈

#14556128

yes

characteristic

176

C₂²

C₂×(C₂×D₇₁)⋊C₈

yes

characteristic

177

C₂²

C₈×Dic₁₄₂

yes

—

178

C₂²

M₃₂×D₇₁

yes

—

179

C₂²

#17986417

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

284

—

183

C₄

—

142

—

184

C₄

#17985905

yes

characteristic

185

C₄

#17985965

yes

characteristic

186

C₄

#17986129

yes

—

187

C₂

#22589276

yes

characteristic

188

C₂

#22589950

yes

characteristic

189

C₂

#22589955

yes

characteristic

190

C₂

#22590246

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator, socle, prime radical	C₂³×C₇₁	C₂²×C₈
Fitting	C₂×#346×C₇₁	C₂
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#346	D₇₁
hypocenter	C₇₁	#13264
LCS	G ▷ C₂³×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₇₁

commutator, socle, prime radical

C₂³×C₇₁

C₂²×C₈

Fitting

C₂×#346×C₇₁

C₂

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂³×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₈ ◁ C₂×#346

N	H ≅ G/N	type
34. C₂×C₁₆×C₇₁	Q₈	semidirect
100. C₇₁	#13264	semidirect
126. C₂×C₁₆	Dic₁₄₂	semidirect

H ≅ G/N

type

34. C₂×C₁₆×C₇₁

Q₈

semidirect

100. C₇₁

#13264

semidirect

126. C₂×C₁₆

Dic₁₄₂

semidirect

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

subgroup	index
C₂×C₈×Dic₁₄₂	2
C₂×C₄⋊(C₇₁⋊C₁₆)	2
#22586417	2
C₂×#346×C₇₁	2
#13264	71

subgroup

index

C₂×C₈×Dic₁₄₂

C₂×C₄⋊(C₇₁⋊C₁₆)

#22586417

C₂×#346×C₇₁

#13264

Subgroups — #31915975

Conjugacy classes of subgroups

Complements

Maximal subgroups