Group #31888619 — subgroups

Subgroups — #31888619

Order 18176.

390 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 195 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	C₄×#14556038	C₂	1	1	yes	maximal, characteristic
2	#22587086	C₂	1	2	yes	maximal
3	#22586623	C₂	1	1	yes	maximal, characteristic
4	#22586172	C₂	1	2	yes	maximal
5	C₇₁×#1450	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	2	yes	—
7	C₂×#14556038	C₄	1	2	yes	—
8	C₂×#14556038	C₄	1	2	yes	—
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄²×Dic₇₁	C₂²	1	1	yes	characteristic
11	#17985751	C₄	1	4	yes	complemented
12	#17985749	C₄	1	4	yes	complemented
13	#17985720	—	2	1	no	—
14	C₈×C₇₁⋊C₈	—	2	1	no	—
15	#431×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#322×C₇₁	—	2	1	no	—
18	C₈²×C₇₁	—	2	1	no	—
19	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₁	—	2	2	no	—
22	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
23	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
24	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
25	#14556038	—	2	4	no	—
26	#14556038	C₈	1	8	yes	complemented
27	C₄×C₇₁⋊C₈	—	2	2	no	—
28	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
29	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₇₁	C₂×C₄	1	4	yes	—
33	C₂²⋊C₈×C₇₁	—	2	2	no	—
34	C₄×C₈×C₇₁	—	2	2	no	—
35	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
36	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
37	C₄×C₈×C₇₁	Q₈	1	2	yes	—
38	C₂²×Dic₇₁	—	2	2	no	—
39	C₂×C₇₁⋊C₈	—	2	2	no	—
40	C₂×C₇₁⋊C₈	—	2	4	no	—
41	C₂×C₇₁⋊C₈	C₄²	1	1	yes	characteristic
42	C₂×C₇₁⋊C₈	C₄²	1	1	yes	characteristic
43	C₂×C₇₁⋊C₈	C₄²	1	2	yes	—
44	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
45	C₄×Dic₇₁	—	2	2	no	—
46	C₄×Dic₇₁	—	4	4	no	—
47	C₄×Dic₇₁	C₂×C₈	1	4	yes	—
48	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	#41	1	2	yes	—
50	C₂×C₈×C₇₁	—	2	4	no	—
51	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
54	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
55	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
56	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
58	C₂³×C₇₁	#117	1	1	yes	characteristic
59	C₂×C₄×C₇₁	—	2	4	no	—
60	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
66	C₂×Dic₇₁	—	2	4	no	—
67	C₂×Dic₇₁	—	4	4	no	—
68	C₈×C₇₁	—	4	4	no	—
69	C₈×C₇₁	C₄⋊C₈	1	4	yes	—
70	C₈×C₇₁	C₄⋊C₈	1	4	yes	—
71	C₇₁⋊C₈	—	2	1	no	—
72	C₇₁⋊C₈	—	2	1	no	—
73	C₇₁⋊C₈	—	4	4	no	—
74	C₇₁⋊C₈	C₄×C₈	1	4	yes	—
75	C₂²×C₇₁	—	2	2	no	—
76	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₇₁	#431	1	2	yes	—
78	C₄×C₇₁	—	2	2	no	—
79	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
80	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
81	C₄×C₇₁	C₂×C₄⋊C₈	1	2	yes	—
82	C₄×C₇₁	#444	1	4	yes	—
83	Dic₇₁	—	4	4	no	—
84	#8268	—	71	1	no	maximal, Sylow
85	C₂×C₇₁	—	2	2	no	—
86	C₂×C₇₁	C₄×C₄⋊C₈	1	1	yes	characteristic
87	C₂×C₇₁	#1773	1	2	yes	—
88	C₄×#97	—	71	1	no	—
89	#1564	—	71	2	no	—
90	#1450	—	71	1	no	—
91	#1450	D₇₁	1	1	yes	characteristic
92	#1276	—	71	2	no	—
93	C₇₁	#8268	1	1	yes	characteristic, complemented, Sylow
94	#431	—	71	2	no	—
95	#431	C₂×D₇₁	1	2	yes	—
96	C₂×#97	—	71	2	no	—
97	C₂×#97	—	71	2	no	—
98	C₂×C₄×C₈	—	71	1	no	—
99	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
100	C₄³	—	71	1	no	—
101	#335	—	71	4	no	—
102	#334	—	71	4	no	—
103	#322	—	2	1	no	—
104	#322	—	142	1	no	—
105	C₈²	—	2	1	no	—
106	C₈²	—	142	1	no	—
107	C₂²×C₈	—	71	1	no	—
108	C₂²×C₈	—	71	1	no	—
109	C₂²×C₈	C₄×D₇₁	1	2	yes	—
110	C₂×C₄²	—	71	2	no	—
111	C₂×C₄²	—	142	2	no	—
112	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
113	C₄⋊C₈	—	71	4	no	—
114	C₄⋊C₈	C₄×D₇₁	1	4	yes	—
115	C₂²⋊C₈	—	2	2	no	—
116	C₂²⋊C₈	—	142	2	no	—
117	#97	—	71	8	no	—
118	#97	—	142	4	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	71	2	no	—
121	C₄×C₈	—	71	2	no	—
122	C₄×C₈	—	142	2	no	—
123	C₄×C₈	Dic₁₄₂	1	2	yes	—
124	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
126	C₂²×C₄	—	142	2	no	—
127	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
128	C₂²×C₄	#11765865	1	2	yes	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	71	2	no	—
131	C₂×C₈	—	71	4	no	—
132	C₂×C₈	—	71	1	no	—
133	C₂×C₈	—	71	1	no	—
134	C₂×C₈	—	142	4	no	—
135	C₂×C₈	—	142	2	no	—
136	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
137	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
138	C₂×C₈	C₈×D₇₁	1	4	yes	—
139	C₂×C₈	#11765848	1	4	yes	—
140	C₄²	—	284	4	no	—
141	C₄²	—	71	4	no	—
142	C₄²	—	142	2	no	—
143	C₄²	C₂×C₄×D₇₁	1	2	yes	—
144	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
145	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
146	C₂³	#14556085	1	1	yes	characteristic
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	284	4	no	—
149	C₂×C₄	—	142	4	no	—
150	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
151	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
152	C₂×C₄	C₄×C₂²⋊D₇₁	1	2	yes	—
153	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
154	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
155	C₂×C₄	C₂×#11765848	1	2	yes	—
156	C₈	—	4	4	no	—
157	C₈	—	284	4	no	—
158	C₈	—	71	4	no	—
159	C₈	—	142	1	no	—
160	C₈	—	142	1	no	—
161	C₈	Dic₇₁⋊C₈	1	4	yes	—
162	C₈	Dic₇₁⋊C₈	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	C₄×Dic₇₁⋊C₄	1	1	yes	characteristic
165	C₂²	#17986050	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	284	4	no	—
168	C₄	C₄×C₈×D₇₁	1	1	yes	characteristic
169	C₄	C₄×#11765848	1	1	yes	characteristic
170	C₄	C₂×Dic₇₁⋊C₈	1	2	yes	—
171	C₄	#17986119	1	4	yes	—
172	C₂	—	2	2	no	—
173	C₂	C₄×Dic₇₁⋊C₈	1	1	yes	characteristic
174	C₂	#22588225	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556038

C₂

yes

maximal, characteristic

#22587086

C₂

yes

maximal

#22586623

C₂

yes

maximal, characteristic

#22586172

C₂

yes

maximal

C₇₁×#1450

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

C₂×#14556038

C₄

yes

—

C₂×#14556038

C₄

yes

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄²×Dic₇₁

C₂²

yes

characteristic

#17985751

C₄

yes

complemented

#17985749

C₄

yes

complemented

#17985720

—

C₈×C₇₁⋊C₈

—

#431×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#322×C₇₁

—

C₈²×C₇₁

—

C₂²×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₁

—

C₂×C₄×Dic₇₁

C₂×C₄

yes

—

C₄⋊(C₇₁⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₇₁⋊C₈)

—

#14556038

—

#14556038

C₈

yes

complemented

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

C₂×C₄

yes

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

Q₈

yes

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄²

yes

characteristic

C₂×C₇₁⋊C₈

C₄²

yes

characteristic

C₂×C₇₁⋊C₈

C₄²

yes

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₂×C₈

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

M₁₆

yes

—

C₄²×C₇₁

C₂²×C₄

yes

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₂³×C₇₁

#117

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

—

C₂×C₄×C₇₁

C₄×Q₈

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₄⋊C₈

yes

—

C₈×C₇₁

C₄⋊C₈

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₄×C₈

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

#431

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄×C₈

yes

characteristic

C₄×C₇₁

C₄×M₁₆

yes

characteristic

C₄×C₇₁

C₂×C₄⋊C₈

yes

—

C₄×C₇₁

#444

yes

—

Dic₇₁

—

#8268

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

C₄×C₄⋊C₈

yes

characteristic

C₂×C₇₁

#1773

yes

—

C₄×#97

—

#1564

—

#1450

—

#1450

D₇₁

yes

characteristic

#1276

—

C₇₁

#8268

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₇₁

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

100

C₄³

—

101

#335

—

102

#334

—

103

#322

—

104

#322

—

142

—

105

C₈²

—

106

C₈²

—

142

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂²×C₈

C₄×D₇₁

yes

—

110

C₂×C₄²

—

111

C₂×C₄²

—

142

—

112

C₂×C₄²

C₂²×D₇₁

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

C₄×D₇₁

yes

—

115

C₂²⋊C₈

—

116

C₂²⋊C₈

—

142

—

117

#97

—

118

#97

—

142

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

142

—

123

C₄×C₈

Dic₁₄₂

yes

—

124

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

125

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

126

C₂²×C₄

—

142

—

127

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

128

C₂²×C₄

#11765865

yes

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

142

—

135

C₂×C₈

—

142

—

136

C₂×C₈

Dic₇₁⋊C₄

yes

—

137

C₂×C₈

Dic₇₁⋊C₄

yes

—

138

C₂×C₈

C₈×D₇₁

yes

—

139

C₂×C₈

#11765848

yes

—

140

C₄²

—

284

—

141

C₄²

—

142

C₄²

—

142

—

143

C₄²

C₂×C₄×D₇₁

yes

—

144

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

145

C₄²

C₂×Dic₁₄₂

yes

characteristic

146

C₂³

#14556085

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

284

—

149

C₂×C₄

—

142

—

150

C₂×C₄

C₄²×D₇₁

yes

characteristic

151

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

152

C₂×C₄

C₄×C₂²⋊D₇₁

yes

—

153

C₂×C₄

C₄×Dic₁₄₂

yes

—

154

C₂×C₄

C₂×C₈×D₇₁

yes

—

155

C₂×C₄

C₂×#11765848

yes

—

156

C₈

—

157

C₈

—

284

—

158

C₈

—

159

C₈

—

142

—

160

C₈

—

142

—

161

C₈

Dic₇₁⋊C₈

yes

—

162

C₈

Dic₇₁⋊C₈

yes

—

163

C₂²

—

164

C₂²

C₄×Dic₇₁⋊C₄

yes

characteristic

165

C₂²

#17986050

yes

—

166

C₄

—

167

C₄

—

284

—

168

C₄

C₄×C₈×D₇₁

yes

characteristic

169

C₄

C₄×#11765848

yes

characteristic

170

C₄

C₂×Dic₇₁⋊C₈

yes

—

171

C₄

#17986119

yes

—

172

C₂

—

173

C₂

C₄×Dic₇₁⋊C₈

yes

characteristic

174

C₂

#22588225

yes

—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1450	C₂
socle	C₂²×C₇₁	C₄×C₄⋊C₄
prime radical	C₂³×C₇₁	#117
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1450	D₇₁
hypocenter	C₇₁	#8268
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1450

subgroup

G/H

center

C₄²

C₂×C₂²⋊D₇₁

commutator

C₄×C₇₁

C₂×C₄×C₈

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1450

C₂

socle

C₂²×C₇₁

C₄×C₄⋊C₄

prime radical

C₂³×C₇₁

#117

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1450

N	H ≅ G/N	type
11. #17985751	C₄	semidirect
12. #17985749	C₄	semidirect
26. #14556038	C₈	semidirect
93. C₇₁	#8268	semidirect

H ≅ G/N

type

11. #17985751

C₄

semidirect

12. #17985749

C₄

semidirect

26. #14556038

C₈

semidirect

93. C₇₁

#8268

semidirect

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

subgroup	index
C₄×#14556038	2
#22587086	2
#22586623	2
#22586172	2
C₇₁×#1450	2
#8268	71

subgroup

index

C₇₁×#1450

Subgroups — #31888619

Conjugacy classes of subgroups

Complements

Maximal subgroups