Group #31916060 — subgroups

Subgroups — #31916060

Order 18176.

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

normal subgroups only

This group has 374 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×#17986401	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985775	C₂	1	1	yes	maximal, characteristic
3	#22586431	C₂	1	4	yes	maximal
4	C₇₁×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556120	—	2	1	no	—
6	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
7	#17986401	—	2	2	no	—
8	#17986401	C₄	1	4	yes	complemented
9	C₂×#14556062	C₂²	1	2	yes	—
10	C₂×#14556060	—	2	1	no	—
11	#17985775	—	2	2	no	—
12	C₂×#107×C₇₁	—	2	1	no	—
13	C₂×#107×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	#384×C₇₁	—	2	1	no	—
16	#14556120	—	2	4	no	—
17	#14556120	—	4	2	no	—
18	C₂×C₄⋊Dic₇₁	—	2	2	no	—
19	C₂×Dic₇₁⋊C₄	—	4	2	no	—
20	C₂×C₇₁⋊C₁₆	—	2	2	no	—
21	C₂×C₇₁⋊C₁₆	—	2	2	no	—
22	C₂×C₇₁⋊C₁₆	Q₈	1	4	yes	—
23	#14556062	—	2	2	no	—
24	#14556062	C₂×C₄	1	4	yes	—
25	#14556060	—	2	4	no	—
26	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₇₁	—	4	2	no	—
30	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
31	#107×C₇₁	—	2	4	no	—
32	#107×C₇₁	—	4	2	no	—
33	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₇₁	D₄	1	2	yes	—
35	C₂²×Dic₇₁	—	4	2	no	—
36	C₄⋊Dic₇₁	—	2	4	no	—
37	C₄⋊Dic₇₁	—	4	2	no	—
38	Dic₇₁⋊C₄	—	4	8	no	—
39	C₇₁⋊C₁₆	—	2	4	no	—
40	C₂²×C₄×C₇₁	—	4	2	no	—
41	C₂²×C₄×C₇₁	D₈	1	2	yes	—
42	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
45	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
46	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₇₁	#41	1	2	yes	—
50	C₄⋊C₄×C₇₁	—	4	2	no	—
51	C₄⋊C₄×C₇₁	—	4	2	no	—
52	C₄⋊C₄×C₇₁	—	8	4	no	—
53	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₇₁	SD₁₆	1	2	yes	—
55	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₇₁	—	2	2	no	—
57	C₂×C₄×C₇₁	—	2	2	no	—
58	C₂×C₄×C₇₁	—	8	2	no	—
59	C₂×C₄×C₇₁	—	8	4	no	—
60	C₂×C₄×C₇₁	#102	1	4	yes	—
61	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×Dic₇₁	—	4	4	no	—
68	C₂×Dic₇₁	—	8	2	no	—
69	C₂×Dic₇₁	—	8	2	no	—
70	C₈×C₇₁	#103	1	4	yes	—
71	C₈×C₇₁	C₄×Q₈	1	2	yes	—
72	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
73	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
74	C₂²×C₇₁	#486	1	1	yes	characteristic
75	C₂²×C₇₁	#500	1	1	yes	characteristic
76	C₂²×C₇₁	#509	1	2	yes	—
77	C₂²×C₇₁	#510	1	2	yes	—
78	C₄×C₇₁	—	2	4	no	—
79	C₄×C₇₁	—	16	2	no	—
80	C₄×C₇₁	#391	1	1	yes	characteristic
81	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
82	C₄×C₇₁	#487	1	1	yes	characteristic
83	C₄×C₇₁	#498	1	1	yes	characteristic
84	Dic₇₁	—	8	4	no	—
85	#13284	—	71	1	no	maximal, Sylow
86	C₂×C₇₁	#1990	1	1	yes	characteristic
87	C₂×C₇₁	#2139	1	2	yes	—
88	C₂×C₇₁	#2244	1	1	yes	characteristic
89	C₂×C₇₁	#2245	1	1	yes	characteristic
90	C₂×C₇₁	#2255	1	2	yes	—
91	C₂×#499	—	71	1	no	—
92	C₂×#346	—	71	1	no	—
93	#1845	D₇₁	1	1	yes	characteristic
94	#1380	—	71	4	no	—
95	C₇₁	#13284	1	1	yes	characteristic, complemented, Sylow
96	C₂×#125	—	142	1	no	—
97	C₂²×C₁₆	—	71	2	no	—
98	#499	—	71	4	no	—
99	#499	—	142	2	no	—
100	C₂×#107	—	2	1	no	—
101	C₂×#107	—	71	2	no	—
102	C₂×#107	C₂×D₇₁	1	2	yes	—
103	C₂×#106	—	142	1	no	—
104	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
105	#384	—	2	1	no	—
106	#346	—	142	2	no	—
107	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
109	#125	—	284	2	no	—
110	#125	—	142	4	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	4	2	no	—
113	C₂×C₄⋊C₄	—	284	2	no	—
114	C₂×C₄⋊C₄	—	142	2	no	—
115	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
116	C₂×C₁₆	—	71	4	no	—
117	C₂×C₁₆	—	142	2	no	—
118	C₂×C₁₆	—	142	2	no	—
119	#107	—	2	4	no	—
120	#107	—	4	2	no	—
121	#107	—	71	4	no	—
122	#107	—	142	2	no	—
123	#106	—	142	4	no	—
124	C₄×C₈	C₄×D₇₁	1	2	yes	—
125	C₄×C₈	D₂₈₄	1	2	yes	—
126	C₂²×C₄	—	4	2	no	—
127	C₂²×C₄	—	284	2	no	—
128	C₂²×C₄	#11765850	1	2	yes	—
129	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
130	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
131	C₂×C₈	#11765853	1	2	yes	—
132	C₂×C₈	#11765855	1	2	yes	—
133	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
134	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₇₁	1	2	yes	—
136	C₂×C₈	#11765864	1	2	yes	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	8	4	no	—
140	C₄⋊C₄	—	284	8	no	—
141	C₄⋊C₄	—	284	2	no	—
142	C₄⋊C₄	—	142	4	no	—
143	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
144	C₄²	#11765852	1	2	yes	—
145	C₁₆	—	142	4	no	—
146	C₂³	C₂×#11765850	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	4	no	—
151	C₂×C₄	—	284	4	no	—
152	C₂×C₄	—	568	2	no	—
153	C₂×C₄	—	568	2	no	—
154	C₂×C₄	#14556048	1	4	yes	—
155	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
156	C₂×C₄	#14556115	1	1	yes	characteristic
157	C₂×C₄	#14556126	1	1	yes	characteristic
158	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
159	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
160	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
161	C₈	#14556051	1	4	yes	—
162	C₈	#14556093	1	2	yes	—
163	C₈	#14556108	1	2	yes	—
164	C₂²	C₂×#14556048	1	1	yes	characteristic
165	C₂²	#17986345	1	1	yes	characteristic
166	C₂²	#17986402	1	1	yes	characteristic
167	C₂²	#17986435	1	2	yes	—
168	C₂²	#17986440	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	16	2	no	—
171	C₄	—	568	4	no	—
172	C₄	#17985904	1	1	yes	characteristic
173	C₄	C₂×#14556051	1	1	yes	characteristic
174	C₄	#17986349	1	1	yes	characteristic
175	C₄	#17986395	1	1	yes	characteristic
176	C₂	#22589300	1	1	yes	characteristic
177	C₂	#22590050	1	2	yes	—
178	C₂	#22590581	1	1	yes	characteristic
179	C₂	#22590585	1	1	yes	characteristic
180	C₂	#22590623	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986401

C₂

yes

maximal, characteristic

C₂×#17985775

C₂

yes

maximal, characteristic

#22586431

C₂

yes

maximal

C₇₁×#1845

C₂

yes

maximal, characteristic

C₂×#14556120

—

C₂²×C₇₁⋊C₁₆

C₂²

yes

—

#17986401

—

#17986401

C₄

yes

complemented

C₂×#14556062

C₂²

yes

—

C₂×#14556060

—

#17985775

—

C₂×#107×C₇₁

—

C₂×#107×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#384×C₇₁

—

#14556120

—

#14556120

—

C₂×C₄⋊Dic₇₁

—

C₂×Dic₇₁⋊C₄

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

Q₈

yes

—

#14556062

—

#14556062

C₂×C₄

yes

—

#14556060

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

characteristic

#107×C₇₁

—

#107×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₈

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

Q₁₆

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂²⋊C₄

yes

—

C₄²×C₇₁

SD₁₆

yes

—

C₂³×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#124

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

#103

yes

—

C₈×C₇₁

C₄×Q₈

yes

—

C₈×C₇₁

C₂²⋊Q₈

yes

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₂²×C₇₁

#510

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×#499

—

C₂×#346

—

#1845

D₇₁

yes

characteristic

#1380

—

C₇₁

#13284

yes

characteristic, complemented, Sylow

C₂×#125

—

142

—

C₂²×C₁₆

—

#499

—

#499

—

142

—

100

C₂×#107

—

101

C₂×#107

—

102

C₂×#107

C₂×D₇₁

yes

—

103

C₂×#106

—

142

—

104

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

105

#384

—

106

#346

—

142

—

107

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

108

C₂²×C₈

C₂²×D₇₁

yes

characteristic

109

#125

—

284

—

110

#125

—

142

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

284

—

114

C₂×C₄⋊C₄

—

142

—

115

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

116

C₂×C₁₆

—

117

C₂×C₁₆

—

142

—

118

C₂×C₁₆

—

142

—

119

#107

—

120

#107

—

121

#107

—

122

#107

—

142

—

123

#106

—

142

—

124

C₄×C₈

C₄×D₇₁

yes

—

125

C₄×C₈

D₂₈₄

yes

—

126

C₂²×C₄

—

127

C₂²×C₄

—

284

—

128

C₂²×C₄

#11765850

yes

—

129

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

130

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

131

C₂×C₈

#11765853

yes

—

132

C₂×C₈

#11765855

yes

—

133

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

134

C₂×C₈

C₂×D₂₈₄

yes

characteristic

135

C₂×C₈

Q₈×D₇₁

yes

—

136

C₂×C₈

#11765864

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

284

—

141

C₄⋊C₄

—

284

—

142

C₄⋊C₄

—

142

—

143

C₄²

(C₂×D₇₁)⋊C₄

yes

—

144

C₄²

#11765852

yes

—

145

C₁₆

—

142

—

146

C₂³

C₂×#11765850

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

284

—

152

C₂×C₄

—

568

—

153

C₂×C₄

—

568

—

154

C₂×C₄

#14556048

yes

—

155

C₂×C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

156

C₂×C₄

#14556115

yes

characteristic

157

C₂×C₄

#14556126

yes

characteristic

158

C₂×C₄

C₂×#11765852

yes

characteristic

159

C₂×C₄

C₂×#11765853

yes

characteristic

160

C₂×C₄

C₂×#11765855

yes

characteristic

161

C₈

#14556051

yes

—

162

C₈

#14556093

yes

—

163

C₈

#14556108

yes

—

164

C₂²

C₂×#14556048

yes

characteristic

165

C₂²

#17986345

yes

characteristic

166

C₂²

#17986402

yes

characteristic

167

C₂²

#17986435

yes

—

168

C₂²

#17986440

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

568

—

172

C₄

#17985904

yes

characteristic

173

C₄

C₂×#14556051

yes

characteristic

174

C₄

#17986349

yes

characteristic

175

C₄

#17986395

yes

characteristic

176

C₂

#22589300

yes

characteristic

177

C₂

#22590050

yes

—

178

C₂

#22590581

yes

characteristic

179

C₂

#22590585

yes

characteristic

180

C₂

#22590623

yes

—

	subgroup	G/H
center	C₂³	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1845	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1845	D₇₁
hypocenter	C₇₁	#13284
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

subgroup

G/H

center

C₂³

C₂×#11765850

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1845

C₂

socle, prime radical

C₂³×C₇₁

C₂×D₈

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

N	H ≅ G/N	type
8. #17986401	C₄	semidirect
95. C₇₁	#13284	semidirect

H ≅ G/N

type

8. #17986401

C₄

semidirect

95. C₇₁

#13284

semidirect

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

subgroup	index
C₂×#17986401	2
C₂×#17985775	2
#22586431	2
C₇₁×#1845	2
#13284	71

subgroup

index

C₇₁×#1845

Subgroups — #31916060

Conjugacy classes of subgroups

Complements

Maximal subgroups