Group #31915518 — subgroups

Subgroups — #31915518

Order 18176.

274 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 123 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×#17985827	C₂	1	1	yes	maximal, characteristic
2	#22589936	C₂	1	1	yes	maximal, characteristic
3	#22589126	C₂	1	2	yes	maximal
4	#22586450	C₂	1	2	yes	maximal
5	C₂×#368×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	1	yes	characteristic
7	#17986041	—	2	2	no	—
8	C₂×#14556062	C₂²	1	2	yes	—
9	#17985954	C₂²	1	1	yes	characteristic
10	#17985827	—	2	1	no	—
11	#17985827	—	2	1	no	—
12	#17985774	—	2	1	no	—
13	C₁₆×Dic₇₁	—	2	1	no	—
14	#17985767	—	2	2	no	—
15	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#108×C₇₁	C₂²	1	2	yes	—
17	#368×C₇₁	C₄	1	4	yes	complemented
18	C₂×#11765849	D₄	1	1	yes	characteristic
19	#14556087	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₇₁	—	2	2	no	—
21	#14556068	—	2	1	no	—
22	#14556068	Q₈	1	2	yes	—
23	C₂×C₇₁⋊C₁₆	—	2	1	no	—
24	#14556062	—	2	2	no	—
25	#14556062	—	2	2	no	—
26	#14556062	—	4	2	no	—
27	#14556060	—	4	2	no	—
28	#14556038	—	2	1	no	—
29	C₈×Dic₇₁	—	2	1	no	—
30	C₂×M₁₆×C₇₁	—	2	2	no	—
31	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
36	#108×C₇₁	—	2	2	no	—
37	#108×C₇₁	C₂×C₄	1	4	yes	—
38	C₂²×Dic₇₁	—	4	2	no	—
39	#11765849	—	2	1	no	—
40	#11765849	Q₁₆	1	2	yes	—
41	C₂×C₇₁⋊C₈	—	2	1	no	—
42	C₄⋊Dic₇₁	—	4	2	no	—
43	C₄⋊Dic₇₁	—	4	2	no	—
44	C₄⋊Dic₇₁	D₈	1	2	yes	—
45	C₂²⋊Dic₇₁	—	2	1	no	—
46	C₄×Dic₇₁	—	2	1	no	—
47	C₇₁⋊C₁₆	—	2	2	no	—
48	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	M₁₆×C₇₁	—	2	4	no	—
50	M₁₆×C₇₁	—	4	2	no	—
51	C₂×C₈×C₇₁	—	4	2	no	—
52	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
57	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
58	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
59	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
60	C₂³×C₇₁	#135	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
67	C₂×Dic₇₁	—	2	2	no	—
68	C₂×Dic₇₁	—	8	2	no	—
69	C₂×Dic₇₁	—	16	2	no	—
70	C₈×C₇₁	—	4	4	no	—
71	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
73	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
74	C₈×C₇₁	#125	1	1	yes	characteristic
75	C₇₁⋊C₈	—	2	2	no	—
76	C₂²×C₇₁	—	2	2	no	—
77	C₂²×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
78	C₂²×C₇₁	#466	1	1	yes	characteristic
79	C₂²×C₇₁	#499	1	1	yes	characteristic
80	C₄×C₇₁	#389	1	1	yes	characteristic
81	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
82	C₄×C₇₁	#467	1	1	yes	characteristic
83	C₄×C₇₁	#500	1	1	yes	characteristic
84	Dic₇₁	—	4	2	no	—
85	Dic₇₁	—	16	2	no	—
86	#13172	—	71	1	no	maximal, Sylow
87	C₂×C₇₁	—	2	1	no	—
88	C₂×C₇₁	—	2	1	no	—
89	C₂×C₇₁	#1942	1	1	yes	characteristic
90	C₂×C₇₁	#2223	1	2	yes	—
91	C₂×#368	D₇₁	1	1	yes	characteristic
92	C₂×#367	—	71	1	no	—
93	#2108	—	71	1	no	—
94	#1952	—	71	2	no	—
95	#1387	—	71	2	no	—
96	C₇₁	#13172	1	1	yes	characteristic, complemented, Sylow
97	C₂×M₃₂	—	71	1	no	—
98	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
99	C₂×#108	C₂×D₇₁	1	2	yes	—
100	#428	—	142	2	no	—
101	C₂×#107	—	71	2	no	—
102	#405	—	71	1	no	—
103	#368	Dic₇₁	1	4	yes	complemented
104	#367	—	142	1	no	—
105	#367	—	142	1	no	—
106	#346	—	142	1	no	—
107	C₄×C₁₆	—	142	1	no	—
108	#343	—	142	2	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	71	1	no	—
111	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
112	#117	—	71	1	no	—
113	C₂×C₄⋊C₄	—	142	2	no	—
114	M₃₂	—	71	2	no	—
115	M₃₂	—	142	1	no	—
116	C₂×C₁₆	—	142	1	no	—
117	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
118	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
119	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
120	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
121	#108	—	2	2	no	—
122	#108	C₂×Dic₇₁	1	4	yes	—
123	#107	—	284	2	no	—
124	#107	—	142	2	no	—
125	#107	—	142	2	no	—
126	#106	—	284	2	no	—
127	#97	—	142	1	no	—
128	C₄×C₈	—	142	1	no	—
129	C₂²×C₄	—	284	2	no	—
130	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
131	M₁₆	—	2	4	no	—
132	M₁₆	—	4	2	no	—
133	M₁₆	—	71	2	no	—
134	M₁₆	—	142	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	142	1	no	—
137	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
138	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
139	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
140	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
141	C₂×C₈	#11765864	1	1	yes	characteristic
142	C₂×C₈	#11765866	1	1	yes	characteristic
143	C₄⋊C₄	—	284	2	no	—
144	C₄⋊C₄	—	284	2	no	—
145	C₄⋊C₄	—	71	2	no	—
146	C₂²⋊C₄	—	142	1	no	—
147	C₄²	—	142	1	no	—
148	C₁₆	—	142	2	no	—
149	C₁₆	C₄⋊Dic₇₁	1	2	yes	—
150	C₁₆	C₄⋊Dic₇₁	1	2	yes	—
151	C₂³	#14556151	1	1	yes	characteristic
152	C₂×C₄	—	568	2	no	—
153	C₂×C₄	—	1136	2	no	—
154	C₂×C₄	—	142	2	no	—
155	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
156	C₂×C₄	#14556103	1	1	yes	characteristic
157	C₂×C₄	#14556127	1	1	yes	characteristic
158	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
159	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
160	C₂×C₄	#14556147	1	1	yes	characteristic
161	C₈	—	4	4	no	—
162	C₈	—	142	2	no	—
163	C₈	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
164	C₈	Q₈×Dic₇₁	1	1	yes	characteristic
165	C₈	#14556108	1	1	yes	characteristic
166	C₈	#14556119	1	1	yes	characteristic
167	C₂²	—	2	2	no	—
168	C₂²	Q₁₆×Dic₇₁	1	1	yes	characteristic
169	C₂²	#17986236	1	1	yes	characteristic
170	C₂²	#17986400	1	1	yes	characteristic
171	C₄	—	284	2	no	—
172	C₄	—	1136	2	no	—
173	C₄	#17985898	1	1	yes	characteristic
174	C₄	D₈×Dic₇₁	1	1	yes	characteristic
175	C₄	#17986241	1	1	yes	characteristic
176	C₄	#17986403	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#22589074	1	1	yes	characteristic
180	C₂	#22590471	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985827

C₂

yes

maximal, characteristic

#22589936

C₂

yes

maximal, characteristic

#22589126

C₂

yes

maximal

#22586450

C₂

yes

maximal

C₂×#368×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

characteristic

#17986041

—

C₂×#14556062

C₂²

yes

—

#17985954

C₂²

yes

—

—

—

C₁₆×Dic₇₁

—

#17985767

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#108×C₇₁

C₂²

yes

—

#368×C₇₁

C₄

yes

complemented

C₂×#11765849

D₄

yes

characteristic

#14556087

D₄

yes

characteristic

C₂×C₄⋊Dic₇₁

—

#14556068

—

#14556068

Q₈

yes

—

C₂×C₇₁⋊C₁₆

—

—

—

—

—

—

C₈×Dic₇₁

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

Q₈

yes

—

#108×C₇₁

—

#108×C₇₁

C₂×C₄

yes

—

C₂²×Dic₇₁

—

#11765849

—

#11765849

Q₁₆

yes

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₈

yes

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

C₂×C₈×C₇₁

yes

C₂×C₈×C₇₁

yes

C₁₆×C₇₁

C₄⋊C₄

yes

—

C₁₆×C₇₁

C₄⋊C₄

yes

—

C₂³×C₇₁

#135

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

#121

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₄×Q₈

yes

characteristic

C₈×C₇₁

C₂²⋊Q₈

yes

characteristic

C₈×C₇₁

#125

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄×Q₁₆

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₄×D₈

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

#13172

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1942

yes

characteristic

C₂×C₇₁

#2223

yes

—

C₂×#368

D₇₁

yes

—

—

—

—

C₇₁

#13172

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

C₂×#108

C₂×D₇₁

yes

—

100

#428

—

142

—

101

C₂×#107

—

102

#405

—

103

#368

Dic₇₁

yes

complemented

104

#367

—

142

—

105

#367

—

142

—

106

#346

—

142

—

107

C₄×C₁₆

—

142

—

108

#343

—

142

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

111

C₂²×C₈

C₂²×D₇₁

yes

characteristic

112

#117

—

113

C₂×C₄⋊C₄

—

142

—

114

M₃₂

—

115

M₃₂

—

142

—

116

C₂×C₁₆

—

142

—

117

C₂×C₁₆

Dic₁₄₂

yes

—

118

C₂×C₁₆

D₂₈₄

yes

characteristic

119

C₂×C₁₆

D₂₈₄

yes

characteristic

120

C₂×C₁₆

C₂×Dic₇₁

yes

—

121

#108

—

122

#108

C₂×Dic₇₁

yes

—

123

#107

—

284

—

124

#107

—

142

—

125

#107

—

142

—

126

#106

—

284

—

127

#97

—

142

—

128

C₄×C₈

—

142

—

129

C₂²×C₄

—

284

—

130

C₂²×C₄

D₄×D₇₁

yes

characteristic

131

M₁₆

—

132

M₁₆

—

133

M₁₆

—

134

M₁₆

—

142

—

135

C₂×C₈

—

136

C₂×C₈

—

142

—

137

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

138

C₂×C₈

C₂×D₂₈₄

yes

characteristic

139

C₂×C₈

Q₈×D₇₁

yes

characteristic

140

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

141

C₂×C₈

#11765864

yes

characteristic

142

C₂×C₈

#11765866

yes

characteristic

143

C₄⋊C₄

—

284

—

144

C₄⋊C₄

—

284

—

145

C₄⋊C₄

—

146

C₂²⋊C₄

—

142

—

147

C₄²

—

142

—

148

C₁₆

—

142

—

149

C₁₆

C₄⋊Dic₇₁

yes

—

150

C₁₆

C₄⋊Dic₇₁

yes

—

151

C₂³

#14556151

yes

characteristic

152

C₂×C₄

—

568

—

153

C₂×C₄

—

1136

—

154

C₂×C₄

—

142

—

155

C₂×C₄

D₄×Dic₇₁

yes

characteristic

156

C₂×C₄

#14556103

yes

characteristic

157

C₂×C₄

#14556127

yes

characteristic

158

C₂×C₄

D₈×D₇₁

yes

characteristic

159

C₂×C₄

Q₁₆×D₇₁

yes

characteristic

160

C₂×C₄

#14556147

yes

characteristic

161

C₈

—

162

C₈

—

142

—

163

C₈

C₂×C₄⋊Dic₇₁

yes

characteristic

164

C₈

Q₈×Dic₇₁

yes

characteristic

165

C₈

#14556108

yes

characteristic

166

C₈

#14556119

yes

characteristic

167

C₂²

—

168

C₂²

Q₁₆×Dic₇₁

yes

characteristic

169

C₂²

#17986236

yes

characteristic

170

C₂²

#17986400

yes

characteristic

171

C₄

—

284

—

172

C₄

—

1136

—

173

C₄

#17985898

yes

characteristic

174

C₄

D₈×Dic₇₁

yes

characteristic

175

C₄

#17986241

yes

characteristic

176

C₄

#17986403

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#22589074

yes

characteristic

180

C₂

#22590471

yes

—

	subgroup	G/H
center	C₂²	#17986236
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#368×C₇₁	C₂
socle	C₂²×C₇₁	#466
prime radical	C₂³×C₇₁	#135
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#368	D₇₁
hypocenter	C₇₁	#13172
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₂²

#17986236

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×#368×C₇₁

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
17. #368×C₇₁	C₄	semidirect
96. C₇₁	#13172	semidirect
103. #368	Dic₇₁	semidirect

H ≅ G/N

type

17. #368×C₇₁

C₄

semidirect

96. C₇₁

#13172

semidirect

103. #368

Dic₇₁

semidirect

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

subgroup	index
C₂×#17985827	2
#22589936	2
#22589126	2
#22586450	2
C₂×#368×C₇₁	2
#13172	71

subgroup

index

C₂×#368×C₇₁

Subgroups — #31915518

Conjugacy classes of subgroups

Complements

Maximal subgroups