Group #31913902 — subgroups

Subgroups — #31913902

Order 18176.

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

normal subgroups only

This group has 306 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	C₂×#17985829	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985828	C₂	1	2	yes	maximal
3	C₂×#17985774	C₂	1	1	yes	maximal, characteristic
4	#22588510	C₂	1	2	yes	maximal
5	C₂×#368×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14556064	C₂²	1	2	yes	—
8	C₂×#14556063	—	2	2	no	—
9	#17986022	—	2	2	no	—
10	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
11	#17985829	—	2	2	no	—
12	#17985828	—	2	4	no	—
13	#17985774	C₄	1	4	yes	—
14	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#108×C₇₁	C₂²	1	2	yes	—
16	#368×C₇₁	—	2	2	no	—
17	C₂×#11765849	—	2	2	no	—
18	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
19	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
20	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
22	C₂×C₇₁⋊C₁₆	Q₈	1	2	yes	—
23	#14556064	—	2	2	no	—
24	#14556064	—	2	2	no	—
25	#14556063	—	4	4	no	—
26	Dic₇₁⋊C₈	—	4	4	no	—
27	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
28	C₈×Dic₇₁	Q₈	1	2	yes	—
29	C₂×M₁₆×C₇₁	—	2	2	no	—
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
33	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
34	#108×C₇₁	—	2	2	no	—
35	#108×C₇₁	—	2	2	no	—
36	C₂²×Dic₇₁	Q₁₆	1	2	yes	—
37	#11765849	—	4	2	no	—
38	#11765849	—	4	2	no	—
39	C₂×C₇₁⋊C₈	—	4	2	no	—
40	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
41	C₂×C₇₁⋊C₈	D₈	1	2	yes	—
42	C₂×C₇₁⋊C₈	Q₁₆	1	2	yes	—
43	C₄×Dic₇₁	C₄⋊C₄	1	2	yes	—
44	C₄×Dic₇₁	D₈	1	2	yes	—
45	C₇₁⋊C₁₆	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	M₁₆×C₇₁	—	4	2	no	—
48	M₁₆×C₇₁	—	4	2	no	—
49	C₂×C₈×C₇₁	—	4	2	no	—
50	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₂×Q₈	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	C₁₆×C₇₁	C₄⋊C₄	1	4	yes	—
57	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×Dic₇₁	—	4	2	no	—
65	C₂×Dic₇₁	#107	1	4	yes	—
66	C₈×C₇₁	—	8	2	no	—
67	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
70	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
71	C₇₁⋊C₈	—	8	2	no	—
72	C₇₁⋊C₈	#107	1	4	yes	—
73	C₂²×C₇₁	—	2	2	no	—
74	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
75	C₂²×C₇₁	#494	1	1	yes	characteristic
76	C₂²×C₇₁	#500	1	1	yes	characteristic
77	C₄×C₇₁	#384	1	1	yes	characteristic
78	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
79	C₄×C₇₁	#493	1	1	yes	characteristic
80	C₄×C₇₁	#500	1	1	yes	characteristic
81	Dic₇₁	—	2	4	no	—
82	#12910	—	71	1	no	maximal, Sylow
83	C₂×C₇₁	—	2	1	no	—
84	C₂×C₇₁	—	2	1	no	—
85	C₂×C₇₁	#1845	1	1	yes	characteristic
86	C₂×C₇₁	#2161	1	2	yes	—
87	C₂×#368	—	71	1	no	—
88	C₂×#368	—	71	2	no	—
89	C₂×#368	D₇₁	1	1	yes	characteristic
90	C₂×#346	—	71	1	no	—
91	#1848	—	71	2	no	—
92	C₇₁	#12910	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₁₆	—	71	1	no	—
94	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
95	C₂×#108	—	71	2	no	—
96	C₂×#108	—	142	2	no	—
97	C₂×#108	C₂×D₇₁	1	2	yes	—
98	#423	—	142	2	no	—
99	C₂×C₄×C₈	—	71	1	no	—
100	#368	—	2	2	no	—
101	#368	—	142	4	no	—
102	#368	—	142	2	no	—
103	#346	—	71	4	no	—
104	C₂×M₁₆	—	2	2	no	—
105	C₂×M₁₆	—	142	2	no	—
106	C₂²×C₈	—	71	1	no	—
107	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
108	C₂×C₄²	—	71	1	no	—
109	C₂×C₁₆	—	71	2	no	—
110	C₂×C₁₆	—	71	2	no	—
111	C₂×C₁₆	—	71	2	no	—
112	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
113	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
114	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
115	#108	—	2	2	no	—
116	#108	—	2	2	no	—
117	#108	—	284	4	no	—
118	#108	—	142	2	no	—
119	#108	—	142	2	no	—
120	C₄⋊C₈	—	284	4	no	—
121	C₄×C₈	—	71	2	no	—
122	C₄×C₈	—	71	2	no	—
123	C₂²×C₄	—	71	2	no	—
124	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
125	M₁₆	—	4	2	no	—
126	M₁₆	—	4	2	no	—
127	M₁₆	—	284	2	no	—
128	M₁₆	—	284	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	284	2	no	—
131	C₂×C₈	—	71	2	no	—
132	C₂×C₈	—	71	2	no	—
133	C₂×C₈	—	71	2	no	—
134	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
135	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
136	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
137	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
139	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
140	C₄²	—	71	2	no	—
141	C₄²	—	71	2	no	—
142	C₁₆	—	71	4	no	—
143	C₁₆	Dic₇₁⋊C₄	1	4	yes	—
144	C₂³	Q₁₆×D₇₁	1	1	yes	characteristic
145	C₂×C₄	—	284	2	no	—
146	C₂×C₄	—	71	4	no	—
147	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
148	C₂×C₄	#14556124	1	1	yes	characteristic
149	C₂×C₄	#14556127	1	1	yes	characteristic
150	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
151	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
152	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
153	C₈	—	8	2	no	—
154	C₈	—	568	2	no	—
155	C₈	—	71	4	no	—
156	C₈	C₄⋊C₄×D₇₁	1	1	yes	characteristic
157	C₈	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
158	C₈	#14556126	1	1	yes	characteristic
159	C₈	#14556127	1	1	yes	characteristic
160	C₂²	—	2	2	no	—
161	C₂²	#107×D₇₁	1	1	yes	characteristic
162	C₂²	#17986377	1	1	yes	characteristic
163	C₂²	#17986403	1	1	yes	characteristic
164	C₄	—	142	4	no	—
165	C₄	#17985868	1	1	yes	characteristic
166	C₄	#107×D₇₁	1	1	yes	characteristic
167	C₄	#17986374	1	1	yes	characteristic
168	C₄	#17986403	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#22588498	1	1	yes	characteristic
172	C₂	#22590133	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985829

C₂

yes

maximal, characteristic

C₂×#17985828

C₂

yes

maximal

C₂×#17985774

C₂

yes

maximal, characteristic

#22588510

C₂

yes

maximal

C₂×#368×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₂×#14556064

C₂²

yes

—

C₂×#14556063

—

#17986022

—

C₂×C₈×Dic₇₁

C₂²

yes

—

—

C₄

yes

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#108×C₇₁

C₂²

yes

—

#368×C₇₁

—

C₂×#11765849

—

C₂²×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

C₂×C₇₁⋊C₁₆

Q₈

yes

—

#14556064

—

#14556064

—

#14556063

—

Dic₇₁⋊C₈

—

C₈×Dic₇₁

C₂×C₄

yes

—

C₈×Dic₇₁

Q₈

yes

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

D₄

yes

—

C₂×C₁₆×C₇₁

Q₈

yes

—

#108×C₇₁

—

#108×C₇₁

—

C₂²×Dic₇₁

Q₁₆

yes

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₁⋊C₈

D₈

yes

—

C₂×C₇₁⋊C₈

Q₁₆

yes

—

C₄×Dic₇₁

C₄⋊C₄

yes

—

C₄×Dic₇₁

D₈

yes

—

C₇₁⋊C₁₆

C₄⋊C₄

yes

—

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×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₁₆×C₇₁

C₄⋊C₄

yes

—

C₂³×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

#107

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₄⋊Q₈

yes

characteristic

C₈×C₇₁

C₄⋊Q₈

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

#107

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

#12910

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1845

yes

characteristic

C₂×C₇₁

#2161

yes

—

C₂×#368

—

C₂×#368

—

C₂×#368

D₇₁

yes

characteristic

C₂×#346

—

#1848

—

C₇₁

#12910

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

C₂×#108

—

C₂×#108

—

142

—

C₂×#108

C₂×D₇₁

yes

—

#423

—

142

—

C₂×C₄×C₈

—

100

#368

—

101

#368

—

142

—

102

#368

—

142

—

103

#346

—

104

C₂×M₁₆

—

105

C₂×M₁₆

—

142

—

106

C₂²×C₈

—

107

C₂²×C₈

C₂²×D₇₁

yes

characteristic

108

C₂×C₄²

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

C₂×C₁₆

Dic₁₄₂

yes

—

113

C₂×C₁₆

C₄×D₇₁

yes

—

114

C₂×C₁₆

C₂²⋊D₇₁

yes

—

115

#108

—

116

#108

—

117

#108

—

284

—

118

#108

—

142

—

119

#108

—

142

—

120

C₄⋊C₈

—

284

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₂²×C₄

—

124

C₂²×C₄

D₄×D₇₁

yes

characteristic

125

M₁₆

—

126

M₁₆

—

127

M₁₆

—

284

—

128

M₁₆

—

284

—

129

C₂×C₈

—

130

C₂×C₈

—

284

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

135

C₂×C₈

D₄×D₇₁

yes

characteristic

136

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

137

C₂×C₈

Q₈×D₇₁

yes

characteristic

138

C₂×C₈

Q₈×D₇₁

yes

characteristic

139

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

140

C₄²

—

141

C₄²

—

142

C₁₆

—

143

C₁₆

Dic₇₁⋊C₄

yes

—

144

C₂³

Q₁₆×D₇₁

yes

characteristic

145

C₂×C₄

—

284

—

146

C₂×C₄

—

147

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

148

C₂×C₄

#14556124

yes

characteristic

149

C₂×C₄

#14556127

yes

characteristic

150

C₂×C₄

D₈×D₇₁

yes

characteristic

151

C₂×C₄

D₈×D₇₁

yes

characteristic

152

C₂×C₄

Q₁₆×D₇₁

yes

characteristic

153

C₈

—

154

C₈

—

568

—

155

C₈

—

156

C₈

C₄⋊C₄×D₇₁

yes

characteristic

157

C₈

C₂×Dic₇₁⋊C₄

yes

characteristic

158

C₈

#14556126

yes

characteristic

159

C₈

#14556127

yes

characteristic

160

C₂²

—

161

C₂²

#107×D₇₁

yes

characteristic

162

C₂²

#17986377

yes

characteristic

163

C₂²

#17986403

yes

characteristic

164

C₄

—

142

—

165

C₄

#17985868

yes

characteristic

166

C₄

#107×D₇₁

yes

characteristic

167

C₄

#17986374

yes

characteristic

168

C₄

#17986403

yes

characteristic

169

C₂

—

170

C₂

—

171

C₂

#22588498

yes

characteristic

172

C₂

#22590133

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#368×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂³×C₇₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#368	D₇₁
hypocenter	C₇₁	#12910
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₂×C₄

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×#368×C₇₁

C₂

socle

C₂²×C₇₁

C₂×#107

prime radical

C₂³×C₇₁

C₂×Q₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
92. C₇₁	#12910	semidirect

H ≅ G/N

type

92. C₇₁

#12910

semidirect

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

subgroup	index
C₂×#17985829	2
C₂×#17985828	2
C₂×#17985774	2
#22588510	2
C₂×#368×C₇₁	2
#12910	71

subgroup

index

C₂×#368×C₇₁

Subgroups — #31913902

Conjugacy classes of subgroups

Complements

Maximal subgroups