Group #45696571 — subgroups

Subgroups — C₁₃₁×#10687 #45696571

Order 33536.

264 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 106 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	#1940×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#1849×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1303×C₁₃₁	C₂	1	2	yes	maximal
4	#1301×C₁₃₁	C₂	1	2	yes	maximal
5	#1275×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	#431×C₁₃₁	C₂²	1	2	yes	—
7	C₂×#108×C₁₃₁	—	2	1	no	—
8	C₂×#108×C₁₃₁	—	2	1	no	—
9	#423×C₁₃₁	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₁₃₁	—	2	1	no	—
11	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×#97×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
14	#336×C₁₃₁	—	2	1	no	—
15	C₂×M₁₆×C₁₃₁	—	2	2	no	—
16	C₂²×C₈×C₁₃₁	—	2	1	no	—
17	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
20	C₂×C₄²×C₁₃₁	—	2	1	no	—
21	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
22	#108×C₁₃₁	—	4	2	no	—
23	#108×C₁₃₁	—	4	2	no	—
24	C₄⋊C₈×C₁₃₁	—	2	2	no	—
25	C₄⋊C₈×C₁₃₁	—	2	4	no	—
26	C₄⋊C₈×C₁₃₁	—	2	2	no	—
27	C₄⋊C₈×C₁₃₁	—	4	2	no	—
28	C₂²⋊C₈×C₁₃₁	—	2	2	no	—
29	#97×C₁₃₁	—	2	2	no	—
30	C₄×C₈×C₁₃₁	—	2	2	no	—
31	#95×C₁₃₁	Q₈	1	1	yes	characteristic
32	#95×C₁₃₁	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₃₁	—	2	1	no	—
34	C₂²×C₄×C₁₃₁	—	2	1	no	—
35	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₁	#41	1	1	yes	characteristic
38	M₁₆×C₁₃₁	—	4	2	no	—
39	M₁₆×C₁₃₁	—	4	2	no	—
40	C₂×C₈×C₁₃₁	—	2	1	no	—
41	C₂×C₈×C₁₃₁	—	2	1	no	—
42	C₂×C₈×C₁₃₁	—	2	1	no	—
43	C₂×C₈×C₁₃₁	—	2	1	no	—
44	C₂×C₈×C₁₃₁	—	4	1	no	—
45	C₂×C₈×C₁₃₁	—	4	2	no	—
46	C₂×C₈×C₁₃₁	—	4	2	no	—
47	C₂×C₈×C₁₃₁	—	4	1	no	—
48	C₂×C₈×C₁₃₁	—	4	1	no	—
49	C₂×C₈×C₁₃₁	D₈	1	2	yes	—
50	C₂×C₈×C₁₃₁	SD₁₆	1	2	yes	—
51	C₄⋊C₄×C₁₃₁	—	4	2	no	—
52	C₄²×C₁₃₁	—	4	1	no	—
53	C₄²×C₁₃₁	—	4	1	no	—
54	C₄²×C₁₃₁	#41	1	2	yes	—
55	C₄²×C₁₃₁	#41	1	2	yes	—
56	C₂³×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃₁	—	2	2	no	—
58	C₂×C₄×C₁₃₁	—	4	1	no	—
59	C₂×C₄×C₁₃₁	—	4	1	no	—
60	C₂×C₄×C₁₃₁	—	4	1	no	—
61	C₂×C₄×C₁₃₁	—	4	1	no	—
62	C₂×C₄×C₁₃₁	—	4	1	no	—
63	C₂×C₄×C₁₃₁	—	4	1	no	—
64	C₂×C₄×C₁₃₁	—	4	1	no	—
65	C₂×C₄×C₁₃₁	#123	1	1	yes	characteristic
66	C₂×C₄×C₁₃₁	#124	1	1	yes	characteristic
67	C₂×C₄×C₁₃₁	#125	1	2	yes	—
68	C₂×C₄×C₁₃₁	#126	1	2	yes	—
69	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₃₁	#137	1	1	yes	characteristic
72	C₂×C₄×C₁₃₁	#137	1	1	yes	characteristic
73	C₈×C₁₃₁	—	2	2	no	—
74	C₈×C₁₃₁	—	2	2	no	—
75	C₈×C₁₃₁	—	8	2	no	—
76	C₈×C₁₃₁	—	8	1	no	—
77	C₈×C₁₃₁	—	8	2	no	—
78	C₂²×C₁₃₁	—	2	1	no	—
79	C₂²×C₁₃₁	—	2	1	no	—
80	C₂²×C₁₃₁	#400	1	1	yes	characteristic
81	C₂²×C₁₃₁	#474	1	1	yes	characteristic
82	C₂²×C₁₃₁	#475	1	1	yes	characteristic
83	C₄×C₁₃₁	—	2	2	no	—
84	C₄×C₁₃₁	—	2	2	no	—
85	C₄×C₁₃₁	—	4	2	no	—
86	C₄×C₁₃₁	—	8	1	no	—
87	C₄×C₁₃₁	#480	1	1	yes	characteristic
88	C₄×C₁₃₁	#483	1	1	yes	characteristic
89	C₄×C₁₃₁	#486	1	1	yes	characteristic
90	C₄×C₁₃₁	#491	1	1	yes	characteristic
91	C₂×C₁₃₁	—	2	1	no	—
92	C₂×C₁₃₁	—	2	1	no	—
93	C₂×C₁₃₁	#2099	1	1	yes	characteristic
94	C₂×C₁₃₁	#2102	1	1	yes	characteristic
95	C₂×C₁₃₁	#2103	1	1	yes	characteristic
96	#10687	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₁₃₁	#10687	1	1	yes	characteristic, direct product factor, Sylow
98	#1940	C₂×C₁₃₁	1	1	yes	characteristic
99	#1849	C₂×C₁₃₁	1	1	yes	characteristic
100	#1303	C₂×C₁₃₁	1	2	yes	—
101	#1301	C₂×C₁₃₁	1	2	yes	—
102	#1275	C₂×C₁₃₁	1	1	yes	characteristic
103	#431	C₂²×C₁₃₁	1	2	yes	—
104	C₂×#108	—	2	1	no	—
105	C₂×#108	—	2	1	no	—
106	#423	C₂²×C₁₃₁	1	2	yes	—
107	C₂×C₄⋊C₈	—	2	1	no	—
108	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	1	yes	characteristic
109	C₂×#97	C₂²×C₁₃₁	1	1	yes	characteristic
110	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
111	#336	—	2	1	no	—
112	C₂×M₁₆	—	2	2	no	—
113	C₂²×C₈	—	2	1	no	—
114	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
115	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	2	1	no	—
117	C₂×C₄²	—	2	1	no	—
118	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
119	#108	—	4	2	no	—
120	#108	—	4	2	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	2	4	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	4	2	no	—
125	C₂²⋊C₈	—	2	2	no	—
126	#97	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	#95	Q₈×C₁₃₁	1	1	yes	characteristic
129	#95	Q₈×C₁₃₁	1	1	yes	characteristic
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
133	C₂²×C₄	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
134	C₂²×C₄	#41×C₁₃₁	1	1	yes	characteristic
135	M₁₆	—	4	2	no	—
136	M₁₆	—	4	2	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	1	no	—
145	C₂×C₈	—	4	1	no	—
146	C₂×C₈	D₈×C₁₃₁	1	2	yes	—
147	C₂×C₈	SD₁₆×C₁₃₁	1	2	yes	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄²	—	4	1	no	—
150	C₄²	—	4	1	no	—
151	C₄²	#41×C₁₃₁	1	2	yes	—
152	C₄²	#41×C₁₃₁	1	2	yes	—
153	C₂³	C₂²⋊Q₈×C₁₃₁	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	#123×C₁₃₁	1	1	yes	characteristic
163	C₂×C₄	#124×C₁₃₁	1	1	yes	characteristic
164	C₂×C₄	#125×C₁₃₁	1	2	yes	—
165	C₂×C₄	#126×C₁₃₁	1	2	yes	—
166	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
167	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
168	C₂×C₄	#137×C₁₃₁	1	1	yes	characteristic
169	C₂×C₄	#137×C₁₃₁	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	—	8	2	no	—
173	C₈	—	8	1	no	—
174	C₈	—	8	2	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	#400×C₁₃₁	1	1	yes	characteristic
178	C₂²	#474×C₁₃₁	1	1	yes	characteristic
179	C₂²	#475×C₁₃₁	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	#480×C₁₃₁	1	1	yes	characteristic
185	C₄	#483×C₁₃₁	1	1	yes	characteristic
186	C₄	#486×C₁₃₁	1	1	yes	characteristic
187	C₄	#491×C₁₃₁	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	2	1	no	—
190	C₂	#2099×C₁₃₁	1	1	yes	characteristic
191	C₂	#2102×C₁₃₁	1	1	yes	characteristic
192	C₂	#2103×C₁₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1940×C₁₃₁

C₂

yes

maximal, characteristic

#1849×C₁₃₁

C₂

yes

maximal, characteristic

#1303×C₁₃₁

C₂

yes

maximal

#1301×C₁₃₁

C₂

yes

maximal

#1275×C₁₃₁

C₂

yes

maximal, characteristic

#431×C₁₃₁

C₂²

yes

—

C₂×#108×C₁₃₁

—

C₂×#108×C₁₃₁

—

#423×C₁₃₁

C₂²

yes

—

C₂×C₄⋊C₈×C₁₃₁

—

C₂×C₄⋊C₈×C₁₃₁

C₂²

yes

characteristic

C₂×#97×C₁₃₁

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₁₃₁

C₂²

yes

characteristic

#336×C₁₃₁

—

C₂×M₁₆×C₁₃₁

—

C₂²×C₈×C₁₃₁

—

C₂²×C₈×C₁₃₁

D₄

yes

characteristic

C₂²×C₈×C₁₃₁

D₄

yes

characteristic

C₂×C₄⋊C₄×C₁₃₁

—

C₂×C₄²×C₁₃₁

—

C₂×C₄²×C₁₃₁

C₂³

yes

characteristic

#108×C₁₃₁

—

#108×C₁₃₁

—

C₄⋊C₈×C₁₃₁

—

C₄⋊C₈×C₁₃₁

—

C₄⋊C₈×C₁₃₁

—

C₄⋊C₈×C₁₃₁

—

C₂²⋊C₈×C₁₃₁

—

#97×C₁₃₁

—

C₄×C₈×C₁₃₁

—

#95×C₁₃₁

Q₈

yes

characteristic

#95×C₁₃₁

Q₈

yes

characteristic

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₃₁

#41

yes

characteristic

M₁₆×C₁₃₁

—

M₁₆×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

D₈

yes

—

C₂×C₈×C₁₃₁

SD₁₆

yes

—

C₄⋊C₄×C₁₃₁

—

C₄²×C₁₃₁

—

C₄²×C₁₃₁

—

C₄²×C₁₃₁

#41

yes

—

C₄²×C₁₃₁

#41

yes

—

C₂³×C₁₃₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

yes

C₂×C₄×C₁₃₁

yes

C₂×C₄×C₁₃₁

yes

—

C₂×C₄×C₁₃₁

#126

yes

—

C₂×C₄×C₁₃₁

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃₁

C₂×SD₁₆

yes

C₂×C₄×C₁₃₁

yes

C₂×C₄×C₁₃₁

yes

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

yes

C₂²×C₁₃₁

yes

C₂²×C₁₃₁

yes

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

yes

C₄×C₁₃₁

yes

C₄×C₁₃₁

yes

C₄×C₁₃₁

yes

C₂×C₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₁

#10687

yes

characteristic, direct product factor, Sylow

#1940

C₂×C₁₃₁

yes

characteristic

#1849

C₂×C₁₃₁

yes

characteristic

100

#1303

C₂×C₁₃₁

yes

—

101

#1301

C₂×C₁₃₁

yes

—

102

#1275

C₂×C₁₃₁

yes

characteristic

103

#431

C₂²×C₁₃₁

yes

—

104

C₂×#108

—

105

C₂×#108

—

106

#423

C₂²×C₁₃₁

yes

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

C₂²×C₁₃₁

yes

characteristic

109

C₂×#97

C₂²×C₁₃₁

yes

characteristic

110

C₄×C₄⋊C₄

C₂²×C₁₃₁

yes

characteristic

111

#336

—

112

C₂×M₁₆

—

113

C₂²×C₈

—

114

C₂²×C₈

D₄×C₁₃₁

yes

characteristic

115

C₂²×C₈

D₄×C₁₃₁

yes

characteristic

116

C₂×C₄⋊C₄

—

117

C₂×C₄²

—

118

C₂×C₄²

C₂³×C₁₃₁

yes

characteristic

119

#108

—

120

#108

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₂²⋊C₈

—

126

#97

—

127

C₄×C₈

—

128

#95

Q₈×C₁₃₁

yes

characteristic

129

#95

Q₈×C₁₃₁

yes

characteristic

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂×D₄×C₁₃₁

yes

characteristic

133

C₂²×C₄

C₂×Q₈×C₁₃₁

yes

characteristic

134

C₂²×C₄

#41×C₁₃₁

yes

characteristic

135

M₁₆

—

136

M₁₆

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

D₈×C₁₃₁

yes

—

147

C₂×C₈

SD₁₆×C₁₃₁

yes

—

148

C₄⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₄²

#41×C₁₃₁

yes

—

152

C₄²

#41×C₁₃₁

yes

—

153

C₂³

C₂²⋊Q₈×C₁₃₁

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

#123×C₁₃₁

yes

characteristic

163

C₂×C₄

#124×C₁₃₁

yes

characteristic

164

C₂×C₄

#125×C₁₃₁

yes

—

165

C₂×C₄

#126×C₁₃₁

yes

—

166

C₂×C₄

C₂×D₈×C₁₃₁

yes

characteristic

167

C₂×C₄

C₂×SD₁₆×C₁₃₁

yes

characteristic

168

C₂×C₄

#137×C₁₃₁

yes

characteristic

169

C₂×C₄

#137×C₁₃₁

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

—

175

C₂²

—

176

C₂²

—

177

C₂²

#400×C₁₃₁

yes

characteristic

178

C₂²

#474×C₁₃₁

yes

characteristic

179

C₂²

#475×C₁₃₁

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#480×C₁₃₁

yes

characteristic

185

C₄

#483×C₁₃₁

yes

characteristic

186

C₄

#486×C₁₃₁

yes

characteristic

187

C₄

#491×C₁₃₁

yes

characteristic

188

C₂

—

189

C₂

—

190

C₂

#2099×C₁₃₁

yes

characteristic

191

C₂

#2102×C₁₃₁

yes

characteristic

192

C₂

#2103×C₁₃₁

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₁	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
socle	C₂²×C₁₃₁	#400
prime radical	C₂³×C₁₃₁	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₃₁

#124

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₃₁

socle

C₂²×C₁₃₁

#400

prime radical

C₂³×C₁₃₁

C₂²⋊Q₈

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

N	H ≅ G/N	type
96. #10687	C₁₃₁	direct (N × H)
97. C₁₃₁	#10687	direct (N × H)

H ≅ G/N

type

96. #10687

C₁₃₁

direct (N × H)

97. C₁₃₁

#10687

direct (N × H)

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

subgroup	index
#1940×C₁₃₁	2
#1849×C₁₃₁	2
#1303×C₁₃₁	2
#1301×C₁₃₁	2
#1275×C₁₃₁	2
#10687	131

subgroup

index

#1940×C₁₃₁

#1849×C₁₃₁

#1303×C₁₃₁

#1301×C₁₃₁

#1275×C₁₃₁

#10687

131

Subgroups — C₁₃₁×#10687 #45696571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C131×#10687 #45696571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#10687 #45696571