Group #45719193 — subgroups

Subgroups — C₁₃₁×#33309 #45719193

Order 33536.

1386 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

normal subgroups only

This group has 1386 conjugacy classes of subgroups — too many to draw. Only its 382 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×#497×C₁₃₁	C₂	1	6	yes	maximal, complemented
2	C₂×#495×C₁₃₁	C₂	1	3	yes	maximal
3	C₂×#494×C₁₃₁	C₂	1	3	yes	maximal, complemented
4	#2658×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#1726×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₁₃₁	C₂²	1	3	yes	—
7	C₂²×Q₁₆×C₁₃₁	C₂²	1	6	yes	—
8	C₂²×SD₁₆×C₁₃₁	C₂²	1	6	yes	—
9	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	6	yes	—
11	C₂×#124×C₁₃₁	C₂²	1	6	yes	—
12	#497×C₁₃₁	—	2	24	no	—
13	#495×C₁₃₁	—	2	12	no	—
14	#494×C₁₃₁	—	2	12	no	—
15	C₂×#97×C₁₃₁	C₂²	1	3	yes	—
16	C₂×C₄×C₈×C₁₃₁	C₂²	1	3	yes	—
17	#399×C₁₃₁	—	2	8	no	—
18	#376×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₁₃₁	C₂³	1	6	yes	—
20	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
21	C₂×Q₁₆×C₁₃₁	—	2	12	no	—
22	C₂×Q₁₆×C₁₃₁	—	2	12	no	—
23	C₂×Q₁₆×C₁₃₁	—	2	6	no	—
24	C₂×Q₁₆×C₁₃₁	—	2	24	no	—
25	C₂×SD₁₆×C₁₃₁	—	2	12	no	—
26	C₂×SD₁₆×C₁₃₁	—	2	24	no	—
27	C₂×D₈×C₁₃₁	—	2	6	no	—
28	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₁₃₁	C₂³	1	3	yes	—
30	C₄⋊Q₈×C₁₃₁	—	2	12	no	—
31	C₄⋊Q₈×C₁₃₁	—	2	12	no	—
32	#124×C₁₃₁	—	2	12	no	—
33	#124×C₁₃₁	—	2	12	no	—
34	C₂×C₄⋊C₄×C₁₃₁	—	2	12	no	—
35	C₂×C₂²⋊C₄×C₁₃₁	—	2	12	no	—
36	C₂×C₄²×C₁₃₁	C₂³	1	3	yes	—
37	#97×C₁₃₁	D₄	1	12	yes	—
38	C₄×C₈×C₁₃₁	D₄	1	12	yes	—
39	#95×C₁₃₁	D₄	1	4	yes	—
40	C₂⁴×C₁₃₁	—	2	2	no	—
41	C₂×Q₈×C₁₃₁	—	2	6	no	—
42	C₂×Q₈×C₁₃₁	—	2	6	no	—
43	C₂×Q₈×C₁₃₁	—	2	12	no	—
44	C₂×Q₈×C₁₃₁	—	2	12	no	—
45	C₂×D₄×C₁₃₁	—	2	6	no	—
46	C₂×D₄×C₁₃₁	—	2	6	no	—
47	C₂²×C₄×C₁₃₁	—	2	6	no	—
48	C₂²×C₄×C₁₃₁	C₂×D₄	1	6	yes	—
49	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
50	Q₁₆×C₁₃₁	—	4	12	no	—
51	Q₁₆×C₁₃₁	—	4	24	no	—
52	SD₁₆×C₁₃₁	—	4	24	no	—
53	D₈×C₁₃₁	—	4	4	no	—
54	C₂×C₈×C₁₃₁	C₂×D₄	1	6	yes	—
55	C₂×C₈×C₁₃₁	C₂×D₄	1	6	yes	—
56	C₂×C₈×C₁₃₁	C₂×D₄	1	12	yes	—
57	C₄⋊C₄×C₁₃₁	—	4	24	no	—
58	C₂²⋊C₄×C₁₃₁	—	4	24	no	—
59	C₄²×C₁₃₁	C₂×D₄	1	6	yes	—
60	C₄²×C₁₃₁	C₂×D₄	1	6	yes	—
61	C₂³×C₁₃₁	—	4	6	no	—
62	C₂³×C₁₃₁	—	4	8	no	—
63	C₂³×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
64	Q₈×C₁₃₁	—	4	6	no	—
65	Q₈×C₁₃₁	—	4	6	no	—
66	Q₈×C₁₃₁	—	4	12	no	—
67	D₄×C₁₃₁	—	4	2	no	—
68	D₄×C₁₃₁	—	4	6	no	—
69	C₂×C₄×C₁₃₁	—	2	12	no	—
70	C₂×C₄×C₁₃₁	—	4	6	no	—
71	C₂×C₄×C₁₃₁	—	4	12	no	—
72	C₂×C₄×C₁₃₁	#127	1	12	yes	—
73	C₂×C₄×C₁₃₁	C₂²×D₄	1	3	yes	—
74	C₂×C₄×C₁₃₁	C₂²×D₄	1	3	yes	—
75	C₈×C₁₃₁	#127	1	4	yes	—
76	C₈×C₁₃₁	#127	1	12	yes	—
77	C₂²×C₁₃₁	—	8	2	no	—
78	C₂²×C₁₃₁	—	8	12	no	—
79	C₂²×C₁₃₁	C₂×#127	1	3	yes	—
80	C₂²×C₁₃₁	#578	1	4	yes	—
81	C₄×C₁₃₁	—	2	12	no	—
82	C₄×C₁₃₁	—	8	6	no	—
83	C₄×C₁₃₁	C₂×#127	1	1	yes	characteristic
84	C₄×C₁₃₁	C₂×#127	1	3	yes	—
85	C₂×C₁₃₁	—	16	2	no	—
86	C₂×C₁₃₁	#2867	1	1	yes	characteristic
87	C₂×C₁₃₁	#3156	1	6	yes	—
88	#33309	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₁₃₁	#33309	1	1	yes	characteristic, direct product factor, Sylow
90	C₂×#497	C₂×C₁₃₁	1	6	yes	complemented
91	C₂×#495	C₂×C₁₃₁	1	3	yes	—
92	C₂×#494	C₂×C₁₃₁	1	3	yes	complemented
93	#2658	C₂×C₁₃₁	1	2	yes	complemented
94	#1726	C₂×C₁₃₁	1	1	yes	characteristic, complemented
95	C₂²×Q₁₆	C₂²×C₁₃₁	1	3	yes	—
96	C₂²×Q₁₆	C₂²×C₁₃₁	1	6	yes	—
97	C₂²×SD₁₆	C₂²×C₁₃₁	1	6	yes	—
98	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
99	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	6	yes	—
100	C₂×#124	C₂²×C₁₃₁	1	6	yes	—
101	#497	—	2	24	no	—
102	#495	—	2	12	no	—
103	#494	—	2	12	no	—
104	C₂×#97	C₂²×C₁₃₁	1	3	yes	—
105	C₂×C₄×C₈	C₂²×C₁₃₁	1	3	yes	—
106	#399	—	2	8	no	—
107	#376	C₂²×C₁₃₁	1	1	yes	characteristic
108	C₂²×Q₈	C₂³×C₁₃₁	1	6	yes	—
109	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
110	C₂×Q₁₆	—	2	12	no	—
111	C₂×Q₁₆	—	2	12	no	—
112	C₂×Q₁₆	—	2	6	no	—
113	C₂×Q₁₆	—	2	24	no	—
114	C₂×SD₁₆	—	2	12	no	—
115	C₂×SD₁₆	—	2	24	no	—
116	C₂×D₈	—	2	6	no	—
117	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
118	C₂²×C₈	C₂³×C₁₃₁	1	3	yes	—
119	C₄⋊Q₈	—	2	12	no	—
120	C₄⋊Q₈	—	2	12	no	—
121	#124	—	2	12	no	—
122	#124	—	2	12	no	—
123	C₂×C₄⋊C₄	—	2	12	no	—
124	C₂×C₂²⋊C₄	—	2	12	no	—
125	C₂×C₄²	C₂³×C₁₃₁	1	3	yes	—
126	#97	D₄×C₁₃₁	1	12	yes	—
127	C₄×C₈	D₄×C₁₃₁	1	12	yes	—
128	#95	D₄×C₁₃₁	1	4	yes	—
129	C₂⁴	—	2	2	no	—
130	C₂×Q₈	—	2	6	no	—
131	C₂×Q₈	—	2	6	no	—
132	C₂×Q₈	—	2	12	no	—
133	C₂×Q₈	—	2	12	no	—
134	C₂×D₄	—	2	6	no	—
135	C₂×D₄	—	2	6	no	—
136	C₂²×C₄	—	2	6	no	—
137	C₂²×C₄	C₂×D₄×C₁₃₁	1	6	yes	—
138	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
139	Q₁₆	—	4	12	no	—
140	Q₁₆	—	4	24	no	—
141	SD₁₆	—	4	24	no	—
142	D₈	—	4	4	no	—
143	C₂×C₈	C₂×D₄×C₁₃₁	1	6	yes	—
144	C₂×C₈	C₂×D₄×C₁₃₁	1	6	yes	—
145	C₂×C₈	C₂×D₄×C₁₃₁	1	12	yes	—
146	C₄⋊C₄	—	4	24	no	—
147	C₂²⋊C₄	—	4	24	no	—
148	C₄²	C₂×D₄×C₁₃₁	1	6	yes	—
149	C₄²	C₂×D₄×C₁₃₁	1	6	yes	—
150	C₂³	—	4	6	no	—
151	C₂³	—	4	8	no	—
152	C₂³	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
153	Q₈	—	4	6	no	—
154	Q₈	—	4	6	no	—
155	Q₈	—	4	12	no	—
156	D₄	—	4	2	no	—
157	D₄	—	4	6	no	—
158	C₂×C₄	—	2	12	no	—
159	C₂×C₄	—	4	6	no	—
160	C₂×C₄	—	4	12	no	—
161	C₂×C₄	#127×C₁₃₁	1	12	yes	—
162	C₂×C₄	C₂²×D₄×C₁₃₁	1	3	yes	—
163	C₂×C₄	C₂²×D₄×C₁₃₁	1	3	yes	—
164	C₈	#127×C₁₃₁	1	4	yes	—
165	C₈	#127×C₁₃₁	1	12	yes	—
166	C₂²	—	8	2	no	—
167	C₂²	—	8	12	no	—
168	C₂²	C₂×#127×C₁₃₁	1	3	yes	—
169	C₂²	#578×C₁₃₁	1	4	yes	—
170	C₄	—	2	12	no	—
171	C₄	—	8	6	no	—
172	C₄	C₂×#127×C₁₃₁	1	1	yes	characteristic
173	C₄	C₂×#127×C₁₃₁	1	3	yes	—
174	C₂	—	16	2	no	—
175	C₂	#2867×C₁₃₁	1	1	yes	characteristic
176	C₂	#3156×C₁₃₁	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#497×C₁₃₁

C₂

yes

maximal, complemented

C₂×#495×C₁₃₁

C₂

yes

maximal

C₂×#494×C₁₃₁

C₂

yes

maximal, complemented

#2658×C₁₃₁

C₂

yes

maximal, complemented

#1726×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₁₃₁

C₂²

yes

—

C₂²×Q₁₆×C₁₃₁

C₂²

yes

—

C₂²×SD₁₆×C₁₃₁

C₂²

yes

—

C₂²×D₈×C₁₃₁

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₁₃₁

C₂²

yes

—

C₂×#124×C₁₃₁

C₂²

yes

—

#497×C₁₃₁

—

#495×C₁₃₁

—

#494×C₁₃₁

—

C₂×#97×C₁₃₁

C₂²

yes

—

C₂×C₄×C₈×C₁₃₁

C₂²

yes

—

#399×C₁₃₁

—

#376×C₁₃₁

C₂²

yes

characteristic

C₂²×Q₈×C₁₃₁

C₂³

yes

—

C₂²×D₄×C₁₃₁

C₂³

yes

—

C₂×Q₁₆×C₁₃₁

—

C₂×Q₁₆×C₁₃₁

—

C₂×Q₁₆×C₁₃₁

—

C₂×Q₁₆×C₁₃₁

—

C₂×SD₁₆×C₁₃₁

—

C₂×SD₁₆×C₁₃₁

—

C₂×D₈×C₁₃₁

—

C₂²×C₈×C₁₃₁

C₂³

yes

characteristic

C₂²×C₈×C₁₃₁

C₂³

yes

—

C₄⋊Q₈×C₁₃₁

—

C₄⋊Q₈×C₁₃₁

—

#124×C₁₃₁

—

#124×C₁₃₁

—

C₂×C₄⋊C₄×C₁₃₁

—

C₂×C₂²⋊C₄×C₁₃₁

—

C₂×C₄²×C₁₃₁

C₂³

yes

—

#97×C₁₃₁

D₄

yes

—

C₄×C₈×C₁₃₁

D₄

yes

—

#95×C₁₃₁

D₄

yes

—

C₂⁴×C₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂×D₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

C₂×D₄

yes

—

C₂²×C₄×C₁₃₁

C₂⁴

yes

characteristic

Q₁₆×C₁₃₁

—

Q₁₆×C₁₃₁

—

SD₁₆×C₁₃₁

—

D₈×C₁₃₁

—

C₂×C₈×C₁₃₁

C₂×D₄

yes

—

C₂×C₈×C₁₃₁

C₂×D₄

yes

—

C₂×C₈×C₁₃₁

C₂×D₄

yes

—

C₄⋊C₄×C₁₃₁

—

C₂²⋊C₄×C₁₃₁

—

C₄²×C₁₃₁

C₂×D₄

yes

—

C₄²×C₁₃₁

C₂×D₄

yes

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

C₂²×D₄

yes

characteristic

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

D₄×C₁₃₁

—

D₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

#127

yes

—

C₂×C₄×C₁₃₁

C₂²×D₄

yes

—

C₂×C₄×C₁₃₁

C₂²×D₄

yes

—

C₈×C₁₃₁

#127

yes

—

C₈×C₁₃₁

#127

yes

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₂×#127

yes

—

C₂²×C₁₃₁

#578

yes

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×#127

yes

characteristic

C₄×C₁₃₁

C₂×#127

yes

—

C₂×C₁₃₁

—

C₂×C₁₃₁

#2867

yes

characteristic

C₂×C₁₃₁

#3156

yes

—

#33309

C₁₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₁

#33309

yes

characteristic, direct product factor, Sylow

C₂×#497

C₂×C₁₃₁

yes

complemented

C₂×#495

C₂×C₁₃₁

yes

—

C₂×#494

C₂×C₁₃₁

yes

complemented

#2658

C₂×C₁₃₁

yes

complemented

#1726

C₂×C₁₃₁

yes

characteristic, complemented

C₂²×Q₁₆

C₂²×C₁₃₁

yes

—

C₂²×Q₁₆

C₂²×C₁₃₁

yes

—

C₂²×SD₁₆

C₂²×C₁₃₁

yes

—

C₂²×D₈

C₂²×C₁₃₁

yes

characteristic

C₂×C₄⋊Q₈

C₂²×C₁₃₁

yes

—

100

C₂×#124

C₂²×C₁₃₁

yes

—

101

#497

—

102

#495

—

103

#494

—

104

C₂×#97

C₂²×C₁₃₁

yes

—

105

C₂×C₄×C₈

C₂²×C₁₃₁

yes

—

106

#399

—

107

#376

C₂²×C₁₃₁

yes

characteristic

108

C₂²×Q₈

C₂³×C₁₃₁

yes

—

109

C₂²×D₄

C₂³×C₁₃₁

yes

—

110

C₂×Q₁₆

—

111

C₂×Q₁₆

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

—

114

C₂×SD₁₆

—

115

C₂×SD₁₆

—

116

C₂×D₈

—

117

C₂²×C₈

C₂³×C₁₃₁

yes

characteristic

118

C₂²×C₈

C₂³×C₁₃₁

yes

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

—

121

#124

—

122

#124

—

123

C₂×C₄⋊C₄

—

124

C₂×C₂²⋊C₄

—

125

C₂×C₄²

C₂³×C₁₃₁

yes

—

126

#97

D₄×C₁₃₁

yes

—

127

C₄×C₈

D₄×C₁₃₁

yes

—

128

#95

D₄×C₁₃₁

yes

—

129

C₂⁴

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×D₄×C₁₃₁

yes

—

138

C₂²×C₄

C₂⁴×C₁₃₁

yes

characteristic

139

Q₁₆

—

140

Q₁₆

—

141

SD₁₆

—

142

D₈

—

143

C₂×C₈

C₂×D₄×C₁₃₁

yes

—

144

C₂×C₈

C₂×D₄×C₁₃₁

yes

—

145

C₂×C₈

C₂×D₄×C₁₃₁

yes

—

146

C₄⋊C₄

—

147

C₂²⋊C₄

—

148

C₄²

C₂×D₄×C₁₃₁

yes

—

149

C₄²

C₂×D₄×C₁₃₁

yes

—

150

C₂³

—

151

C₂³

—

152

C₂³

C₂²×D₄×C₁₃₁

yes

characteristic

153

Q₈

—

154

Q₈

—

155

Q₈

—

156

D₄

—

157

D₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

#127×C₁₃₁

yes

—

162

C₂×C₄

C₂²×D₄×C₁₃₁

yes

—

163

C₂×C₄

C₂²×D₄×C₁₃₁

yes

—

164

C₈

#127×C₁₃₁

yes

—

165

C₈

#127×C₁₃₁

yes

—

166

C₂²

—

167

C₂²

—

168

C₂²

C₂×#127×C₁₃₁

yes

—

169

C₂²

#578×C₁₃₁

yes

—

170

C₄

—

171

C₄

—

172

C₄

C₂×#127×C₁₃₁

yes

characteristic

173

C₄

C₂×#127×C₁₃₁

yes

—

174

C₂

—

175

C₂

#2867×C₁₃₁

yes

characteristic

176

C₂

#3156×C₁₃₁

yes

—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂²×D₈×C₁₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ #376×C₁₃₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₃₁

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₁₃₁

prime radical

C₂²×D₈×C₁₃₁

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃₁ ◁ #376×C₁₃₁ ◁ G

N	H ≅ G/N	type
1. C₂×#497×C₁₃₁	C₂	semidirect
3. C₂×#494×C₁₃₁	C₂	semidirect
4. #2658×C₁₃₁	C₂	semidirect
5. #1726×C₁₃₁	C₂	semidirect
88. #33309	C₁₃₁	direct (N × H)
89. C₁₃₁	#33309	direct (N × H)
90. C₂×#497	C₂×C₁₃₁	semidirect
92. C₂×#494	C₂×C₁₃₁	semidirect
93. #2658	C₂×C₁₃₁	semidirect
94. #1726	C₂×C₁₃₁	semidirect

H ≅ G/N

type

1. C₂×#497×C₁₃₁

C₂

semidirect

3. C₂×#494×C₁₃₁

C₂

semidirect

4. #2658×C₁₃₁

C₂

semidirect

5. #1726×C₁₃₁

C₂

semidirect

88. #33309

C₁₃₁

direct (N × H)

89. C₁₃₁

#33309

direct (N × H)

90. C₂×#497

C₂×C₁₃₁

semidirect

92. C₂×#494

C₂×C₁₃₁

semidirect

93. #2658

C₂×C₁₃₁

semidirect

94. #1726

C₂×C₁₃₁

semidirect

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

subgroup	index
C₂×#497×C₁₃₁	2
C₂×#495×C₁₃₁	2
C₂×#494×C₁₃₁	2
#2658×C₁₃₁	2
#1726×C₁₃₁	2
#33309	131

subgroup

index

C₂×#497×C₁₃₁

C₂×#495×C₁₃₁

C₂×#494×C₁₃₁

#2658×C₁₃₁

#1726×C₁₃₁

#33309

131

Subgroups — C₁₃₁×#33309 #45719193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C131×#33309 #45719193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33309 #45719193