Group #50314697 — subgroups

Subgroups — C₁₅₁×#9772 #50314697

Order 38656.

478 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 174 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×#494×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#1848×C₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#1546×C₁₅₁	C₂	1	4	yes	maximal
4	#1450×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₁₅₁	—	2	2	no	—
6	C₂×C₄⋊Q₈×C₁₅₁	C₂²	1	2	yes	—
7	#494×C₁₅₁	—	2	1	no	—
8	#494×C₁₅₁	—	2	1	no	—
9	#494×C₁₅₁	C₄	1	4	yes	—
10	#431×C₁₅₁	C₂²	1	2	yes	—
11	C₂×#108×C₁₅₁	—	2	2	no	—
12	#423×C₁₅₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
14	#332×C₁₅₁	—	2	8	no	—
15	#322×C₁₅₁	—	2	1	no	—
16	C₈²×C₁₅₁	—	2	1	no	—
17	C₂²×Q₈×C₁₅₁	—	2	2	no	—
18	C₂×Q₁₆×C₁₅₁	—	2	4	no	—
19	C₂×Q₁₆×C₁₅₁	—	2	4	no	—
20	C₂×Q₁₆×C₁₅₁	—	4	4	no	—
21	C₂×Q₁₆×C₁₅₁	—	4	4	no	—
22	C₂×M₁₆×C₁₅₁	—	2	2	no	—
23	C₂²×C₈×C₁₅₁	D₄	1	2	yes	—
24	C₄⋊Q₈×C₁₅₁	—	2	2	no	—
25	C₄⋊Q₈×C₁₅₁	—	2	4	no	—
26	C₄⋊Q₈×C₁₅₁	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₁₅₁	—	2	4	no	—
28	C₂×C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
29	#108×C₁₅₁	—	4	4	no	—
30	C₄⋊C₈×C₁₅₁	—	2	4	no	—
31	C₄⋊C₈×C₁₅₁	D₄	1	4	yes	—
32	C₂²⋊C₈×C₁₅₁	—	2	2	no	—
33	C₄×C₈×C₁₅₁	—	2	2	no	—
34	C₄×C₈×C₁₅₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
36	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
37	C₂×Q₈×C₁₅₁	—	2	4	no	—
38	C₂×Q₈×C₁₅₁	—	4	2	no	—
39	C₂×Q₈×C₁₅₁	—	4	4	no	—
40	C₂×Q₈×C₁₅₁	—	4	4	no	—
41	C₂²×C₄×C₁₅₁	—	4	2	no	—
42	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₅₁	#41	1	2	yes	—
44	Q₁₆×C₁₅₁	—	4	8	no	—
45	Q₁₆×C₁₅₁	—	4	8	no	—
46	M₁₆×C₁₅₁	—	4	4	no	—
47	C₂×C₈×C₁₅₁	—	2	4	no	—
48	C₂×C₈×C₁₅₁	—	4	2	no	—
49	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₁₅₁	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₁₅₁	D₈	1	4	yes	—
52	C₂×C₈×C₁₅₁	SD₁₆	1	4	yes	—
53	C₄⋊C₄×C₁₅₁	—	2	8	no	—
54	C₄⋊C₄×C₁₅₁	—	4	4	no	—
55	C₄²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₁₅₁	C₂×D₄	1	2	yes	—
58	C₂³×C₁₅₁	#124	1	1	yes	characteristic
59	Q₈×C₁₅₁	—	4	8	no	—
60	Q₈×C₁₅₁	—	8	2	no	—
61	Q₈×C₁₅₁	—	8	2	no	—
62	C₂×C₄×C₁₅₁	—	2	4	no	—
63	C₂×C₄×C₁₅₁	—	4	4	no	—
64	C₂×C₄×C₁₅₁	—	8	4	no	—
65	C₂×C₄×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₅₁	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₁₅₁	#121	1	2	yes	—
68	C₂×C₄×C₁₅₁	#127	1	1	yes	characteristic
69	C₂×C₄×C₁₅₁	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₁₅₁	C₂×SD₁₆	1	2	yes	—
71	C₈×C₁₅₁	—	4	4	no	—
72	C₈×C₁₅₁	—	8	2	no	—
73	C₈×C₁₅₁	#102	1	4	yes	—
74	C₈×C₁₅₁	#102	1	4	yes	—
75	C₂²×C₁₅₁	—	2	2	no	—
76	C₂²×C₁₅₁	#390	1	1	yes	characteristic
77	C₂²×C₁₅₁	#486	1	2	yes	—
78	C₄×C₁₅₁	—	2	2	no	—
79	C₄×C₁₅₁	—	8	4	no	—
80	C₄×C₁₅₁	C₂×#102	1	2	yes	—
81	C₄×C₁₅₁	#473	1	4	yes	—
82	C₄×C₁₅₁	#492	1	1	yes	characteristic
83	C₄×C₁₅₁	#493	1	1	yes	characteristic
84	C₂×C₁₅₁	—	2	2	no	—
85	C₂×C₁₅₁	#1957	1	1	yes	characteristic
86	C₂×C₁₅₁	#1979	1	2	yes	—
87	#9772	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₁₅₁	#9772	1	1	yes	characteristic, direct product factor, Sylow
89	C₂×#494	C₂×C₁₅₁	1	1	yes	characteristic
90	#1848	C₂×C₁₅₁	1	1	yes	characteristic
91	#1546	C₂×C₁₅₁	1	4	yes	—
92	#1450	C₂×C₁₅₁	1	1	yes	characteristic
93	C₂²×Q₁₆	—	2	2	no	—
94	C₂×C₄⋊Q₈	C₂²×C₁₅₁	1	2	yes	—
95	#494	—	2	1	no	—
96	#494	—	2	1	no	—
97	#494	C₄×C₁₅₁	1	4	yes	—
98	#431	C₂²×C₁₅₁	1	2	yes	—
99	C₂×#108	—	2	2	no	—
100	#423	C₂²×C₁₅₁	1	2	yes	—
101	C₂×C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
102	#332	—	2	8	no	—
103	#322	—	2	1	no	—
104	C₈²	—	2	1	no	—
105	C₂²×Q₈	—	2	2	no	—
106	C₂×Q₁₆	—	2	4	no	—
107	C₂×Q₁₆	—	2	4	no	—
108	C₂×Q₁₆	—	4	4	no	—
109	C₂×Q₁₆	—	4	4	no	—
110	C₂×M₁₆	—	2	2	no	—
111	C₂²×C₈	D₄×C₁₅₁	1	2	yes	—
112	C₄⋊Q₈	—	2	4	no	—
113	C₄⋊Q₈	—	2	2	no	—
114	C₄⋊Q₈	C₂×C₄×C₁₅₁	1	4	yes	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
117	#108	—	4	4	no	—
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	D₄×C₁₅₁	1	4	yes	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₂×C₄×C₁₅₁	1	2	yes	—
123	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
124	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
125	C₂×Q₈	—	2	4	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	—	4	4	no	—
128	C₂×Q₈	—	4	4	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
131	C₂²×C₄	#41×C₁₅₁	1	2	yes	—
132	Q₁₆	—	4	8	no	—
133	Q₁₆	—	4	8	no	—
134	M₁₆	—	4	4	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	2	yes	—
138	C₂×C₈	C₂²⋊C₄×C₁₅₁	1	2	yes	—
139	C₂×C₈	D₈×C₁₅₁	1	4	yes	—
140	C₂×C₈	SD₁₆×C₁₅₁	1	4	yes	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	C₂²×C₄×C₁₅₁	1	1	yes	characteristic
144	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
145	C₄²	C₂×D₄×C₁₅₁	1	2	yes	—
146	C₂³	#124×C₁₅₁	1	1	yes	characteristic
147	Q₈	—	4	8	no	—
148	Q₈	—	8	2	no	—
149	Q₈	—	8	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	C₂×C₂²⋊C₄×C₁₅₁	1	1	yes	characteristic
154	C₂×C₄	C₄×D₄×C₁₅₁	1	2	yes	—
155	C₂×C₄	#121×C₁₅₁	1	2	yes	—
156	C₂×C₄	#127×C₁₅₁	1	1	yes	characteristic
157	C₂×C₄	C₂×D₈×C₁₅₁	1	2	yes	—
158	C₂×C₄	C₂×SD₁₆×C₁₅₁	1	2	yes	—
159	C₈	—	4	4	no	—
160	C₈	—	8	2	no	—
161	C₈	#102×C₁₅₁	1	4	yes	—
162	C₈	#102×C₁₅₁	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	#390×C₁₅₁	1	1	yes	characteristic
165	C₂²	#486×C₁₅₁	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	8	4	no	—
168	C₄	C₂×#102×C₁₅₁	1	2	yes	—
169	C₄	#473×C₁₅₁	1	4	yes	—
170	C₄	#492×C₁₅₁	1	1	yes	characteristic
171	C₄	#493×C₁₅₁	1	1	yes	characteristic
172	C₂	—	2	2	no	—
173	C₂	#1957×C₁₅₁	1	1	yes	characteristic
174	C₂	#1979×C₁₅₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#494×C₁₅₁

C₂

yes

maximal, characteristic

#1848×C₁₅₁

C₂

yes

maximal, characteristic

#1546×C₁₅₁

C₂

yes

maximal

#1450×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₁₅₁

—

C₂×C₄⋊Q₈×C₁₅₁

C₂²

yes

—

#494×C₁₅₁

—

#494×C₁₅₁

—

#494×C₁₅₁

C₄

yes

—

#431×C₁₅₁

C₂²

yes

—

C₂×#108×C₁₅₁

—

#423×C₁₅₁

C₂²

yes

—

C₂×C₄×C₈×C₁₅₁

C₂²

yes

characteristic

#332×C₁₅₁

—

#322×C₁₅₁

—

C₈²×C₁₅₁

—

C₂²×Q₈×C₁₅₁

—

C₂×Q₁₆×C₁₅₁

—

C₂×Q₁₆×C₁₅₁

—

C₂×Q₁₆×C₁₅₁

—

C₂×Q₁₆×C₁₅₁

—

C₂×M₁₆×C₁₅₁

—

C₂²×C₈×C₁₅₁

D₄

yes

—

C₄⋊Q₈×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄²×C₁₅₁

C₂³

yes

characteristic

#108×C₁₅₁

—

C₄⋊C₈×C₁₅₁

—

C₄⋊C₈×C₁₅₁

D₄

yes

—

C₂²⋊C₈×C₁₅₁

—

C₄×C₈×C₁₅₁

—

C₄×C₈×C₁₅₁

C₂×C₄

yes

—

C₄×C₈×C₁₅₁

D₄

yes

characteristic

C₄×C₈×C₁₅₁

D₄

yes

characteristic

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₅₁

#41

yes

—

Q₁₆×C₁₅₁

—

Q₁₆×C₁₅₁

—

M₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

C₂²⋊C₄

yes

—

C₂×C₈×C₁₅₁

C₂²⋊C₄

yes

—

C₂×C₈×C₁₅₁

D₈

yes

—

C₂×C₈×C₁₅₁

SD₁₆

yes

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂²×C₄

yes

characteristic

C₄²×C₁₅₁

C₂×D₄

yes

characteristic

C₄²×C₁₅₁

C₂×D₄

yes

—

C₂³×C₁₅₁

#124

yes

characteristic

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₄×D₄

yes

—

C₂×C₄×C₁₅₁

#121

yes

—

C₂×C₄×C₁₅₁

#127

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₈

yes

—

C₂×C₄×C₁₅₁

C₂×SD₁₆

yes

—

C₈×C₁₅₁

—

C₈×C₁₅₁

—

C₈×C₁₅₁

#102

yes

—

C₈×C₁₅₁

#102

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#390

yes

characteristic

C₂²×C₁₅₁

#486

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×#102

yes

—

C₄×C₁₅₁

#473

yes

—

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

C₂×C₁₅₁

—

C₂×C₁₅₁

#1957

yes

characteristic

C₂×C₁₅₁

#1979

yes

—

#9772

C₁₅₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₅₁

#9772

yes

characteristic, direct product factor, Sylow

C₂×#494

C₂×C₁₅₁

yes

characteristic

#1848

C₂×C₁₅₁

yes

characteristic

#1546

C₂×C₁₅₁

yes

—

#1450

C₂×C₁₅₁

yes

characteristic

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂²×C₁₅₁

yes

—

#494

—

#494

—

#494

C₄×C₁₅₁

yes

—

#431

C₂²×C₁₅₁

yes

—

C₂×#108

—

100

#423

C₂²×C₁₅₁

yes

—

101

C₂×C₄×C₈

C₂²×C₁₅₁

yes

characteristic

102

#332

—

103

#322

—

104

C₈²

—

105

C₂²×Q₈

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

108

C₂×Q₁₆

—

109

C₂×Q₁₆

—

110

C₂×M₁₆

—

111

C₂²×C₈

D₄×C₁₅₁

yes

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂×C₄×C₁₅₁

yes

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂³×C₁₅₁

yes

characteristic

117

#108

—

118

C₄⋊C₈

—

119

C₄⋊C₈

D₄×C₁₅₁

yes

—

120

C₂²⋊C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₂×C₄×C₁₅₁

yes

—

123

C₄×C₈

D₄×C₁₅₁

yes

characteristic

124

C₄×C₈

D₄×C₁₅₁

yes

characteristic

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×D₄×C₁₅₁

yes

characteristic

131

C₂²×C₄

#41×C₁₅₁

yes

—

132

Q₁₆

—

133

Q₁₆

—

134

M₁₆

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

C₂²⋊C₄×C₁₅₁

yes

—

138

C₂×C₈

C₂²⋊C₄×C₁₅₁

yes

—

139

C₂×C₈

D₈×C₁₅₁

yes

—

140

C₂×C₈

SD₁₆×C₁₅₁

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄²

C₂²×C₄×C₁₅₁

yes

characteristic

144

C₄²

C₂×D₄×C₁₅₁

yes

characteristic

145

C₄²

C₂×D₄×C₁₅₁

yes

—

146

C₂³

#124×C₁₅₁

yes

characteristic

147

Q₈

—

148

Q₈

—

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×C₂²⋊C₄×C₁₅₁

yes

characteristic

154

C₂×C₄

C₄×D₄×C₁₅₁

yes

—

155

C₂×C₄

#121×C₁₅₁

yes

—

156

C₂×C₄

#127×C₁₅₁

yes

characteristic

157

C₂×C₄

C₂×D₈×C₁₅₁

yes

—

158

C₂×C₄

C₂×SD₁₆×C₁₅₁

yes

—

159

C₈

—

160

C₈

—

161

C₈

#102×C₁₅₁

yes

—

162

C₈

#102×C₁₅₁

yes

—

163

C₂²

—

164

C₂²

#390×C₁₅₁

yes

characteristic

165

C₂²

#486×C₁₅₁

yes

—

166

C₄

—

167

C₄

—

168

C₄

C₂×#102×C₁₅₁

yes

—

169

C₄

#473×C₁₅₁

yes

—

170

C₄

#492×C₁₅₁

yes

characteristic

171

C₄

#493×C₁₅₁

yes

characteristic

172

C₂

—

173

C₂

#1957×C₁₅₁

yes

characteristic

174

C₂

#1979×C₁₅₁

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₅₁	#390
commutator	C₄²	C₂²×C₄×C₁₅₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₅₁
prime radical	C₂³×C₁₅₁	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₂×C₄²×C₁₅₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₅₁

#390

commutator

C₄²

C₂²×C₄×C₁₅₁

Frattini, cosocle

C₂×C₄²

C₂³×C₁₅₁

C₂³×C₁₅₁

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₁₅₁ ◁ C₂×C₄²×C₁₅₁ ◁ G

N	H ≅ G/N	type
87. #9772	C₁₅₁	direct (N × H)
88. C₁₅₁	#9772	direct (N × H)

H ≅ G/N

type

87. #9772

C₁₅₁

direct (N × H)

88. C₁₅₁

#9772

direct (N × H)

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

subgroup	index
C₂×#494×C₁₅₁	2
#1848×C₁₅₁	2
#1546×C₁₅₁	2
#1450×C₁₅₁	2
#9772	151

subgroup

index

C₂×#494×C₁₅₁

#1848×C₁₅₁

#1546×C₁₅₁

#1450×C₁₅₁

#9772

151

Subgroups — C₁₅₁×#9772 #50314697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C151×#9772 #50314697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#9772 #50314697