Group #21467742 — subgroups

Subgroups — C₃₁×#10983 #21467742

Order 7936.

1318 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 1318 conjugacy classes of subgroups — too many to draw. Only its 286 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	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#101	C₂	1	4	yes	maximal
3	C₃₁×#1728	C₂	1	2	yes	maximal
4	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
5	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
6	C₂×C₃₁×C₄⋊Q₈	—	2	8	no	—
7	C₂×C₃₁×C₄⋊Q₈	—	2	4	no	—
8	C₂²×C₄⋊C₄×C₃₁	C₄	1	4	yes	—
9	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#101	—	2	8	no	—
11	C₂×C₃₁×#101	—	2	16	no	—
12	C₃₁×#336	—	2	4	no	—
13	C₂²×Q₈×C₃₁	—	2	8	no	—
14	C₂²×Q₈×C₃₁	—	2	4	no	—
15	C₂²×Q₈×C₃₁	—	2	4	no	—
16	C₂²×Q₈×C₃₁	—	2	8	no	—
17	C₂²×Q₈×C₃₁	D₄	1	8	yes	—
18	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₃₁	—	2	4	no	—
22	C₂×M₁₆×C₃₁	—	2	8	no	—
23	C₂×M₁₆×C₃₁	—	2	8	no	—
24	C₂×M₁₆×C₃₁	D₄	1	8	yes	—
25	C₂²×C₈×C₃₁	—	2	4	no	—
26	C₃₁×C₄⋊Q₈	—	2	32	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
29	C₂×C₄²×C₃₁	—	2	4	no	—
30	C₂×C₄²×C₃₁	D₄	1	4	yes	—
31	C₃₁×#101	—	2	32	no	—
32	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×Q₈×C₃₁	—	2	8	no	—
34	C₂×Q₈×C₃₁	—	2	16	no	—
35	C₂×Q₈×C₃₁	—	2	16	no	—
36	C₂×Q₈×C₃₁	—	2	8	no	—
37	C₂×Q₈×C₃₁	—	4	4	no	—
38	C₂×Q₈×C₃₁	—	4	8	no	—
39	C₂×Q₈×C₃₁	—	4	8	no	—
40	C₂×Q₈×C₃₁	—	4	4	no	—
41	C₂×Q₈×C₃₁	—	4	4	no	—
42	C₂×Q₈×C₃₁	—	4	4	no	—
43	C₂²×C₄×C₃₁	—	2	4	no	—
44	C₂²×C₄×C₃₁	—	2	8	no	—
45	C₂²×C₄×C₃₁	—	2	4	no	—
46	C₂²×C₄×C₃₁	—	2	4	no	—
47	C₂²×C₄×C₃₁	—	2	16	no	—
48	C₂²×C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
49	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
51	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
52	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₃₁	#41	1	4	yes	—
54	M₁₆×C₃₁	—	2	16	no	—
55	M₁₆×C₃₁	—	4	8	no	—
56	C₂×C₈×C₃₁	—	2	8	no	—
57	C₂×C₈×C₃₁	—	4	4	no	—
58	C₂×C₈×C₃₁	—	4	4	no	—
59	C₄⋊C₄×C₃₁	—	2	32	no	—
60	C₄²×C₃₁	—	2	8	no	—
61	#10983	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₃₁	—	2	4	no	—
63	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
64	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂³×C₃₁	C₄×D₄	1	4	yes	—
66	Q₈×C₃₁	—	4	16	no	—
67	Q₈×C₃₁	—	4	16	no	—
68	C₂×C₄×C₃₁	—	2	16	no	—
69	C₂×C₄×C₃₁	—	2	4	no	—
70	C₂×C₄×C₃₁	—	2	4	no	—
71	C₂×C₄×C₃₁	—	2	8	no	—
72	C₂×C₄×C₃₁	—	2	16	no	—
73	C₂×C₄×C₃₁	—	4	8	no	—
74	C₂×C₄×C₃₁	—	4	8	no	—
75	C₂×C₄×C₃₁	#101	1	8	yes	complemented
76	C₂×C₄×C₃₁	#120	1	4	yes	—
77	C₂×C₄×C₃₁	#120	1	4	yes	—
78	C₂×C₄×C₃₁	#121	1	8	yes	—
79	C₂×C₄×C₃₁	#121	1	4	yes	—
80	C₂×C₄×C₃₁	#123	1	4	yes	—
81	C₂×C₄×C₃₁	#124	1	2	yes	—
82	C₂×C₄×C₃₁	#127	1	2	yes	—
83	C₈×C₃₁	—	4	8	no	—
84	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
85	C₂²×#101	C₂×C₃₁	1	4	yes	—
86	#1728	C₂×C₃₁	1	2	yes	—
87	C₂²×C₃₁	—	2	8	no	—
88	C₂²×C₃₁	—	2	4	no	—
89	C₂²×C₃₁	#379	1	1	yes	characteristic
90	C₂²×C₃₁	#386	1	4	yes	—
91	C₂²×C₃₁	#390	1	2	yes	—
92	C₂²×C₃₁	C₂×#101	1	4	yes	—
93	C₄×C₃₁	—	2	8	no	—
94	C₄×C₃₁	—	4	8	no	—
95	C₄×C₃₁	#392	1	4	yes	—
96	C₄×C₃₁	#394	1	4	yes	—
97	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
98	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
99	C₂×C₄⋊Q₈	—	2	8	no	—
100	C₂×C₄⋊Q₈	—	2	4	no	—
101	C₂²×C₄⋊C₄	C₄×C₃₁	1	4	yes	—
102	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
103	C₂×#101	—	2	8	no	—
104	C₂×#101	—	2	16	no	—
105	#336	—	2	4	no	—
106	C₂×C₃₁	—	2	4	no	—
107	C₂×C₃₁	#1438	1	1	yes	characteristic
108	C₂×C₃₁	#1784	1	2	yes	—
109	C₂×C₃₁	#1966	1	4	yes	—
110	C₂²×Q₈	—	2	4	no	—
111	C₂²×Q₈	—	2	4	no	—
112	C₂²×Q₈	—	2	8	no	—
113	C₂²×Q₈	—	2	8	no	—
114	C₂²×Q₈	D₄×C₃₁	1	8	yes	—
115	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
116	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
117	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
118	C₂×M₁₆	—	2	4	no	—
119	C₂×M₁₆	—	2	8	no	—
120	C₂×M₁₆	—	2	8	no	—
121	C₂×M₁₆	D₄×C₃₁	1	8	yes	—
122	C₂²×C₈	—	2	4	no	—
123	C₄⋊Q₈	—	2	32	no	—
124	C₂×C₄⋊C₄	—	2	16	no	—
125	C₂×C₄⋊C₄	—	2	8	no	—
126	C₂×C₄²	—	2	4	no	—
127	C₂×C₄²	D₄×C₃₁	1	4	yes	—
128	#101	—	2	32	no	—
129	C₃₁	#10983	1	1	yes	characteristic, direct product factor, Sylow
130	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
131	C₂×Q₈	—	2	8	no	—
132	C₂×Q₈	—	2	16	no	—
133	C₂×Q₈	—	2	16	no	—
134	C₂×Q₈	—	2	8	no	—
135	C₂×Q₈	—	4	4	no	—
136	C₂×Q₈	—	4	8	no	—
137	C₂×Q₈	—	4	8	no	—
138	C₂×Q₈	—	4	4	no	—
139	C₂×Q₈	—	4	4	no	—
140	C₂×Q₈	—	4	4	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	8	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	2	16	no	—
146	C₂²×C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
147	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
148	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
149	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
150	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
151	C₂²×C₄	#41×C₃₁	1	4	yes	—
152	M₁₆	—	2	16	no	—
153	M₁₆	—	4	8	no	—
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	—	4	4	no	—
156	C₂×C₈	—	4	4	no	—
157	C₄⋊C₄	—	2	32	no	—
158	C₄²	—	2	8	no	—
159	C₂³	—	2	4	no	—
160	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
161	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
162	C₂³	C₄×D₄×C₃₁	1	4	yes	—
163	Q₈	—	4	16	no	—
164	Q₈	—	4	16	no	—
165	C₂×C₄	—	2	16	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	16	no	—
170	C₂×C₄	—	4	8	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	C₃₁×#101	1	8	yes	complemented
173	C₂×C₄	C₃₁×#120	1	4	yes	—
174	C₂×C₄	C₃₁×#120	1	4	yes	—
175	C₂×C₄	C₃₁×#121	1	8	yes	—
176	C₂×C₄	C₃₁×#121	1	4	yes	—
177	C₂×C₄	C₃₁×#123	1	4	yes	—
178	C₂×C₄	C₃₁×#124	1	2	yes	—
179	C₂×C₄	C₃₁×#127	1	2	yes	—
180	C₈	—	4	8	no	—
181	C₂²	—	2	8	no	—
182	C₂²	—	2	4	no	—
183	C₂²	C₃₁×#379	1	1	yes	characteristic
184	C₂²	C₃₁×#386	1	4	yes	—
185	C₂²	C₃₁×#390	1	2	yes	—
186	C₂²	C₂×C₃₁×#101	1	4	yes	—
187	C₄	—	2	8	no	—
188	C₄	—	4	8	no	—
189	C₄	C₃₁×#392	1	4	yes	—
190	C₄	C₃₁×#394	1	4	yes	—
191	C₂	—	2	4	no	—
192	C₂	C₃₁×#1438	1	1	yes	characteristic
193	C₂	C₃₁×#1784	1	2	yes	—
194	C₂	C₃₁×#1966	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₃₁×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×C₃₁×#101

C₂

yes

maximal

C₃₁×#1728

C₂

yes

maximal

C₂³×Q₈×C₃₁

C₂²

yes

—

C₂²×M₁₆×C₃₁

C₂²

yes

—

C₂×C₃₁×C₄⋊Q₈

—

C₂×C₃₁×C₄⋊Q₈

—

C₂²×C₄⋊C₄×C₃₁

C₄

yes

—

C₂²×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₃₁×#101

—

C₂×C₃₁×#101

—

C₃₁×#336

—

C₂²×Q₈×C₃₁

—

C₂²×Q₈×C₃₁

—

C₂²×Q₈×C₃₁

—

C₂²×Q₈×C₃₁

—

C₂²×Q₈×C₃₁

D₄

yes

—

C₂³×C₄×C₃₁

C₂×C₄

yes

—

C₂³×C₄×C₃₁

C₂×C₄

yes

—

C₂³×C₄×C₃₁

C₂³

yes

characteristic

C₂×M₁₆×C₃₁

—

C₂×M₁₆×C₃₁

—

C₂×M₁₆×C₃₁

—

C₂×M₁₆×C₃₁

D₄

yes

—

C₂²×C₈×C₃₁

—

C₃₁×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

—

C₂×C₄²×C₃₁

D₄

yes

—

C₃₁×#101

—

C₂⁴×C₃₁

C₂²×C₄

yes

characteristic

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂²×C₄×C₃₁

#41

yes

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

—

#10983

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

—

C₂³×C₃₁

C₂×C₂²⋊C₄

yes

—

C₂³×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₃₁

C₄×D₄

yes

—

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#101

yes

complemented

C₂×C₄×C₃₁

#120

yes

—

C₂×C₄×C₃₁

#120

yes

—

C₂×C₄×C₃₁

#121

yes

—

C₂×C₄×C₃₁

#121

yes

—

C₂×C₄×C₃₁

#123

yes

—

C₂×C₄×C₃₁

#124

yes

—

C₂×C₄×C₃₁

#127

yes

—

C₈×C₃₁

—

C₂²×C₄⋊Q₈

C₂×C₃₁

yes

characteristic

C₂²×#101

C₂×C₃₁

yes

—

#1728

C₂×C₃₁

yes

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#379

yes

characteristic

C₂²×C₃₁

#386

yes

—

C₂²×C₃₁

#390

yes

—

C₂²×C₃₁

C₂×#101

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#392

yes

—

C₄×C₃₁

#394

yes

—

C₂³×Q₈

C₂²×C₃₁

yes

—

C₂²×M₁₆

C₂²×C₃₁

yes

—

C₂×C₄⋊Q₈

—

100

C₂×C₄⋊Q₈

—

101

C₂²×C₄⋊C₄

C₄×C₃₁

yes

—

102

C₂²×C₄²

C₂²×C₃₁

yes

characteristic

103

C₂×#101

—

104

C₂×#101

—

105

#336

—

106

C₂×C₃₁

—

107

C₂×C₃₁

#1438

yes

characteristic

108

C₂×C₃₁

#1784

yes

—

109

C₂×C₃₁

#1966

yes

—

110

C₂²×Q₈

—

111

C₂²×Q₈

—

112

C₂²×Q₈

—

113

C₂²×Q₈

—

114

C₂²×Q₈

D₄×C₃₁

yes

—

115

C₂³×C₄

C₂×C₄×C₃₁

yes

—

116

C₂³×C₄

C₂×C₄×C₃₁

yes

—

117

C₂³×C₄

C₂³×C₃₁

yes

characteristic

118

C₂×M₁₆

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

121

C₂×M₁₆

D₄×C₃₁

yes

—

122

C₂²×C₈

—

123

C₄⋊Q₈

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄²

—

127

C₂×C₄²

D₄×C₃₁

yes

—

128

#101

—

129

C₃₁

#10983

yes

characteristic, direct product factor, Sylow

130

C₂⁴

C₂²×C₄×C₃₁

yes

characteristic

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂²⋊C₄×C₃₁

yes

—

147

C₂²×C₄

C₂²⋊C₄×C₃₁

yes

—

148

C₂²×C₄

C₂×D₄×C₃₁

yes

—

149

C₂²×C₄

C₂×D₄×C₃₁

yes

—

150

C₂²×C₄

C₂×D₄×C₃₁

yes

—

151

C₂²×C₄

#41×C₃₁

yes

—

152

M₁₆

—

153

M₁₆

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₄⋊C₄

—

158

C₄²

—

159

C₂³

—

160

C₂³

C₂×C₂²⋊C₄×C₃₁

yes

—

161

C₂³

C₂×C₂²⋊C₄×C₃₁

yes

characteristic

162

C₂³

C₄×D₄×C₃₁

yes

—

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

C₃₁×#101

yes

complemented

173

C₂×C₄

C₃₁×#120

yes

—

174

C₂×C₄

C₃₁×#120

yes

—

175

C₂×C₄

C₃₁×#121

yes

—

176

C₂×C₄

C₃₁×#121

yes

—

177

C₂×C₄

C₃₁×#123

yes

—

178

C₂×C₄

C₃₁×#124

yes

—

179

C₂×C₄

C₃₁×#127

yes

—

180

C₈

—

181

C₂²

—

182

C₂²

—

183

C₂²

C₃₁×#379

yes

characteristic

184

C₂²

C₃₁×#386

yes

—

185

C₂²

C₃₁×#390

yes

—

186

C₂²

C₂×C₃₁×#101

yes

—

187

C₄

—

188

C₄

—

189

C₄

C₃₁×#392

yes

—

190

C₄

C₃₁×#394

yes

—

191

C₂

—

192

C₂

C₃₁×#1438

yes

characteristic

193

C₂

C₃₁×#1784

yes

—

194

C₂

C₃₁×#1966

yes

—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₃₁

C₂×C₂²⋊C₄

commutator

C₂⁴

C₂²×C₄×C₃₁

Frattini, cosocle

C₂³×C₄

C₂³×C₃₁

prime radical

C₂⁴×C₃₁

C₂²×C₄

LCS

G ▷ C₂⁴ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
61. #10983	C₃₁	direct (N × H)
75. C₂×C₄×C₃₁	#101	semidirect
129. C₃₁	#10983	direct (N × H)
172. C₂×C₄	C₃₁×#101	semidirect

H ≅ G/N

type

61. #10983

C₃₁

direct (N × H)

75. C₂×C₄×C₃₁

#101

semidirect

129. C₃₁

#10983

direct (N × H)

172. C₂×C₄

C₃₁×#101

semidirect

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

subgroup	index
C₂²×C₃₁×C₄⋊Q₈	2
C₂²×C₃₁×#101	2
C₃₁×#1728	2
#10983	31

subgroup

index

C₂²×C₃₁×C₄⋊Q₈

C₂²×C₃₁×#101

C₃₁×#1728

#10983

Subgroups — C₃₁×#10983 #21467742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#10983 #21467742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10983 #21467742