Group #21464508 — subgroups

Subgroups — C₃₁×#7749 #21464508

Order 7936.

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

normal subgroups only

This group has 588 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	C₃₁×#2811	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2014	C₂	1	3	yes	maximal
3	C₃₁×#1272	C₂	1	3	yes	maximal, complemented
4	C₂²×SD₁₆×C₃₁	—	2	3	no	—
5	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
6	C₂×C₃₁×#102	—	2	3	no	—
7	C₃₁×#395	—	2	3	no	—
8	C₃₁×#391	—	2	3	no	—
9	C₃₁×#391	C₂²	1	3	yes	—
10	C₃₁×#336	C₂²	1	3	yes	—
11	C₂²×Q₈×C₃₁	—	2	1	no	—
12	C₂²×Q₈×C₃₁	D₄	1	3	yes	—
13	C₂²×D₄×C₃₁	—	2	3	no	—
14	C₂×SD₁₆×C₃₁	—	4	6	no	—
15	C₂×SD₁₆×C₃₁	—	4	6	no	—
16	C₂×SD₁₆×C₃₁	—	4	6	no	—
17	C₂²×C₈×C₃₁	—	2	3	no	—
18	C₃₁×#127	—	4	2	no	—
19	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
20	C₂×C₄⋊C₄×C₃₁	D₄	1	3	yes	—
21	C₂×C₄²×C₃₁	—	2	1	no	—
22	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
23	C₃₁×#102	—	4	6	no	—
24	C₃₁×#95	—	2	3	no	—
25	C₃₁×#95	—	2	6	no	—
26	C₂⁴×C₃₁	—	4	1	no	—
27	C₂×Q₈×C₃₁	—	2	3	no	—
28	C₂×Q₈×C₃₁	—	4	3	no	—
29	C₂×Q₈×C₃₁	—	4	3	no	—
30	C₂×Q₈×C₃₁	—	8	3	no	—
31	C₂×Q₈×C₃₁	SD₁₆	1	6	yes	complemented
32	C₂×D₄×C₃₁	—	2	6	no	—
33	C₂×D₄×C₃₁	—	4	3	no	—
34	C₂×D₄×C₃₁	—	4	3	no	—
35	C₂×D₄×C₃₁	—	4	3	no	—
36	C₂×D₄×C₃₁	—	4	3	no	—
37	C₂×D₄×C₃₁	—	4	3	no	—
38	C₂²×C₄×C₃₁	—	2	3	no	—
39	C₂²×C₄×C₃₁	—	2	3	no	—
40	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
41	SD₁₆×C₃₁	—	4	24	no	—
42	C₂×C₈×C₃₁	—	2	6	no	—
43	C₂×C₈×C₃₁	—	4	3	no	—
44	C₂×C₈×C₃₁	—	4	3	no	—
45	C₄⋊C₄×C₃₁	—	4	3	no	—
46	C₄⋊C₄×C₃₁	—	8	3	no	—
47	C₄²×C₃₁	—	4	1	no	—
48	C₄²×C₃₁	—	8	1	no	—
49	#7749	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₃₁	—	4	2	no	—
51	C₂³×C₃₁	—	8	3	no	—
52	C₂³×C₃₁	—	8	3	no	—
53	C₂³×C₃₁	#120	1	1	yes	characteristic
54	Q₈×C₃₁	—	4	6	no	—
55	Q₈×C₃₁	—	8	3	no	—
56	Q₈×C₃₁	—	8	4	no	—
57	D₄×C₃₁	—	4	6	no	—
58	D₄×C₃₁	—	4	6	no	—
59	D₄×C₃₁	—	8	3	no	—
60	D₄×C₃₁	—	8	3	no	—
61	C₂×C₄×C₃₁	—	2	3	no	—
62	C₂×C₄×C₃₁	—	2	3	no	—
63	C₂×C₄×C₃₁	—	2	6	no	—
64	C₂×C₄×C₃₁	—	4	3	no	—
65	C₂×C₄×C₃₁	—	4	3	no	—
66	C₂×C₄×C₃₁	—	4	3	no	—
67	C₂×C₄×C₃₁	—	4	6	no	—
68	C₂×C₄×C₃₁	C₂×SD₁₆	1	3	yes	—
69	C₂×C₄×C₃₁	#137	1	3	yes	—
70	C₈×C₃₁	—	4	6	no	—
71	#2811	C₂×C₃₁	1	1	yes	characteristic, complemented
72	#2014	C₂×C₃₁	1	3	yes	—
73	#1272	C₂×C₃₁	1	3	yes	complemented
74	C₂²×C₃₁	—	8	6	no	—
75	C₂²×C₃₁	—	16	3	no	—
76	C₂²×C₃₁	—	16	1	no	—
77	C₂²×C₃₁	#448	1	3	yes	—
78	C₂²×C₃₁	#453	1	3	yes	—
79	C₂²×C₃₁	B(4,𝔽₂)	1	1	yes	characteristic
80	C₄×C₃₁	—	2	3	no	—
81	C₄×C₃₁	—	2	3	no	—
82	C₄×C₃₁	—	4	6	no	—
83	C₄×C₃₁	—	8	3	no	—
84	C₂²×SD₁₆	—	2	3	no	—
85	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
86	C₂×#102	—	2	3	no	—
87	#395	—	2	3	no	—
88	#391	—	2	3	no	—
89	#391	C₂²×C₃₁	1	3	yes	—
90	#336	C₂²×C₃₁	1	3	yes	—
91	C₂×C₃₁	—	16	2	no	—
92	C₂×C₃₁	#1596	1	3	yes	—
93	C₂×C₃₁	#1623	1	3	yes	—
94	C₂×C₃₁	#1655	1	1	yes	characteristic
95	C₂²×Q₈	—	2	1	no	—
96	C₂²×Q₈	D₄×C₃₁	1	3	yes	—
97	C₂²×D₄	—	2	3	no	—
98	C₂×SD₁₆	—	4	6	no	—
99	C₂×SD₁₆	—	4	6	no	—
100	C₂×SD₁₆	—	4	6	no	—
101	C₂²×C₈	—	2	3	no	—
102	#127	—	4	2	no	—
103	C₂×C₄⋊C₄	—	2	3	no	—
104	C₂×C₄⋊C₄	D₄×C₃₁	1	3	yes	—
105	C₂×C₄²	—	2	1	no	—
106	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
107	#102	—	4	6	no	—
108	#95	—	2	3	no	—
109	#95	—	2	6	no	—
110	C₃₁	#7749	1	1	yes	characteristic, direct product factor, Sylow
111	C₂⁴	—	4	1	no	—
112	C₂×Q₈	—	2	3	no	—
113	C₂×Q₈	—	4	3	no	—
114	C₂×Q₈	—	4	3	no	—
115	C₂×Q₈	—	8	3	no	—
116	C₂×Q₈	SD₁₆×C₃₁	1	6	yes	complemented
117	C₂×D₄	—	2	6	no	—
118	C₂×D₄	—	4	3	no	—
119	C₂×D₄	—	4	3	no	—
120	C₂×D₄	—	4	3	no	—
121	C₂×D₄	—	4	3	no	—
122	C₂×D₄	—	4	3	no	—
123	C₂²×C₄	—	2	3	no	—
124	C₂²×C₄	—	2	3	no	—
125	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
126	SD₁₆	—	4	24	no	—
127	C₂×C₈	—	2	6	no	—
128	C₂×C₈	—	4	3	no	—
129	C₂×C₈	—	4	3	no	—
130	C₄⋊C₄	—	4	3	no	—
131	C₄⋊C₄	—	8	3	no	—
132	C₄²	—	4	1	no	—
133	C₄²	—	8	1	no	—
134	C₂³	—	4	2	no	—
135	C₂³	—	8	3	no	—
136	C₂³	—	8	3	no	—
137	C₂³	C₃₁×#120	1	1	yes	characteristic
138	Q₈	—	4	6	no	—
139	Q₈	—	8	3	no	—
140	Q₈	—	8	4	no	—
141	D₄	—	4	6	no	—
142	D₄	—	4	6	no	—
143	D₄	—	8	3	no	—
144	D₄	—	8	3	no	—
145	C₂×C₄	—	2	3	no	—
146	C₂×C₄	—	2	3	no	—
147	C₂×C₄	—	2	6	no	—
148	C₂×C₄	—	4	3	no	—
149	C₂×C₄	—	4	3	no	—
150	C₂×C₄	—	4	3	no	—
151	C₂×C₄	—	4	6	no	—
152	C₂×C₄	C₂×SD₁₆×C₃₁	1	3	yes	—
153	C₂×C₄	C₃₁×#137	1	3	yes	—
154	C₈	—	4	6	no	—
155	C₂²	—	8	6	no	—
156	C₂²	—	16	3	no	—
157	C₂²	—	16	1	no	—
158	C₂²	C₃₁×#448	1	3	yes	—
159	C₂²	C₃₁×#453	1	3	yes	—
160	C₂²	C₃₁×B(4,𝔽₂)	1	1	yes	characteristic
161	C₄	—	2	3	no	—
162	C₄	—	2	3	no	—
163	C₄	—	4	6	no	—
164	C₄	—	8	3	no	—
165	C₂	—	16	2	no	—
166	C₂	C₃₁×#1596	1	3	yes	—
167	C₂	C₃₁×#1623	1	3	yes	—
168	C₂	C₃₁×#1655	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#2811

C₂

yes

maximal, characteristic, complemented

C₃₁×#2014

C₂

yes

maximal

C₃₁×#1272

C₂

yes

maximal, complemented

C₂²×SD₁₆×C₃₁

—

C₂×C₃₁×#127

C₂²

yes

characteristic

C₂×C₃₁×#102

—

C₃₁×#395

—

C₃₁×#391

—

C₃₁×#391

C₂²

yes

—

C₃₁×#336

C₂²

yes

—

C₂²×Q₈×C₃₁

—

C₂²×Q₈×C₃₁

D₄

yes

—

C₂²×D₄×C₃₁

—

C₂×SD₁₆×C₃₁

—

C₂×SD₁₆×C₃₁

—

C₂×SD₁₆×C₃₁

—

C₂²×C₈×C₃₁

—

C₃₁×#127

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

D₄

yes

—

C₂×C₄²×C₃₁

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×#102

—

C₃₁×#95

—

C₃₁×#95

—

C₂⁴×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

SD₁₆

yes

complemented

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

SD₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

—

#7749

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

#120

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×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

C₂×SD₁₆

yes

—

C₂×C₄×C₃₁

#137

yes

—

C₈×C₃₁

—

#2811

C₂×C₃₁

yes

characteristic, complemented

#2014

C₂×C₃₁

yes

—

#1272

C₂×C₃₁

yes

complemented

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#448

yes

—

C₂²×C₃₁

#453

yes

—

C₂²×C₃₁

B(4,𝔽₂)

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₂²×SD₁₆

—

C₂×#127

C₂²×C₃₁

yes

—

—

—

C₂²×C₃₁

yes

—

#336

C₂²×C₃₁

yes

—

C₂×C₃₁

—

C₂×C₃₁

#1596

yes

—

C₂×C₃₁

#1623

yes

—

C₂×C₃₁

#1655

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

D₄×C₃₁

yes

—

C₂²×D₄

—

C₂×SD₁₆

—

C₂×SD₁₆

—

100

C₂×SD₁₆

—

101

C₂²×C₈

—

102

#127

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

D₄×C₃₁

yes

—

105

C₂×C₄²

—

106

C₂×C₄²

C₂³×C₃₁

yes

characteristic

107

#102

—

108

#95

—

109

#95

—

110

C₃₁

#7749

yes

characteristic, direct product factor, Sylow

111

C₂⁴

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

SD₁₆×C₃₁

yes

complemented

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂×D₄×C₃₁

yes

—

126

SD₁₆

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄²

—

133

C₄²

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

C₃₁×#120

yes

characteristic

138

Q₈

—

139

Q₈

—

140

Q₈

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

C₂×SD₁₆×C₃₁

yes

—

153

C₂×C₄

C₃₁×#137

yes

—

154

C₈

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₃₁×#448

yes

—

159

C₂²

C₃₁×#453

yes

—

160

C₂²

C₃₁×B(4,𝔽₂)

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₂

—

166

C₂

C₃₁×#1596

yes

—

167

C₂

C₃₁×#1623

yes

—

168

C₂

C₃₁×#1655

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₃₁

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

prime radical

C₂×C₃₁×#127

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
1. C₃₁×#2811	C₂	semidirect
3. C₃₁×#1272	C₂	semidirect
31. C₂×Q₈×C₃₁	SD₁₆	semidirect
49. #7749	C₃₁	direct (N × H)
71. #2811	C₂×C₃₁	semidirect
73. #1272	C₂×C₃₁	semidirect
110. C₃₁	#7749	direct (N × H)
116. C₂×Q₈	SD₁₆×C₃₁	semidirect

H ≅ G/N

type

1. C₃₁×#2811

C₂

semidirect

3. C₃₁×#1272

C₂

semidirect

31. C₂×Q₈×C₃₁

SD₁₆

semidirect

49. #7749

C₃₁

direct (N × H)

71. #2811

C₂×C₃₁

semidirect

73. #1272

C₂×C₃₁

semidirect

110. C₃₁

#7749

direct (N × H)

116. C₂×Q₈

SD₁₆×C₃₁

semidirect

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

subgroup	index
C₃₁×#2811	2
C₃₁×#2014	2
C₃₁×#1272	2
#7749	31

subgroup

index

C₃₁×#2811

C₃₁×#2014

C₃₁×#1272

#7749

Subgroups — C₃₁×#7749 #21464508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#7749 #21464508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7749 #21464508