Group #31831502 — subgroups

Subgroups — C₇₁×#10630 #31831502

Order 18176.

272 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₇₁×#1995	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1832	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1542	C₂	1	2	yes	maximal
4	C₇₁×#1301	C₂	1	2	yes	maximal
5	C₇₁×#1274	C₂	1	1	yes	maximal, characteristic
6	#533×C₇₁	C₂²	1	1	yes	characteristic
7	#431×C₇₁	C₂²	1	2	yes	—
8	#423×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
10	#376×C₇₁	C₂²	1	1	yes	characteristic
11	#339×C₇₁	—	2	2	no	—
12	#336×C₇₁	—	2	2	no	—
13	#333×C₇₁	—	2	2	no	—
14	#332×C₇₁	—	2	2	no	—
15	C₂×M₁₆×C₇₁	—	2	1	no	—
16	C₂²×C₈×C₇₁	—	2	2	no	—
17	C₄⋊Q₈×C₇₁	D₄	1	2	yes	—
18	#125×C₇₁	—	2	1	no	—
19	C₂²⋊Q₈×C₇₁	—	2	2	no	—
20	C₄×Q₈×C₇₁	—	2	2	no	—
21	#117×C₇₁	—	2	1	no	—
22	C₂×C₄²×C₇₁	D₄	1	2	yes	—
23	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₈×C₇₁	—	2	2	no	—
25	C₄⋊C₈×C₇₁	—	2	2	no	—
26	C₄⋊C₈×C₇₁	—	2	2	no	—
27	C₄⋊C₈×C₇₁	—	2	2	no	—
28	C₂²⋊C₈×C₇₁	—	2	2	no	—
29	C₄×C₈×C₇₁	—	2	2	no	—
30	#95×C₇₁	—	2	2	no	—
31	C₂×Q₈×C₇₁	—	2	2	no	—
32	C₂²×C₄×C₇₁	—	2	2	no	—
33	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	#41	1	2	yes	—
35	M₁₆×C₇₁	—	4	2	no	—
36	C₂×C₈×C₇₁	—	4	2	no	—
37	C₂×C₈×C₇₁	—	4	2	no	—
38	C₂×C₈×C₇₁	—	4	1	no	—
39	C₂×C₈×C₇₁	—	4	2	no	—
40	C₂×C₈×C₇₁	—	4	2	no	—
41	C₄⋊C₄×C₇₁	—	2	2	no	—
42	C₄⋊C₄×C₇₁	—	2	2	no	—
43	C₄⋊C₄×C₇₁	—	4	2	no	—
44	C₂²⋊C₄×C₇₁	—	4	1	no	—
45	C₄²×C₇₁	—	2	2	no	—
46	C₄²×C₇₁	—	2	2	no	—
47	C₄²×C₇₁	—	4	1	no	—
48	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₇₁	#41	1	1	yes	characteristic
50	C₄²×C₇₁	#41	1	2	yes	—
51	C₂³×C₇₁	#123	1	1	yes	characteristic
52	Q₈×C₇₁	—	8	2	no	—
53	C₂×C₄×C₇₁	—	4	2	no	—
54	C₂×C₄×C₇₁	—	4	4	no	—
55	C₂×C₄×C₇₁	—	4	2	no	—
56	C₂×C₄×C₇₁	—	4	2	no	—
57	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
58	C₂×C₄×C₇₁	#123	1	2	yes	—
59	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#126	1	2	yes	—
61	C₂×C₄×C₇₁	#135	1	2	yes	—
62	C₂×C₄×C₇₁	#137	1	2	yes	—
63	C₈×C₇₁	—	8	2	no	—
64	C₈×C₇₁	—	8	2	no	—
65	C₈×C₇₁	—	8	1	no	—
66	C₂²×C₇₁	—	2	2	no	—
67	C₂²×C₇₁	#397	1	1	yes	characteristic
68	C₂²×C₇₁	#485	1	2	yes	—
69	C₄×C₇₁	—	2	2	no	—
70	C₄×C₇₁	—	2	2	no	—
71	C₄×C₇₁	—	8	2	no	—
72	C₄×C₇₁	—	8	2	no	—
73	C₄×C₇₁	#464	1	2	yes	—
74	C₄×C₇₁	#488	1	1	yes	characteristic
75	C₄×C₇₁	#491	1	1	yes	characteristic
76	#10630	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×C₇₁	—	2	2	no	—
78	C₂×C₇₁	#2074	1	1	yes	characteristic
79	C₂×C₇₁	#2080	1	2	yes	—
80	#1995	C₂×C₇₁	1	1	yes	characteristic
81	#1832	C₂×C₇₁	1	1	yes	characteristic
82	#1542	C₂×C₇₁	1	2	yes	—
83	#1301	C₂×C₇₁	1	2	yes	—
84	#1274	C₂×C₇₁	1	1	yes	characteristic
85	C₇₁	#10630	1	1	yes	characteristic, direct product factor, Sylow
86	#533	C₂²×C₇₁	1	1	yes	characteristic
87	#431	C₂²×C₇₁	1	2	yes	—
88	#423	C₂²×C₇₁	1	1	yes	characteristic
89	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
90	#376	C₂²×C₇₁	1	1	yes	characteristic
91	#339	—	2	2	no	—
92	#336	—	2	2	no	—
93	#333	—	2	2	no	—
94	#332	—	2	2	no	—
95	C₂×M₁₆	—	2	1	no	—
96	C₂²×C₈	—	2	2	no	—
97	C₄⋊Q₈	D₄×C₇₁	1	2	yes	—
98	#125	—	2	1	no	—
99	C₂²⋊Q₈	—	2	2	no	—
100	C₄×Q₈	—	2	2	no	—
101	#117	—	2	1	no	—
102	C₂×C₄²	D₄×C₇₁	1	2	yes	—
103	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₂²⋊C₈	—	2	2	no	—
109	C₄×C₈	—	2	2	no	—
110	#95	—	2	2	no	—
111	C₂×Q₈	—	2	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
114	C₂²×C₄	#41×C₇₁	1	2	yes	—
115	M₁₆	—	4	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	4	2	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₂²⋊C₄	—	4	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	4	1	no	—
128	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
129	C₄²	#41×C₇₁	1	1	yes	characteristic
130	C₄²	#41×C₇₁	1	2	yes	—
131	C₂³	#123×C₇₁	1	1	yes	characteristic
132	Q₈	—	8	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
138	C₂×C₄	#123×C₇₁	1	2	yes	—
139	C₂×C₄	#124×C₇₁	1	1	yes	characteristic
140	C₂×C₄	#126×C₇₁	1	2	yes	—
141	C₂×C₄	#135×C₇₁	1	2	yes	—
142	C₂×C₄	#137×C₇₁	1	2	yes	—
143	C₈	—	8	2	no	—
144	C₈	—	8	2	no	—
145	C₈	—	8	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#397×C₇₁	1	1	yes	characteristic
148	C₂²	#485×C₇₁	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	8	2	no	—
152	C₄	—	8	2	no	—
153	C₄	#464×C₇₁	1	2	yes	—
154	C₄	#488×C₇₁	1	1	yes	characteristic
155	C₄	#491×C₇₁	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	C₇₁×#2074	1	1	yes	characteristic
158	C₂	C₇₁×#2080	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₁×#1995

C₂

yes

maximal, characteristic

C₇₁×#1832

C₂

yes

maximal, characteristic

C₇₁×#1542

C₂

yes

maximal

C₇₁×#1301

C₂

yes

maximal

C₇₁×#1274

C₂

yes

maximal, characteristic

#533×C₇₁

C₂²

yes

characteristic

#431×C₇₁

C₂²

yes

—

#423×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₇₁

C₂²

yes

—

#376×C₇₁

C₂²

yes

characteristic

#339×C₇₁

—

#336×C₇₁

—

#333×C₇₁

—

#332×C₇₁

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

—

C₄⋊Q₈×C₇₁

D₄

yes

—

#125×C₇₁

—

C₂²⋊Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

#117×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

—

#95×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

C₄²×C₇₁

yes

C₄²×C₇₁

yes

—

C₂³×C₇₁

#123

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#121

yes

characteristic

C₂×C₄×C₇₁

#123

yes

—

C₂×C₄×C₇₁

#124

yes

characteristic

C₂×C₄×C₇₁

#126

yes

—

C₂×C₄×C₇₁

#135

yes

—

C₂×C₄×C₇₁

#137

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#397

yes

characteristic

C₂²×C₇₁

#485

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#464

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

—

C₂×C₇₁

#2074

yes

characteristic

C₂×C₇₁

#2080

yes

—

#1995

C₂×C₇₁

yes

characteristic

#1832

C₂×C₇₁

yes

characteristic

#1542

C₂×C₇₁

yes

—

#1301

C₂×C₇₁

yes

—

#1274

C₂×C₇₁

yes

characteristic

C₇₁

#10630

yes

characteristic, direct product factor, Sylow

#533

C₂²×C₇₁

yes

characteristic

#431

C₂²×C₇₁

yes

—

#423

C₂²×C₇₁

yes

characteristic

C₂×C₄⋊C₈

C₂²×C₇₁

yes

—

#376

C₂²×C₇₁

yes

—

—

—

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

D₄×C₇₁

yes

—

#125

—

C₂²⋊Q₈

—

100

C₄×Q₈

—

101

#117

—

102

C₂×C₄²

D₄×C₇₁

yes

—

103

C₂×C₄²

C₂³×C₇₁

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₂²⋊C₈

—

109

C₄×C₈

—

110

#95

—

111

C₂×Q₈

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂×D₄×C₇₁

yes

characteristic

114

C₂²×C₄

#41×C₇₁

yes

—

115

M₁₆

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂×D₄×C₇₁

yes

characteristic

129

C₄²

#41×C₇₁

yes

characteristic

130

C₄²

#41×C₇₁

yes

—

131

C₂³

#123×C₇₁

yes

characteristic

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#121×C₇₁

yes

characteristic

138

C₂×C₄

#123×C₇₁

yes

—

139

C₂×C₄

#124×C₇₁

yes

characteristic

140

C₂×C₄

#126×C₇₁

yes

—

141

C₂×C₄

#135×C₇₁

yes

—

142

C₂×C₄

#137×C₇₁

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₂²

—

147

C₂²

#397×C₇₁

yes

characteristic

148

C₂²

#485×C₇₁

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#464×C₇₁

yes

—

154

C₄

#488×C₇₁

yes

characteristic

155

C₄

#491×C₇₁

yes

characteristic

156

C₂

—

157

C₂

C₇₁×#2074

yes

characteristic

158

C₂

C₇₁×#2080

yes

—

	subgroup	G/H
center	C₂×C₄×C₇₁	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
socle	C₂²×C₇₁	#397
prime radical	C₂³×C₇₁	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₇₁

#124

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₇₁

C₂²×C₇₁

C₂³×C₇₁

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

N	H ≅ G/N	type
76. #10630	C₇₁	direct (N × H)
85. C₇₁	#10630	direct (N × H)

H ≅ G/N

type

76. #10630

C₇₁

direct (N × H)

85. C₇₁

#10630

direct (N × H)

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

subgroup	index
C₇₁×#1995	2
C₇₁×#1832	2
C₇₁×#1542	2
C₇₁×#1301	2
C₇₁×#1274	2
#10630	71

subgroup

index

C₇₁×#1995

C₇₁×#1832

C₇₁×#1542

C₇₁×#1301

C₇₁×#1274

#10630

Subgroups — C₇₁×#10630 #31831502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#10630 #31831502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#10630 #31831502