Group #31917646 — subgroups

Subgroups — #31917646

Order 18176.

330 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 155 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×#17985833	C₂	1	2	yes	maximal, complemented
2	C₂²⋊(C₇₁⋊C₃₂)	C₂	1	4	yes	maximal, complemented
3	C₂³×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	#17985833	—	2	4	no	—
5	C₂×C₇₁⋊C₃₂	C₂²	1	4	yes	complemented
6	C₂³×C₈×C₇₁	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₇₁	—	2	4	no	—
8	C₂²×C₁₆×C₇₁	C₄	1	1	yes	characteristic
9	C₂²×C₁₆×C₇₁	C₄	1	2	yes	—
10	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
12	C₇₁⋊C₃₂	—	2	4	no	—
13	C₂³×C₄×C₇₁	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₇₁	—	2	2	no	—
15	C₂²×C₈×C₇₁	—	2	2	no	—
16	C₂²×C₈×C₇₁	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₇₁	C₈	1	2	yes	—
18	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₇₁	—	2	2	no	—
21	C₂×C₁₆×C₇₁	—	2	4	no	—
22	C₂×C₁₆×C₇₁	—	2	4	no	—
23	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₁	D₄	1	4	yes	—
26	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
27	C₂⁴×C₇₁	C₁₆	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	—	2	2	no	—
29	C₂²×C₄×C₇₁	—	2	2	no	—
30	C₂²×C₄×C₇₁	C₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	C₁₆	1	2	yes	—
32	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
33	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₇₁	—	2	2	no	—
35	C₂×C₈×C₇₁	—	2	4	no	—
36	C₂×C₈×C₇₁	—	2	2	no	—
37	C₂×C₈×C₇₁	—	2	2	no	—
38	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
42	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
43	C₁₆×C₇₁	—	2	2	no	—
44	C₁₆×C₇₁	C₂²⋊C₄	1	2	yes	—
45	C₁₆×C₇₁	C₂×D₄	1	2	yes	—
46	C₂³×C₇₁	—	2	2	no	—
47	C₂³×C₇₁	—	2	4	no	—
48	C₂³×C₇₁	C₂×C₁₆	1	1	yes	characteristic
49	C₂³×C₇₁	C₂×C₁₆	1	2	yes	—
50	C₂×C₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	—	2	4	no	—
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	—	2	2	no	—
54	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
56	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₁	M₃₂	1	2	yes	—
58	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₇₁	—	2	2	no	—
60	C₈×C₇₁	C₂²⋊C₈	1	2	yes	—
61	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
63	C₂²×C₇₁	—	2	8	no	—
64	C₂²×C₇₁	—	2	2	no	—
65	C₂²×C₇₁	—	2	2	no	—
66	C₂²×C₇₁	—	2	2	no	—
67	C₂²×C₇₁	C₂²⋊C₁₆	1	2	yes	—
68	C₂²×C₇₁	M₆₄	1	4	yes	complemented
69	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
70	C₄×C₇₁	—	2	2	no	—
71	C₄×C₇₁	C₂²⋊C₁₆	1	2	yes	—
72	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
73	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
74	#13553	—	71	1	no	maximal, Sylow
75	C₂×C₇₁	—	2	2	no	—
76	C₂×C₇₁	—	2	4	no	—
77	C₂×C₇₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
78	C₂×C₇₁	C₂×M₆₄	1	2	yes	—
79	C₂³×C₁₆	D₇₁	1	1	yes	characteristic
80	C₂×M₆₄	—	71	2	no	—
81	C₂²⋊C₃₂	—	71	4	no	—
82	C₇₁	#13553	1	1	yes	characteristic, complemented, Sylow
83	C₂³×C₈	Dic₇₁	1	1	yes	characteristic
84	C₂²×C₁₆	—	2	4	no	—
85	C₂²×C₁₆	Dic₇₁	1	1	yes	characteristic
86	C₂²×C₁₆	Dic₇₁	1	2	yes	—
87	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
88	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
89	M₆₄	—	142	4	no	—
90	C₂×C₃₂	—	71	4	no	—
91	C₂³×C₄	C₇₁⋊C₈	1	1	yes	characteristic
92	C₂²×C₈	—	2	2	no	—
93	C₂²×C₈	—	2	2	no	—
94	C₂²×C₈	C₇₁⋊C₈	1	1	yes	characteristic
95	C₂²×C₈	C₇₁⋊C₈	1	2	yes	—
96	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
97	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
98	C₂×C₁₆	—	2	2	no	—
99	C₂×C₁₆	—	2	4	no	—
100	C₂×C₁₆	—	2	4	no	—
101	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
102	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
103	C₂×C₁₆	C₂²⋊D₇₁	1	4	yes	—
104	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
105	C₃₂	—	142	4	no	—
106	C₂⁴	C₇₁⋊C₁₆	1	1	yes	characteristic
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	C₇₁⋊C₁₆	1	1	yes	characteristic
110	C₂²×C₄	C₇₁⋊C₁₆	1	2	yes	—
111	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
112	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
118	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
119	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
120	C₂×C₈	#11765849	1	2	yes	—
121	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
122	C₁₆	—	2	2	no	—
123	C₁₆	C₂²⋊Dic₇₁	1	2	yes	—
124	C₁₆	C₂×C₂²⋊D₇₁	1	2	yes	—
125	C₂³	—	2	4	no	—
126	C₂³	—	2	2	no	—
127	C₂³	C₂×C₇₁⋊C₁₆	1	2	yes	—
128	C₂³	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
134	C₂×C₄	C₂×C₇₁⋊C₁₆	1	2	yes	—
135	C₂×C₄	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
136	C₂×C₄	#14556068	1	2	yes	—
137	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
140	C₈	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
141	C₈	C₂×#11765849	1	1	yes	characteristic
142	C₂²	—	2	8	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂²⋊(C₇₁⋊C₁₆)	1	2	yes	—
147	C₂²	#17985833	1	4	yes	complemented
148	C₂²	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	C₂²⋊(C₇₁⋊C₁₆)	1	2	yes	—
151	C₄	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
152	C₄	C₂×#14556068	1	1	yes	characteristic
153	C₂	—	2	4	no	—
154	C₂	—	2	2	no	—
155	C₂	C₂×C₂²⋊(C₇₁⋊C₁₆)	1	1	yes	characteristic
156	C₂	C₂×#17985833	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985833

C₂

yes

maximal, complemented

C₂²⋊(C₇₁⋊C₃₂)

C₂

yes

maximal, complemented

C₂³×C₁₆×C₇₁

C₂

yes

maximal, characteristic

#17985833

—

C₂×C₇₁⋊C₃₂

C₂²

yes

complemented

C₂³×C₈×C₇₁

C₄

yes

characteristic

C₂²×C₁₆×C₇₁

—

C₂²×C₁₆×C₇₁

C₄

yes

characteristic

C₂²×C₁₆×C₇₁

C₄

yes

—

C₂²×C₁₆×C₇₁

C₂²

yes

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₇₁⋊C₃₂

—

C₂³×C₄×C₇₁

C₈

yes

characteristic

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

C₈

yes

characteristic

C₂²×C₈×C₇₁

C₈

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

—

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₂⁴×C₇₁

C₁₆

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₁₆

yes

characteristic

C₂²×C₄×C₇₁

C₁₆

yes

—

C₂²×C₄×C₇₁

C₂×C₈

yes

—

C₂²×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

M₁₆

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

C₂²⋊C₄

yes

—

C₁₆×C₇₁

C₂×D₄

yes

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

C₂×C₁₆

yes

characteristic

C₂³×C₇₁

C₂×C₁₆

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₈

yes

—

C₂×C₄×C₇₁

C₂×C₁₆

yes

—

C₂×C₄×C₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

M₃₂

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

C₂²⋊C₈

yes

—

C₈×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²⋊C₁₆

yes

—

C₂²×C₇₁

M₆₄

yes

complemented

C₂²×C₇₁

C₂²×C₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

C₂²⋊C₁₆

yes

—

C₄×C₇₁

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₇₁

C₂×M₃₂

yes

characteristic

#13553

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₂×C₂²⋊C₁₆

yes

characteristic

C₂×C₇₁

C₂×M₆₄

yes

—

C₂³×C₁₆

D₇₁

yes

characteristic

C₂×M₆₄

—

C₂²⋊C₃₂

—

C₇₁

#13553

yes

characteristic, complemented, Sylow

C₂³×C₈

Dic₇₁

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

Dic₇₁

yes

characteristic

C₂²×C₁₆

Dic₇₁

yes

—

C₂²×C₁₆

C₂×D₇₁

yes

—

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

M₆₄

—

142

—

C₂×C₃₂

—

C₂³×C₄

C₇₁⋊C₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₇₁⋊C₈

yes

characteristic

C₂²×C₈

C₇₁⋊C₈

yes

—

C₂²×C₈

C₂×Dic₇₁

yes

—

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

C₂×Dic₇₁

yes

—

102

C₂×C₁₆

C₂×Dic₇₁

yes

characteristic

103

C₂×C₁₆

C₂²⋊D₇₁

yes

—

104

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

105

C₃₂

—

142

—

106

C₂⁴

C₇₁⋊C₁₆

yes

characteristic

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₇₁⋊C₁₆

yes

characteristic

110

C₂²×C₄

C₇₁⋊C₁₆

yes

—

111

C₂²×C₄

C₂×C₇₁⋊C₈

yes

—

112

C₂²×C₄

C₂×C₇₁⋊C₈

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

C₂²⋊Dic₇₁

yes

—

118

C₂×C₈

C₂×C₇₁⋊C₈

yes

—

119

C₂×C₈

C₂×C₇₁⋊C₈

yes

characteristic

120

C₂×C₈

#11765849

yes

—

121

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

122

C₁₆

—

123

C₁₆

C₂²⋊Dic₇₁

yes

—

124

C₁₆

C₂×C₂²⋊D₇₁

yes

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₂×C₇₁⋊C₁₆

yes

—

128

C₂³

C₂×C₇₁⋊C₁₆

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂²⋊(C₇₁⋊C₈)

yes

—

134

C₂×C₄

C₂×C₇₁⋊C₁₆

yes

—

135

C₂×C₄

C₂×C₇₁⋊C₁₆

yes

characteristic

136

C₂×C₄

#14556068

yes

—

137

C₂×C₄

C₂²×C₇₁⋊C₈

yes

characteristic

138

C₈

—

139

C₈

C₂²⋊(C₇₁⋊C₈)

yes

—

140

C₈

C₂×C₂²⋊Dic₇₁

yes

characteristic

141

C₈

C₂×#11765849

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂²⋊(C₇₁⋊C₁₆)

yes

—

147

C₂²

#17985833

yes

complemented

148

C₂²

C₂²×C₇₁⋊C₁₆

yes

characteristic

149

C₄

—

150

C₄

C₂²⋊(C₇₁⋊C₁₆)

yes

—

151

C₄

C₂×C₂²⋊(C₇₁⋊C₈)

yes

characteristic

152

C₄

C₂×#14556068

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

C₂×C₂²⋊(C₇₁⋊C₁₆)

yes

characteristic

156

C₂

C₂×#17985833

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂²×C₁₆
Fitting	C₂³×C₁₆×C₇₁	C₂
prime radical	C₂⁴×C₇₁	C₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	C₂³×C₁₆	D₇₁
hypocenter	C₇₁	#13553
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂³×C₁₆

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇₁

commutator, socle

C₂²×C₇₁

C₂²×C₁₆

Fitting

C₂³×C₁₆×C₇₁

C₂

prime radical

C₂⁴×C₇₁

C₁₆

cosocle

C₂×C₁₆×C₇₁

C₂³

hypercenter

C₂³×C₁₆

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₁₆ ◁ C₂³×C₁₆

N	H ≅ G/N	type
1. C₂×#17985833	C₂	semidirect
2. C₂²⋊(C₇₁⋊C₃₂)	C₂	semidirect
5. C₂×C₇₁⋊C₃₂	C₂²	semidirect
68. C₂²×C₇₁	M₆₄	semidirect
82. C₇₁	#13553	semidirect
147. C₂²	#17985833	semidirect

H ≅ G/N

type

1. C₂×#17985833

C₂

semidirect

2. C₂²⋊(C₇₁⋊C₃₂)

C₂

semidirect

5. C₂×C₇₁⋊C₃₂

C₂²

semidirect

68. C₂²×C₇₁

M₆₄

semidirect

82. C₇₁

#13553

semidirect

147. C₂²

#17985833

semidirect

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

subgroup	index
C₂×#17985833	2
C₂²⋊(C₇₁⋊C₃₂)	2
C₂³×C₁₆×C₇₁	2
#13553	71

subgroup

index

C₂×#17985833

C₂²⋊(C₇₁⋊C₃₂)

C₂³×C₁₆×C₇₁

#13553

Subgroups — #31917646

Conjugacy classes of subgroups

Complements

Maximal subgroups