Group #31917648 — subgroups

Subgroups — #31917648

Order 18176.

298 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 123 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×#17986415	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985833	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₇₁)⋊C₃₂	C₂	1	4	yes	maximal, complemented
4	C₂²×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14556134	C₄	1	1	yes	characteristic
6	#17986415	—	2	2	no	—
7	#17986415	—	2	2	no	—
8	C₂×#14556068	C₄	1	1	yes	characteristic
9	C₂×#14556067	C₄	1	2	yes	—
10	C₂×C₁₆×D₇₁	C₂²	1	2	yes	—
11	#17985833	—	2	2	no	—
12	C₂×C₇₁⋊C₃₂	C₂²	1	2	yes	—
13	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₂×C₇₁	—	2	2	no	—
15	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
16	C₂×#11765865	C₈	1	1	yes	characteristic
17	#14556134	—	2	2	no	—
18	#14556134	—	2	2	no	—
19	C₂×#11765849	C₈	1	1	yes	characteristic
20	C₂×#11765848	C₈	1	2	yes	—
21	C₂×C₈×D₇₁	C₂×C₄	1	2	yes	—
22	#14556068	—	2	2	no	—
23	#14556067	—	2	4	no	—
24	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
25	C₁₆×D₇₁	—	2	2	no	—
26	C₁₆×D₇₁	—	2	2	no	—
27	C₇₁⋊C₃₂	—	2	2	no	—
28	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
31	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
32	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
33	C₃₂×C₇₁	—	2	2	no	—
34	#11765865	—	2	2	no	—
35	#11765865	—	2	2	no	—
36	C₂×Dic₁₄₂	C₁₆	1	1	yes	characteristic
37	C₂×D₂₈₄	C₁₆	1	1	yes	characteristic
38	C₂×C₂²⋊D₇₁	C₁₆	1	2	yes	—
39	C₂×C₄×D₇₁	C₂×C₈	1	2	yes	—
40	#11765849	—	2	2	no	—
41	#11765848	—	2	4	no	—
42	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
43	C₈×D₇₁	—	2	2	no	—
44	C₈×D₇₁	—	2	2	no	—
45	C₇₁⋊C₁₆	—	2	2	no	—
46	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
50	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₁₆×C₇₁	C₂²⋊C₄	1	2	yes	—
52	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂³×C₇₁	C₂×C₁₆	1	1	yes	characteristic
55	C₂²×D₇₁	C₂×C₁₆	1	2	yes	—
56	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
58	C₂×C₄×C₇₁	M₃₂	1	2	yes	—
59	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
60	C₂²⋊D₇₁	—	2	4	no	—
61	C₂×Dic₇₁	C₂×C₁₆	1	2	yes	—
62	D₂₈₄	—	2	2	no	—
63	C₄×D₇₁	—	2	2	no	—
64	C₄×D₇₁	—	2	2	no	—
65	Dic₁₄₂	—	2	2	no	—
66	C₈×C₇₁	C₂²⋊C₈	1	2	yes	—
67	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
69	C₇₁⋊C₈	—	2	2	no	—
70	C₂²×C₇₁	—	2	2	no	—
71	C₂²×C₇₁	C₂²⋊C₁₆	1	2	yes	—
72	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
73	C₂×D₇₁	—	2	2	no	—
74	C₂×D₇₁	—	2	4	no	—
75	C₄×C₇₁	C₂²⋊C₁₆	1	2	yes	—
76	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
78	Dic₇₁	—	2	2	no	—
79	#13554	—	71	1	no	maximal, Sylow
80	C₂×C₇₁	—	2	2	no	—
81	C₂×C₇₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
82	C₂×C₇₁	#2258	1	2	yes	—
83	D₇₁	—	4	2	no	—
84	C₂×#504	—	71	1	no	—
85	C₂×M₆₄	—	71	1	no	—
86	C₂²×C₃₂	D₇₁	1	1	yes	characteristic, complemented
87	C₂²⋊C₃₂	—	71	4	no	—
88	C₇₁	#13554	1	1	yes	characteristic, complemented, Sylow
89	C₂×#131	—	71	1	no	—
90	#504	—	142	2	no	—
91	#504	—	142	2	no	—
92	C₂×M₃₂	—	71	1	no	—
93	C₂×M₃₂	—	71	2	no	—
94	C₂²×C₁₆	—	71	2	no	—
95	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
96	M₆₄	—	142	2	no	—
97	C₂×C₃₂	—	2	2	no	—
98	C₂×C₃₂	—	71	2	no	—
99	C₂×C₃₂	C₂×D₇₁	1	2	yes	—
100	C₂×#41	—	71	1	no	—
101	#131	—	142	2	no	—
102	#131	—	142	2	no	—
103	C₂×M₁₆	—	71	1	no	—
104	C₂×M₁₆	—	71	2	no	—
105	C₂²×C₈	—	71	2	no	—
106	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
107	M₃₂	—	142	2	no	—
108	M₃₂	—	142	4	no	—
109	C₂×C₁₆	—	71	2	no	—
110	C₂×C₁₆	—	142	2	no	—
111	C₂×C₁₆	—	142	2	no	—
112	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
113	C₂×C₁₆	D₂₈₄	1	2	yes	—
114	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
115	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
116	C₃₂	—	2	2	no	—
117	C₃₂	—	142	2	no	—
118	#41	—	142	2	no	—
119	#41	—	142	2	no	—
120	C₂×Q₈	—	71	1	no	—
121	C₂×D₄	—	71	2	no	—
122	C₂×D₄	—	71	1	no	—
123	C₂²×C₄	—	71	2	no	—
124	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
125	M₁₆	—	142	2	no	—
126	M₁₆	—	142	4	no	—
127	C₂×C₈	—	71	2	no	—
128	C₂×C₈	—	142	2	no	—
129	C₂×C₈	—	142	2	no	—
130	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
131	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
132	C₂×C₈	#11765848	1	2	yes	—
133	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
134	C₁₆	—	142	2	no	—
135	C₁₆	(C₂×D₇₁)⋊C₄	1	2	yes	—
136	C₁₆	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
137	C₁₆	C₂×D₂₈₄	1	1	yes	characteristic
138	C₂³	—	71	2	no	—
139	C₂³	C₁₆×D₇₁	1	1	yes	characteristic
140	Q₈	—	142	2	no	—
141	D₄	—	142	4	no	—
142	D₄	—	142	2	no	—
143	C₂×C₄	—	71	2	no	—
144	C₂×C₄	—	142	2	no	—
145	C₂×C₄	—	142	2	no	—
146	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
147	C₂×C₄	C₁₆×D₇₁	1	1	yes	characteristic
148	C₂×C₄	#14556067	1	2	yes	—
149	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
150	C₈	—	142	2	no	—
151	C₈	(C₂×D₇₁)⋊C₈	1	2	yes	—
152	C₈	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
153	C₈	C₂×#11765848	1	1	yes	characteristic
154	C₂²	—	2	2	no	—
155	C₂²	—	142	4	no	—
156	C₂²	—	142	2	no	—
157	C₂²	(C₂×D₇₁)⋊C₁₆	1	2	yes	—
158	C₂²	C₂×C₁₆×D₇₁	1	1	yes	characteristic
159	C₄	—	142	2	no	—
160	C₄	(C₂×D₇₁)⋊C₁₆	1	2	yes	—
161	C₄	C₂×(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
162	C₄	C₂×#14556067	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	284	2	no	—
165	C₂	C₂×(C₂×D₇₁)⋊C₁₆	1	1	yes	characteristic
166	C₂	#22590633	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986415

C₂

yes

maximal, characteristic

C₂×#17985833

C₂

yes

maximal, characteristic, complemented

(C₂×D₇₁)⋊C₃₂

C₂

yes

maximal, complemented

C₂²×C₃₂×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#14556134

C₄

yes

—

—

C₄

yes

characteristic

C₂×#14556067

C₄

yes

—

C₂×C₁₆×D₇₁

C₂²

yes

—

#17985833

—

C₂×C₇₁⋊C₃₂

C₂²

yes

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₃₂×C₇₁

—

C₂×C₃₂×C₇₁

C₂²

yes

—

C₂×#11765865

C₈

yes

—

—

C₈

yes

characteristic

C₂×#11765848

C₈

yes

—

C₂×C₈×D₇₁

C₂×C₄

yes

—

#14556068

—

#14556067

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

C₁₆×D₇₁

—

C₁₆×D₇₁

—

C₇₁⋊C₃₂

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

—

C₂×C₁₆×C₇₁

D₄

yes

—

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₃₂×C₇₁

—

#11765865

—

#11765865

—

C₂×Dic₁₄₂

C₁₆

yes

characteristic

C₂×D₂₈₄

C₁₆

yes

characteristic

C₂×C₂²⋊D₇₁

C₁₆

yes

—

C₂×C₄×D₇₁

C₂×C₈

yes

—

#11765849

—

#11765848

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

—

C₈×D₇₁

—

C₈×D₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

M₁₆

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

C₂²⋊C₄

yes

—

C₁₆×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

C₂×D₄

yes

characteristic

C₂³×C₇₁

C₂×C₁₆

yes

characteristic

C₂²×D₇₁

C₂×C₁₆

yes

—

C₂×C₄×C₇₁

C₂²⋊C₈

yes

—

C₂×C₄×C₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

M₃₂

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂²⋊D₇₁

—

C₂×Dic₇₁

C₂×C₁₆

yes

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

C₈×C₇₁

C₂²⋊C₈

yes

—

C₈×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²⋊C₁₆

yes

—

C₂²×C₇₁

C₂²×C₁₆

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

C₂²⋊C₁₆

yes

—

C₄×C₇₁

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₇₁

C₂×M₃₂

yes

characteristic

Dic₇₁

—

#13554

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

C₂×C₂²⋊C₁₆

yes

characteristic

C₂×C₇₁

#2258

yes

—

D₇₁

—

C₂×#504

—

C₂×M₆₄

—

C₂²×C₃₂

D₇₁

yes

characteristic, complemented

C₂²⋊C₃₂

—

C₇₁

#13554

yes

characteristic, complemented, Sylow

C₂×#131

—

#504

—

142

—

#504

—

142

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

M₆₄

—

142

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₁

yes

—

100

C₂×#41

—

101

#131

—

142

—

102

#131

—

142

—

103

C₂×M₁₆

—

104

C₂×M₁₆

—

105

C₂²×C₈

—

106

C₂²×C₈

C₄×D₇₁

yes

characteristic

107

M₃₂

—

142

—

108

M₃₂

—

142

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

142

—

111

C₂×C₁₆

—

142

—

112

C₂×C₁₆

C₄×D₇₁

yes

characteristic

113

C₂×C₁₆

D₂₈₄

yes

—

114

C₂×C₁₆

C₂²⋊D₇₁

yes

—

115

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

116

C₃₂

—

117

C₃₂

—

142

—

118

#41

—

142

—

119

#41

—

142

—

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₈×D₇₁

yes

characteristic

125

M₁₆

—

142

—

126

M₁₆

—

142

—

127

C₂×C₈

—

128

C₂×C₈

—

142

—

129

C₂×C₈

—

142

—

130

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

131

C₂×C₈

C₈×D₇₁

yes

characteristic

132

C₂×C₈

#11765848

yes

—

133

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

134

C₁₆

—

142

—

135

C₁₆

(C₂×D₇₁)⋊C₄

yes

—

136

C₁₆

C₂×C₂²⋊D₇₁

yes

characteristic

137

C₁₆

C₂×D₂₈₄

yes

characteristic

138

C₂³

—

139

C₂³

C₁₆×D₇₁

yes

characteristic

140

Q₈

—

142

—

141

D₄

—

142

—

142

D₄

—

142

—

143

C₂×C₄

—

144

C₂×C₄

—

142

—

145

C₂×C₄

—

142

—

146

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

—

147

C₂×C₄

C₁₆×D₇₁

yes

characteristic

148

C₂×C₄

#14556067

yes

—

149

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

150

C₈

—

142

—

151

C₈

(C₂×D₇₁)⋊C₈

yes

—

152

C₈

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

153

C₈

C₂×#11765848

yes

characteristic

154

C₂²

—

155

C₂²

—

142

—

156

C₂²

—

142

—

157

C₂²

(C₂×D₇₁)⋊C₁₆

yes

—

158

C₂²

C₂×C₁₆×D₇₁

yes

characteristic

159

C₄

—

142

—

160

C₄

(C₂×D₇₁)⋊C₁₆

yes

—

161

C₄

C₂×(C₂×D₇₁)⋊C₈

yes

characteristic

162

C₄

C₂×#14556067

yes

characteristic

163

C₂

—

164

C₂

—

284

—

165

C₂

C₂×(C₂×D₇₁)⋊C₁₆

yes

characteristic

166

C₂

#22590633

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂²×C₁₆
Fitting	C₂²×C₃₂×C₇₁	C₂
prime radical	C₂×#11765865	C₈
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	C₂²×C₃₂	D₇₁
hypocenter	C₇₁	#13554
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇₁

commutator, socle

C₂²×C₇₁

C₂²×C₁₆

Fitting

C₂²×C₃₂×C₇₁

C₂

prime radical

C₂×#11765865

C₈

cosocle

C₂×C₁₆×C₇₁

C₂³

hypercenter

C₂²×C₃₂

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂

N	H ≅ G/N	type
2. C₂×#17985833	C₂	semidirect
3. (C₂×D₇₁)⋊C₃₂	C₂	semidirect
4. C₂²×C₃₂×C₇₁	C₂	semidirect
86. C₂²×C₃₂	D₇₁	semidirect
88. C₇₁	#13554	semidirect

H ≅ G/N

type

2. C₂×#17985833

C₂

semidirect

3. (C₂×D₇₁)⋊C₃₂

C₂

semidirect

4. C₂²×C₃₂×C₇₁

C₂

semidirect

86. C₂²×C₃₂

D₇₁

semidirect

88. C₇₁

#13554

semidirect

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

subgroup	index
C₂×#17986415	2
C₂×#17985833	2
(C₂×D₇₁)⋊C₃₂	2
C₂²×C₃₂×C₇₁	2
#13554	71

subgroup

index

C₂×#17986415

C₂×#17985833

(C₂×D₇₁)⋊C₃₂

C₂²×C₃₂×C₇₁

#13554

Subgroups — #31917648

Conjugacy classes of subgroups

Complements

Maximal subgroups