Group #31884352 — subgroups

Subgroups — #31884352

Order 18176.

216 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22590139	C₂	1	1	yes	maximal, characteristic
2	#22586629	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
4	#17986052	C₄	1	1	yes	characteristic
5	C₄×#11765849	C₄	1	1	yes	characteristic
6	C₄⋊(C₇₁⋊C₁₆)	C₄	1	2	yes	—
7	C₂²⋊(C₇₁⋊C₁₆)	—	2	1	no	—
8	C₄×C₇₁⋊C₁₆	—	2	1	no	—
9	#17985721	—	2	1	no	—
10	#17985720	—	2	1	no	—
11	C₂×M₃₂×C₇₁	—	2	1	no	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	#346×C₇₁	C₄	1	2	yes	—
14	C₄×C₁₆×C₇₁	—	2	1	no	—
15	C₂×#11765849	C₈	1	2	yes	—
16	C₂×C₇₁⋊C₁₆	—	2	2	no	—
17	C₄⋊(C₇₁⋊C₈)	C₈	1	2	yes	—
18	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
19	#14556039	—	2	1	no	—
20	C₄×C₇₁⋊C₈	—	2	1	no	—
21	C₄×C₇₁⋊C₈	—	2	1	no	—
22	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
25	M₃₂×C₇₁	—	2	4	no	—
26	C₂×C₁₆×C₇₁	—	2	2	no	—
27	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
31	#11765849	—	4	2	no	—
32	C₂×C₇₁⋊C₈	—	2	2	no	—
33	C₂×C₇₁⋊C₈	—	2	2	no	—
34	C₇₁⋊C₁₆	—	4	2	no	—
35	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
38	C₂×C₈×C₇₁	—	2	2	no	—
39	C₂×C₈×C₇₁	—	2	2	no	—
40	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₇₁	C₄²	1	1	yes	characteristic
45	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
48	C₁₆×C₇₁	—	2	4	no	—
49	C₂³×C₇₁	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₁	—	2	1	no	—
51	C₂×C₄×C₇₁	—	2	1	no	—
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	—	2	2	no	—
54	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₇₁	—	2	1	no	—
61	C₈×C₇₁	—	2	1	no	—
62	C₈×C₇₁	#104	1	2	yes	—
63	C₈×C₇₁	#108	1	2	yes	—
64	C₇₁⋊C₈	—	4	2	no	—
65	C₇₁⋊C₈	—	4	2	no	—
66	C₂²×C₇₁	—	2	1	no	—
67	C₂²×C₇₁	—	2	1	no	—
68	C₂²×C₇₁	—	2	1	no	—
69	C₂²×C₇₁	#336	1	1	yes	characteristic
70	C₄×C₇₁	—	2	1	no	—
71	C₄×C₇₁	—	2	1	no	—
72	C₄×C₇₁	—	2	1	no	—
73	C₄×C₇₁	#339	1	1	yes	characteristic
74	C₄×C₇₁	#341	1	1	yes	characteristic
75	C₄×C₇₁	#349	1	2	yes	—
76	C₄×C₇₁	#350	1	2	yes	—
77	#7166	—	71	1	no	maximal, Sylow
78	C₂×C₇₁	—	4	1	no	—
79	C₂×C₇₁	#1278	1	1	yes	characteristic
80	C₂×C₇₁	#1377	1	1	yes	characteristic
81	C₂×C₇₁	#1379	1	1	yes	characteristic
82	#2162	—	71	1	no	—
83	C₄×M₃₂	D₇₁	1	1	yes	characteristic
84	#1451	—	71	1	no	—
85	C₇₁	#7166	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₃₂	—	2	1	no	—
87	#431	—	71	1	no	—
88	C₄×M₁₆	—	71	1	no	—
89	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
90	C₄⋊C₁₆	—	71	2	no	—
91	C₂²⋊C₁₆	—	142	1	no	—
92	#346	Dic₇₁	1	2	yes	—
93	C₄×C₁₆	—	2	1	no	—
94	C₄×C₁₆	—	142	1	no	—
95	#322	—	142	1	no	—
96	#322	—	142	1	no	—
97	C₂×M₁₆	—	71	2	no	—
98	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
99	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
100	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
101	M₃₂	—	2	4	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	142	2	no	—
104	C₄⋊C₈	—	71	2	no	—
105	C₂²⋊C₈	—	142	1	no	—
106	#97	—	142	1	no	—
107	C₄×C₈	—	142	1	no	—
108	C₄×C₈	—	142	1	no	—
109	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
110	C₄×C₈	D₂₈₄	1	1	yes	characteristic
111	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
113	C₂²×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
114	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
115	C₂²×C₄	#11765848	1	1	yes	characteristic
116	M₁₆	—	284	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	142	2	no	—
120	C₂×C₈	—	142	2	no	—
121	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
123	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
124	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
125	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
126	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
127	C₄²	C₈×D₇₁	1	1	yes	characteristic
128	C₄²	#11765848	1	1	yes	characteristic
129	C₁₆	—	2	4	no	—
130	C₁₆	—	284	2	no	—
131	C₂³	Dic₇₁⋊C₈	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
137	C₂×C₄	C₈×Dic₇₁	1	1	yes	characteristic
138	C₂×C₄	#14556038	1	1	yes	characteristic
139	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
140	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
141	C₂×C₄	Dic₇₁⋊C₈	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	284	2	no	—
145	C₈	—	284	2	no	—
146	C₈	#14556056	1	2	yes	—
147	C₈	#14556064	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#17985753	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	#17985759	1	1	yes	characteristic
156	C₄	#17985764	1	1	yes	characteristic
157	C₄	#17985780	1	2	yes	—
158	C₄	#17985784	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#22586177	1	1	yes	characteristic
161	C₂	#22586423	1	1	yes	characteristic
162	C₂	#22586430	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590139

C₂

yes

maximal, characteristic

#22586629

C₂

yes

maximal, characteristic

C₄×M₃₂×C₇₁

C₂

yes

maximal, characteristic

#17986052

C₄

yes

characteristic

C₄×#11765849

C₄

yes

characteristic

C₄⋊(C₇₁⋊C₁₆)

C₄

yes

—

C₂²⋊(C₇₁⋊C₁₆)

—

C₄×C₇₁⋊C₁₆

—

#17985721

—

#17985720

—

C₂×M₃₂×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#346×C₇₁

C₄

yes

—

C₄×C₁₆×C₇₁

—

C₂×#11765849

C₈

yes

—

C₂×C₇₁⋊C₁₆

—

C₄⋊(C₇₁⋊C₈)

C₈

yes

—

C₂²⋊(C₇₁⋊C₈)

—

#14556039

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

Q₈

yes

characteristic

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₁

M₁₆

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₄²×C₇₁

C₄²

yes

characteristic

C₄²×C₇₁

C₂²⋊C₄

yes

characteristic

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₄²×C₇₁

M₁₆

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#95

yes

characteristic

C₂×C₄×C₇₁

C₄×C₈

yes

characteristic

C₂×C₄×C₇₁

#97

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#104

yes

—

C₈×C₇₁

#108

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#336

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₄×C₇₁

#350

yes

—

#7166

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₄×M₃₂

D₇₁

yes

characteristic

#1451

—

C₇₁

#7166

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

142

—

#346

Dic₇₁

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

142

—

#322

—

142

—

#322

—

142

—

C₂×M₁₆

—

C₂²×C₈

Dic₁₄₂

yes

characteristic

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

100

C₂×C₄²

C₄×D₇₁

yes

characteristic

101

M₃₂

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

142

—

104

C₄⋊C₈

—

105

C₂²⋊C₈

—

142

—

106

#97

—

142

—

107

C₄×C₈

—

142

—

108

C₄×C₈

—

142

—

109

C₄×C₈

C₄×D₇₁

yes

characteristic

110

C₄×C₈

D₂₈₄

yes

characteristic

111

C₄×C₈

C₂×Dic₇₁

yes

characteristic

112

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

113

C₂²×C₄

Dic₇₁⋊C₄

yes

characteristic

114

C₂²×C₄

C₈×D₇₁

yes

characteristic

115

C₂²×C₄

#11765848

yes

characteristic

116

M₁₆

—

284

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

142

—

120

C₂×C₈

—

142

—

121

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

122

C₂×C₈

C₂²⋊Dic₇₁

yes

characteristic

123

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

124

C₂×C₈

C₄⋊Dic₇₁

yes

characteristic

125

C₄²

C₄×Dic₇₁

yes

characteristic

126

C₄²

(C₂×D₇₁)⋊C₄

yes

characteristic

127

C₄²

C₈×D₇₁

yes

characteristic

128

C₄²

#11765848

yes

characteristic

129

C₁₆

—

130

C₁₆

—

284

—

131

C₂³

Dic₇₁⋊C₈

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₇₁⋊#95

yes

characteristic

137

C₂×C₄

C₈×Dic₇₁

yes

characteristic

138

C₂×C₄

#14556038

yes

characteristic

139

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

characteristic

140

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

characteristic

141

C₂×C₄

Dic₇₁⋊C₈

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

284

—

145

C₈

—

284

—

146

C₈

#14556056

yes

—

147

C₈

#14556064

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#17985753

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#17985759

yes

characteristic

156

C₄

#17985764

yes

characteristic

157

C₄

#17985780

yes

—

158

C₄

#17985784

yes

—

159

C₂

—

160

C₂

#22586177

yes

characteristic

161

C₂

#22586423

yes

characteristic

162

C₂

#22586430

yes

characteristic

	subgroup	G/H
center	C₄²	(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₄×M₃₂×C₇₁	C₂
socle	C₂²×C₇₁	#336
prime radical	C₂³×C₇₁	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	C₄×M₃₂	D₇₁
hypocenter	C₇₁	#7166
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

subgroup

G/H

center

C₄²

(C₂×D₇₁)⋊C₄

commutator

C₂×C₄×C₇₁

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₁

Fitting

C₄×M₃₂×C₇₁

C₂

socle

C₂²×C₇₁

#336

prime radical

C₂³×C₇₁

C₄⋊C₈

cosocle

C₂×C₄×C₈×C₇₁

C₂²

hypercenter

C₄×M₃₂

D₇₁

hypocenter

C₇₁

#7166

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

N	H ≅ G/N	type
85. C₇₁	#7166	semidirect

H ≅ G/N

type

85. C₇₁

#7166

semidirect

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

subgroup	index
#22590139	2
#22586629	2
C₄×M₃₂×C₇₁	2
#7166	71

subgroup

index

#22590139

#22586629

C₄×M₃₂×C₇₁

#7166

Subgroups — #31884352

Conjugacy classes of subgroups

Complements

Maximal subgroups